Gr 10 Math June P1 Hard
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Questions and Answers

Which set of numbers includes negative values?

  • Natural Numbers (N)
  • Integers (Z) (correct)
  • Rational Numbers (Q)
  • Whole Numbers (N0)

What characterizes irrational numbers?

  • Their decimal expansions are terminating.
  • They cannot be expressed as a simple fraction. (correct)
  • They can be expressed as a fraction of two integers.
  • They include zero.

Which of the following is a property of rational numbers?

  • They can be expressed as a fraction of two integers. (correct)
  • They can only be positive.
  • They include all whole numbers and their negative counterparts.
  • They include only integers.

Which subset of real numbers includes the number zero?

<p>Whole Numbers (N0) (D)</p> Signup and view all the answers

Which of the following statements about real numbers is true?

<p>Real numbers include all rational and irrational numbers. (D)</p> Signup and view all the answers

Which of the following sets contains only non-real numbers?

<p>Imaginary Numbers (C)</p> Signup and view all the answers

Which property characterizes an irrational number?

<p>It has a non-repeating, non-terminating decimal expansion. (B)</p> Signup and view all the answers

What is the first step to convert a recurring decimal into a rational number?

<p>Multiply the decimal by a power of 10. (C)</p> Signup and view all the answers

Which of the following describes surds?

<p>They cannot be expressed as fractions and involve roots. (C)</p> Signup and view all the answers

When rounding the number 3.476 to two decimal places, what is the resulting number?

<p>3.48 (B)</p> Signup and view all the answers

What key component defines a rational number?

<p>It can be written as $\frac{a}{b}$ where $b \neq 0$. (A)</p> Signup and view all the answers

What distinguishes a terminating decimal from a repeating decimal?

<p>Terminating decimals have a fixed number of decimal places. (D)</p> Signup and view all the answers

Which method is used for estimating the value of a surd?

<p>Identifying the nearest perfect squares or cubes. (C)</p> Signup and view all the answers

In the rounding process, what happens if the digit to be rounded up is a 9?

<p>It changes to 0 and the preceding digit is increased by 1. (D)</p> Signup and view all the answers

Which part of a mathematical expression represents the power to which a base is raised?

<p>Exponent (D)</p> Signup and view all the answers

What is the maximum number of solutions for a quadratic equation?

<p>Two (D)</p> Signup and view all the answers

Which step is NOT involved in solving a linear equation?

<p>Use the quadratic formula (A)</p> Signup and view all the answers

When solving simultaneous equations, what is the primary goal of substitution?

<p>To express one variable in terms of another (A)</p> Signup and view all the answers

What should be done to ensure that a quadratic equation is solvable?

<p>Rewrite it in the form $ax^2 + bx + c = 0$ (A)</p> Signup and view all the answers

Which characteristic differentiates linear equations from quadratic equations?

<p>Maximal exponent of the variable (D)</p> Signup and view all the answers

Which of the following is crucial when solving any equation?

<p>Perform the same operation on both sides (C)</p> Signup and view all the answers

After factoring a quadratic equation to find its roots, what is the next step?

<p>Set each factor equal to zero (A)</p> Signup and view all the answers

Which method is effective for eliminating a variable in simultaneous equations?

<p>Elimination by adjusting coefficients (A)</p> Signup and view all the answers

What are the potential outcomes of a quadratic equation?

<p>One, two, or no solutions (A)</p> Signup and view all the answers

In the method of solving linear equations, what is the purpose of grouping like terms?

<p>To simplify and combine similar terms (C)</p> Signup and view all the answers

What is the effect of increasing the value of the gradient, m, in the equation y = mx + c?

<p>The slope of the graph increases. (D)</p> Signup and view all the answers

When solving linear inequalities, what must occur if both sides of the inequality are divided by a negative number?

<p>The inequality sign must be reversed. (A)</p> Signup and view all the answers

In the context of literal equations, what does changing the subject of the formula involve?

<p>Rearranging the equation to solve for a specific variable. (B)</p> Signup and view all the answers

What is required for graphically solving a system of simultaneous equations?

<p>Finding a point of intersection of two lines. (A)</p> Signup and view all the answers

In the solving process of word problems, what is the first step to take after reading the problem?

<p>Determine what is being solved for. (C)</p> Signup and view all the answers

When isolating the unknown variable in a literal equation, what must be considered?

<p>The operations that connect the unknown to other terms. (D)</p> Signup and view all the answers

What characteristic defines linear functions of the form y = mx + c?

<p>The slope and intercept are constant. (A)</p> Signup and view all the answers

What happens to the graph of a linear function when c is less than 0?

<p>The graph shifts vertically downwards. (B)</p> Signup and view all the answers

In solving equations, what represents the solution to the problem stated in a word problem?

<p>The set of equations that represent the scenario. (C)</p> Signup and view all the answers

What is a critical aspect to remember when taking the square root of both sides in an equation?

<p>We must consider both positive and negative solutions. (C)</p> Signup and view all the answers

What is the result of multiplying the monomial 3 and the binomial (x + 4)?

<p>3x + 12 (D)</p> Signup and view all the answers

Which expression correctly represents the product of the binomials (2x + 3) and (x + 5)?

<p>2x^2 + 13x + 15 (C)</p> Signup and view all the answers

Which of these is true about the factorization of the expression x^2 - 9?

<p>It can be expressed as (x + 3)(x - 3) (B)</p> Signup and view all the answers

What is the correct method to factor the trinomial 2x^2 + 8x + 6?

<p>By grouping terms and factoring out common factors. (D)</p> Signup and view all the answers

What is the sum of the cubes x^3 + 27 factored?

<p>(x + 3)(x^2 - 3x + 9) (B)</p> Signup and view all the answers

When simplifying the fraction $ rac{x^2 - 4}{x^2 + 2x}$, what is the first step?

<p>Factor both the numerator and the denominator. (D)</p> Signup and view all the answers

Which expression correctly describes the difference of cubes x^3 - y^3?

<p>(x - y)(x^2 - xy + y^2) (B)</p> Signup and view all the answers

What does the expression (4x + 2)(x^2 + 3x - 2) yield when simplified?

<p>4x^3 + 6x^2 + 14x - 4 (B)</p> Signup and view all the answers

Which of the following explains the process of factoring by grouping?

<p>Separating terms into pairs and factoring each pair. (B)</p> Signup and view all the answers

In the operation of dividing two fractions $ rac{a}{b}$ and $ rac{c}{d}$, which is the correct procedure?

<p>Multiply by the reciprocal of the second fraction. (C)</p> Signup and view all the answers

What effect does a greater value of $a$ have on the graph of a parabola?

<p>It changes the steepness of the branches. (C)</p> Signup and view all the answers

What is the domain of the tangent function defined in the content?

<p>$ heta: 0° o 360°, heta eq 90°, 270°$ (B)</p> Signup and view all the answers

What is the result of simplifying the expression \( rac{a^5}{a^2} imes a^3 \ ?

<p>a^6 (D)</p> Signup and view all the answers

How do you determine the vertical shift in the equation of a hyperbola?

<p>By observing the y-intercept. (C)</p> Signup and view all the answers

In determining the equation of a parabola, what role does the y-intercept play?

<p>It is used to solve for $q$. (D)</p> Signup and view all the answers

Which law states that ( (ab)^n = a^n b^n )?

<p>Raising a Product to a Power (C)</p> Signup and view all the answers

How can you express the square root of a as a rational exponent?

<p>a^{1/2} (B)</p> Signup and view all the answers

What is the significance of asymptotes in the context of hyperbolas?

<p>They define the limits of the graphs. (D)</p> Signup and view all the answers

How can the amplitude of a sine or cosine function be determined?

<p>By observing the value of $a$. (C)</p> Signup and view all the answers

Which expression correctly applies the zero exponent rule?

<p>a^0 = 1 (B)</p> Signup and view all the answers

What is the formula for calculating the points of intersection between two graphs?

<p>Equate the expressions and solve for $x$ and $y$. (D)</p> Signup and view all the answers

When solving the equation ( 2^x = 8 ), which method is appropriate?

<p>Equating the bases (C)</p> Signup and view all the answers

What will be the result of applying the laws of exponents to the expression ( a^{3/4} imes a^{1/2} )?

<p>a^{7/4} (A)</p> Signup and view all the answers

What information do the quadrants of a hyperbola provide when determining its equation?

<p>They help in determining the sign of $a$. (B)</p> Signup and view all the answers

Which of the following correctly exemplifies the law for negative exponents?

<p>a^{-n} = 1/a^n (B)</p> Signup and view all the answers

What method is used to solve for $a$ in the equation of a hyperbola?

<p>Using simultaneous equations from points. (B)</p> Signup and view all the answers

What is the primary use of the distance formula in interpreting graphs?

<p>To calculate intercept distances. (C)</p> Signup and view all the answers

To simplify ( \frac{a^4 b^2}{a^2 b} ), what is the first step?

<p>Rewrite as a^{4-2} b^{2-1} (A)</p> Signup and view all the answers

In rational exponents, how do you express ( \sqrt[3]{x} )?

<p>x^{1/3} (B)</p> Signup and view all the answers

What is the range of the function if $a < 0$ in the form $f(x) = ax^2 + q$?

<p>$(- ext{infinity}; q]$ (D)</p> Signup and view all the answers

Which statement correctly describes the characteristics of hyperbolic functions in the form $y = \frac{a}{x} + q$?

<p>There is no y-intercept for the function. (A)</p> Signup and view all the answers

For the hyperbolic function $y = \frac{a}{x} + q$, what is the equation of the horizontal asymptote?

<p>$y = q$ (D)</p> Signup and view all the answers

If $a > 0$ in the function $y = ax^2 + q$, how is the graph described?

<p>It has a minimum turning point. (A)</p> Signup and view all the answers

Which characteristics are true for the domain of the hyperbolic functions?

<p>All real numbers except zero. (C)</p> Signup and view all the answers

What can be said about the turning points of the function $f(x) = ax^2 + q$ when $a < 0$?

<p>It has a maximum turning point. (D)</p> Signup and view all the answers

In the function $y = ab^x + q$, what is the sign of $a$ if the range is described as $, {f(x) > q}$?

<p>$a &gt; 0$ (B)</p> Signup and view all the answers

What are the axes of symmetry for the hyperbolic function $y = \frac{a}{x} + q$?

<p>$y = x + q$ and $y = -x + q$ (D)</p> Signup and view all the answers

What is the effect of $q$ on a hyperbolic graph with $q < 0$?

<p>It shifts the graph downwards by $q$ units. (D)</p> Signup and view all the answers

How do the signs of $a$ affect the graph of the function $y = \frac{a}{x} + q$?

<p>Determine the quadrants in which the graph resides. (A)</p> Signup and view all the answers

How does the sign of the coefficient 'a' affect the graph of an exponential function of the form $y = ab^x + q$?

<p>It determines whether the graph curves upwards or downwards. (C)</p> Signup and view all the answers

What is the effect of changing the value of 'b' in the exponential function $y = ab^x + q$?

<p>It increases or decreases the curvature rate of the graph. (A)</p> Signup and view all the answers

For a sine function of the form $y = a ext{sin} heta + q$, what change occurs when $a < 0$?

<p>The graph reflects about the x-axis. (D)</p> Signup and view all the answers

Which statement correctly describes the range of the function $y = a ext{cos} heta + q$ when $a > 0$?

<p>The range is $[q - |a|, q + |a|]$. (D)</p> Signup and view all the answers

Which of the following points represent the maximum turning point of the sine function $y = ext{sin} heta$?

<p>(90°, 1) (D)</p> Signup and view all the answers

What is the horizontal asymptote of the exponential function $y = ab^x + q$?

<p>The line $y = q$. (B)</p> Signup and view all the answers

What happens to the graph of the function $y = a ext{tan} heta + q$ if $q < 0$?

<p>The graph shifts downwards by $q$ units. (D)</p> Signup and view all the answers

How does the transformation $y = 2 ext{sin} heta + 1$ affect the sine wave?

<p>It stretches the graph vertically and shifts it upwards. (A)</p> Signup and view all the answers

For the tangent function $y = an heta$, where are the vertical asymptotes located?

<p>At $ heta = 90°, 270°$. (C)</p> Signup and view all the answers

What defines the period of the sine and cosine functions?

<p>360° (A)</p> Signup and view all the answers

What is the relationship between the sign of the coefficient $a$ and the shape of the parabolic graph?

<p>If $a &gt; 0$, the graph 'smiles' and has a minimum turning point. (B)</p> Signup and view all the answers

How does the value of $q$ influence the graph of the equation $y = ax^2 + q$?

<p>The value of $q$ causes a vertical shift of the entire graph. (A)</p> Signup and view all the answers

For the linear equation $y = mx + c$, what does the y-intercept represent?

<p>The value of $y$ when $x = 0$. (A)</p> Signup and view all the answers

Which statement correctly defines the range of the function $f(x) = mx + c$?

<p>The range is all real numbers. (C)</p> Signup and view all the answers

What is the effect of increasing the absolute value of $a$ in the equation $y = ax^2 + q$?

<p>The parabola becomes narrower as $|a|$ increases. (B)</p> Signup and view all the answers

When sketching a straight-line graph, which two points can be used effectively?

<p>The x-intercept and y-intercept. (A)</p> Signup and view all the answers

If the coefficient $m$ is negative in the equation $y = mx + c$, what type of slope does the line have?

<p>A downward slope from left to right. (B)</p> Signup and view all the answers

Which of the following explains why the domain of $f(x) = mx + c$ is all real numbers?

<p>There is no value of $x$ that makes the function undefined. (A)</p> Signup and view all the answers

If $q < 0$ in the quadratic function $y = ax^2 + q$, where does the vertex lie in relation to the x-axis?

<p>The vertex lies below the x-axis. (A)</p> Signup and view all the answers

Which of the following numbers is classified as an irrational number?

<p>$ rac{3}{5} + rac{ ext{sqrt{2}}}{2}$ (D)</p> Signup and view all the answers

Which set of numbers includes only whole numbers?

<p>$N_0$ (C)</p> Signup and view all the answers

What defines a rational number?

<p>A number expressible as the ratio of two integers (A)</p> Signup and view all the answers

Which of the following is NOT a subset of rational numbers?

<p>Irrational numbers (C)</p> Signup and view all the answers

What is the primary characteristic of integers?

<p>They include zero and negative counterparts. (D)</p> Signup and view all the answers

What foundational property applies to the set of real numbers?

<p>They consist of both rational and irrational numbers. (D)</p> Signup and view all the answers

What is the defining feature of irrational numbers compared to rational numbers?

<p>Irrational numbers have non-repeating, non-terminating decimal expansions. (A)</p> Signup and view all the answers

Which statement is true about terminating decimals?

<p>Terminating decimals can be expressed as fractions with whole number numerators and denominators. (D)</p> Signup and view all the answers

When estimating the value of a surd, what is the first step?

<p>Identify the perfect powers surrounding the given surd. (C)</p> Signup and view all the answers

Which of the following is a correct characteristic of surds?

<p>Surds include roots that cannot simplify to rational numbers. (A)</p> Signup and view all the answers

In the rounding process, what occurs if the digit to be rounded up is 9?

<p>It changes to 0 and increments the preceding digit by 1. (C)</p> Signup and view all the answers

Which process is used to convert a recurring decimal into a rational number?

<p>Multiply by a power of 10 to align the repeating parts. (A)</p> Signup and view all the answers

How are rational numbers classified in terms of their decimal representation?

<p>Rational numbers may either have terminating or repeating decimal expansions. (B)</p> Signup and view all the answers

What role does the coefficient play in a mathematical expression?

<p>It serves as the numerical factor in a term. (C)</p> Signup and view all the answers

Which of these represents a defining characteristic of rational numbers?

<p>They can be expressed as a fraction with whole number numerator and non-zero denominator. (A)</p> Signup and view all the answers

What is the maximum number of solutions for a linear equation?

<p>One (D)</p> Signup and view all the answers

When solving a quadratic equation, what form must it be in?

<p>ax^2 + bx + c = 0 (A)</p> Signup and view all the answers

Which of the following strategies is NOT a method for solving simultaneous equations?

<p>Factorization (A)</p> Signup and view all the answers

What should be done first when checking solutions for a quadratic equation?

<p>Substitute the solutions into the original equation (A)</p> Signup and view all the answers

What is the implication of balancing an equation when solving?

<p>Operations must be performed equally on both sides (C)</p> Signup and view all the answers

Which statement about the nature of roots in a quadratic equation is accurate?

<p>There can be two or no roots (B)</p> Signup and view all the answers

In the method of solving linear equations, what does isolating the variable accomplish?

<p>It provides the solution to the equation (D)</p> Signup and view all the answers

Which method effectively reduces the complexity in solving simultaneous equations?

<p>Using substitution to express one variable in terms of the other (D)</p> Signup and view all the answers

When factorizing a quadratic equation in the form $ax^2 + bx + c = 0$, what must be ensured?

<p>Common factors should be identified and divided first (D)</p> Signup and view all the answers

What is the result of multiplying the expression ( (3x + 2)(x + 5) )?

<p>( 3x^2 + 17x + 10 ) (D)</p> Signup and view all the answers

What expression represents the factorization of the quadratic trinomial ( 2x^2 + 7x + 3 )?

<p>( (2x + 1)(x + 3) ) (A)</p> Signup and view all the answers

Using the identity for the difference of two squares, how would you factor ( 16x^2 - 25 )?

<p>( (4x + 5)(4x - 5) ) (B), ( (4x - 5)(4x + 5) ) (D)</p> Signup and view all the answers

What is the first step required when simplifying the fraction ( \frac{2x^2 - 8}{2x} )?

<p>Factor the numerator to ( 2(x^2 - 4) ). (A)</p> Signup and view all the answers

When using the method of factorising by grouping, what is the purpose of grouping terms?

<p>To identify common factors within pairs of terms. (D)</p> Signup and view all the answers

What is the final form of the difference of cubes ( x^3 - 8 ) when fully factored?

<p>( (x - 2)(x^2 + 2x + 4) ) (D)</p> Signup and view all the answers

What is a key difference in simplifying the expression ( \frac{a^2 - b^2}{a + b} ) as opposed to ( a + b )?

<p>The difference of squares can be canceled out. (C)</p> Signup and view all the answers

To simplify the algebraic fraction ( \frac{x^2 + 4x + 4}{x + 2} ), what process must be applied first?

<p>Factor the numerator to ( (x + 2)(x + 2) ). (C)</p> Signup and view all the answers

What does the following multiplication yield: ( a(b + c) + d(b + c) )?

<p>( ab + ac + db + dc ) (B)</p> Signup and view all the answers

What is the range of the function given that the coefficient $a$ is negative?

<p>($- ext{infinity}; q]$ (B)</p> Signup and view all the answers

Which of the following describes the turning point of the function when $a$ is greater than zero?

<p>Minimum at $(0; q)$ (D)</p> Signup and view all the answers

What characterizes the domain of the function $y = rac{a}{x} + q$?

<p>All real numbers except zero (D)</p> Signup and view all the answers

What will happen to the graph of $y = rac{a}{x} + q$ if $q$ is set to zero?

<p>The horizontal asymptote will shift to the y-axis (C)</p> Signup and view all the answers

How is the x-intercept of the hyperbolic function $y = rac{a}{x} + q$ determined?

<p>By setting $y = 0$ (A)</p> Signup and view all the answers

Which of the following statements about the axis of symmetry for the function $f(x) = ax^2 + q$ is true?

<p>It is the vertical axis, $x = 0$ (D)</p> Signup and view all the answers

For the exponential function $y = ab^x + q$, what can be inferred if $a$ is less than zero?

<p>The graph shifts downward by $q$ units (B)</p> Signup and view all the answers

What determines if the hyperbolic graph lies in the first and third quadrants?

<p>When $a$ is greater than zero (A)</p> Signup and view all the answers

What determines the slope of the graph in the linear equation $y = mx + c$?

<p>The value of $m$ (C)</p> Signup and view all the answers

What is true about the y-intercept of the hyperbolic function $y = rac{a}{x} + q$?

<p>It does not exist (C)</p> Signup and view all the answers

Which of the following statements about solving linear inequalities is true?

<p>The product of an inequality remains an inequality if multiplied by a negative number. (C)</p> Signup and view all the answers

What is the primary goal of the problem-solving strategy in word problems?

<p>To assign a variable to the unknown quantity. (A)</p> Signup and view all the answers

What happens to the graph of a linear equation if $c$ is increased?

<p>The graph shifts vertically upwards. (B)</p> Signup and view all the answers

In the context of literal equations, what does isolating the unknown variable involve?

<p>Identifying what the variable is connected to and the type of operation performed. (D)</p> Signup and view all the answers

What should be done if the unknown variable in a literal equation appears in more than one term?

<p>Take it as a common factor. (D)</p> Signup and view all the answers

What is a key difference between solving graphical systems of equations and algebraic methods?

<p>Graphical methods often rely on estimation rather than precise calculations. (B)</p> Signup and view all the answers

When solving word problems, what is the process to write the equations?

<p>Translate the quantitative information into algebraic expressions. (C)</p> Signup and view all the answers

Why is it important to check the solution after solving an equation?

<p>To ensure the solution meets the original problem’s requirements. (B)</p> Signup and view all the answers

In the process of solving simultaneous equations graphically, what represents the solution?

<p>The coordinates of the point of intersection. (B)</p> Signup and view all the answers

What is the effect on the shape of the parabola when the value of $a$ in the equation $y = ax^2 + q$ is greater than 1?

<p>The parabola becomes narrower and remains a 'smile'. (D)</p> Signup and view all the answers

If the y-intercept of a linear function is given by the equation $y = mx + c$ as $c = -4$, which of these statements is correct?

<p>The graph will intersect the x-axis at $(0, -4)$. (B)</p> Signup and view all the answers

In the equation of a straight line $y = mx + c$, what can be inferred if both $m$ and $c$ are equal to zero?

<p>The line coincides with the origin. (D)</p> Signup and view all the answers

How does changing $q$ from positive to negative in the quadratic function $y = ax^2 + q$ affect the graph?

<p>It shifts the graph downwards. (A)</p> Signup and view all the answers

What do the signs of $m$ and $c$ indicate about the straight line in the equation $y = mx + c$ when both are positive?

<p>The line increases and intersects the y-axis above the origin. (D)</p> Signup and view all the answers

In a typical function $f(x) = mx + c$, if $m < 0$, what characteristic can be expected from the graph?

<p>The graph will slant downwards to the right. (D)</p> Signup and view all the answers

For the quadratic function $y = ax^2 + q$, what structure does the graph take if $a$ is negative and $q$ is positive?

<p>The graph will be a downward-opening parabola with a maximum point. (C)</p> Signup and view all the answers

For an exponential function of the form $y = ab^x + q$, if $a < 0$ and $b > 1$, how does the graph behave?

<p>The graph curves downwards. (B)</p> Signup and view all the answers

In the context of linear equations, what is the primary interpretation of the gradient $m$?

<p>It measures the steepness and direction of the line. (B)</p> Signup and view all the answers

Which characteristic best describes the range of a sine function $y = a ext{sin} heta + q$ when $|a| > 1$?

<p>The range can be expressed as $[q - |a|, q + |a|]$. (A)</p> Signup and view all the answers

What results from the inequality $c < 0$ when analyzing the properties of the line $y = mx + c$?

<p>The line will intersect the y-axis below the origin. (D)</p> Signup and view all the answers

In the graph of the function $y = a ext{cos} heta + q$, what does the value of $a < 0$ imply about the amplitude?

<p>The graph is reflected about the x-axis. (B)</p> Signup and view all the answers

Which statement accurately describes the behavior of the function $y = a an heta + q$ regarding its vertical shifts?

<p>$q &gt; 0$ causes the graph to shift upwards by $q$ units. (C)</p> Signup and view all the answers

Which statement is valid for the range of the function $f(x) = ax^2 + q$ when $a > 0$?

<p>The range is from $[q; ext{infinity})$. (D)</p> Signup and view all the answers

What happens to the horizontal asymptote of an exponential function $y = ab^x + q$ when $q < 0$?

<p>The horizontal asymptote is the line $y = q$. (D)</p> Signup and view all the answers

When sketching the graph of $y = a ext{sin} heta + q$, how does an amplitude of $|a| < 1$ affect the graph?

<p>It results in a vertical compression of the graph. (D)</p> Signup and view all the answers

In the context of trigonometric function transformations, what does a positive value of $q$ do to the cosine graph?

<p>It shifts the graph upwards. (C)</p> Signup and view all the answers

How does the decay of a function of the form $y = ab^x + q$ manifest when $0 < b < 1$?

<p>The graph approaches a horizontal line at $y = q$. (D)</p> Signup and view all the answers

What is the impact on the function $y = a ext{tan} heta + q$ if $|a| > 1$?

<p>The vertical stretch results in wider oscillations of the graph. (D)</p> Signup and view all the answers

When simplifying the expression $\frac{a^5}{a^2} \times a^3$, what is the correct final result?

<p>$a^{6}$ (B)</p> Signup and view all the answers

For the exponential equation $2^x = 8$, what method would effectively lead to finding the value of x?

<p>Rewriting both sides with the same base (A)</p> Signup and view all the answers

In the expression $(ab)^{m/n}$, how can this be further simplified according to the exponent laws?

<p>$a^{m/n} b^{m/n}$ (D)</p> Signup and view all the answers

What is the result when applying the negative exponent law to the term $a^{-3}$?

<p>$\frac{1}{a^3}$ (B)</p> Signup and view all the answers

How would you correctly express the product $a^{1/2} \times a^{2/3}$?

<p>$a^{7/6}$ (D)</p> Signup and view all the answers

Which method is NOT correct for simplifying an exponential expression?

<p>Use square roots to eliminate bases (C)</p> Signup and view all the answers

What is the result of the expression $(x^3y^{-2})^2$ when simplified?

<p>$x^6 y^{-4}$ (A)</p> Signup and view all the answers

Which of the following represents the process of solving the exponential equation $3^x = 27$?

<p>Rewriting 27 as $3^3$ (D)</p> Signup and view all the answers

What is the equivalent expression for $ rac{a^{1/3}}{a^{1/2}}$?

<p>$a^{1/6}$ (B)</p> Signup and view all the answers

When expressing the roots of an expression with a fractional exponent, like $\sqrt[4]{x}$, what is its fractional equivalent?

<p>$x^{1/4}$ (A)</p> Signup and view all the answers

What is the y-intercept of the tangent function when expressed in the form $y = a \tan \theta + q$?

<p>$q$ (A)</p> Signup and view all the answers

How can you determine the direction and width of a parabola given the equation $y = ax^2 + q$?

<p>By considering both $a$ and $q$. (A)</p> Signup and view all the answers

In determining the equation of a hyperbola, what does the value of $q$ affect?

<p>The vertical shift of the graph. (D)</p> Signup and view all the answers

Which of the following correctly identifies the asymptotes of the function $y = \frac{a}{x} + q$?

<p>$\theta = 90°$ and $\theta = 270°$ (C)</p> Signup and view all the answers

What is the significance of the points where $\theta = 90°$ and $\theta = 270°$ for the tangent function?

<p>They are vertical asymptotes. (B)</p> Signup and view all the answers

To determine the equation of a trigonometric function, which of the following methods should NOT be used?

<p>Identifying the amplitude and phase shift. (C)</p> Signup and view all the answers

What can be deduced about the domain of the function given by $y = a \tan \theta + q$?

<p>${ \theta : 0° \leq \theta \leq 360°, \theta \neq 90°, 270° }$ (C)</p> Signup and view all the answers

In interpreting the graphs of parabolas, what is the necessary action to find the x-intercepts?

<p>Set $y = 0$ and solve for $x$. (B)</p> Signup and view all the answers

How does changing the value of $a$ in the equation of a hyperbola $y = \frac{a}{x} + q$ affect the graph?

<p>It modifies the steepness of the curves. (A)</p> Signup and view all the answers

Which of the following statements correctly describes a characteristic of natural numbers?

<p>Natural numbers start from 1. (B)</p> Signup and view all the answers

What best defines the set of integers?

<p>Integers are whole numbers that can also be negative. (D)</p> Signup and view all the answers

Which of the following numbers is classified as an irrational number?

<p>π (B)</p> Signup and view all the answers

In which scenario would a number not belong to the set of rational numbers?

<p>A non-terminating, non-repeating decimal. (B)</p> Signup and view all the answers

What distinguishes whole numbers from natural numbers?

<p>Whole numbers encompass both natural numbers and zero. (B)</p> Signup and view all the answers

Which set correctly represents real numbers?

<p>Both rational and irrational numbers. (C)</p> Signup and view all the answers

What is a correct characteristic of irrational numbers?

<p>They have non-repeating, non-terminating decimal expansions. (D)</p> Signup and view all the answers

Which type of decimal can be classified as rational?

<p>A decimal that contains a repeating sequence of numbers. (C)</p> Signup and view all the answers

How can the square root of a number be classified in terms of its properties?

<p>It is rational only if the input is a perfect square. (A)</p> Signup and view all the answers

What is the process involved in converting a terminating decimal into a rational number?

<p>Express it as a fraction by identifying its place value. (A)</p> Signup and view all the answers

What step is essential when estimating the value of a surd?

<p>Determine surrounding perfect squares or cubes. (B)</p> Signup and view all the answers

What happens during the rounding process if the digit after the place you are rounding is a 5?

<p>Round up the chosen digit by one. (C)</p> Signup and view all the answers

When comparing the decimal expansions of rational and irrational numbers, which statement is true?

<p>Rational numbers may exhibit both terminating and repeating decimals. (C)</p> Signup and view all the answers

Which of the following is NOT true about surds?

<p>They can be expressed in fractional form. (B)</p> Signup and view all the answers

To convert a recurring decimal into a rational number, which method is employed?

<p>Align the recurring portion after multiplication by a power of ten. (D)</p> Signup and view all the answers

What is the effect of increasing the gradient, $m$, in the equation $y = mx + c$?

<p>The slope of the graph becomes steeper. (B)</p> Signup and view all the answers

What is the result when multiplying the binomial (2x + 3) by the binomial (x + 5)?

<p>2x^2 + 13x + 15 (B)</p> Signup and view all the answers

When solving linear inequalities, which of the following statements is true if one side is divided by a negative number?

<p>The inequality sign must be switched. (C)</p> Signup and view all the answers

In the context of literal equations, what does 'changing the subject of the formula' entail?

<p>Rearranging the equation to isolate the desired variable. (C)</p> Signup and view all the answers

Which identity correctly factors the expression x^3 - 64?

<p>(x - 4)(x^2 + 4x + 16) (A), (x - 4)(x^2 + 4x + 16) (C)</p> Signup and view all the answers

What expression represents the product of a monomial 'a' and the binomial (x + y)?

<p>a(x + y) (A)</p> Signup and view all the answers

What is the first step in solving word problems using equations?

<p>Read the whole question carefully. (D)</p> Signup and view all the answers

How is the solution to a system of simultaneous equations represented graphically?

<p>By the coordinates of their point of intersection. (A)</p> Signup and view all the answers

What is the first step when simplifying the fraction ( \frac{x^2 - 4}{x^2 + 2x} )?

<p>Factor both numerator and denominator. (A)</p> Signup and view all the answers

Which method is crucial for isolating an unknown variable in a literal equation?

<p>Applying the operation opposite to each term it is joined with. (D)</p> Signup and view all the answers

What is the correct result of using the identity for the sum of two cubes on the expression x^3 + 125?

<p>(x + 5)(x^2 + 5x + 25) (B)</p> Signup and view all the answers

Which method can be used to factor the expression x^2 + 5x + 6?

<p>Finding two binomials with products matching a and c. (D)</p> Signup and view all the answers

What happens to the graph of a linear function when the y-intercept, $c$, is negative?

<p>The graph shifts vertically downwards. (B)</p> Signup and view all the answers

When substituting into one of the original equations after using elimination, what is the goal?

<p>To find the value of the remaining variable. (C)</p> Signup and view all the answers

When multiplying the expression (3x + 4) by the trinomial (2x + 5 + 1), which term results from the expansion?

<p>6x^2 + 12x + 5 (C)</p> Signup and view all the answers

What does the expression ( \frac{a^5}{a^2} \cdot a^3 ) simplify to?

<p>a^6 (C)</p> Signup and view all the answers

Which principle must be remembered when taking the square root of both sides of an equation?

<p>The results must consider both positive and negative answers. (D)</p> Signup and view all the answers

What is an essential characteristic of linear functions of the form $y = mx + c$?

<p>They graphically represent a straight line. (C)</p> Signup and view all the answers

Which of the following describes a common method used for factorization of quadratic expressions?

<p>Converting into binomial products. (C)</p> Signup and view all the answers

In arriving at the expression ( (x + 2)(x - 2) ), what type of factorization is employed?

<p>Difference of two squares (A)</p> Signup and view all the answers

What expression results from applying the negative exponent law to $a^{-3}$?

<p>$\frac{1}{a^3}$ (C)</p> Signup and view all the answers

When simplifying the expression $\frac{a^5}{a^2 \times a^{-3}}$, what is the final result?

<p>$a^{2}$ (B)</p> Signup and view all the answers

Which of the following correctly describes the simplification of the expression $(2x^3y^{-2})^2$?

<p>$4x^6y^{-4}$ (B)</p> Signup and view all the answers

What is the value of $\left(\frac{a^3}{b^2}\right)^{2/3}$ when applying the exponent rules?

<p>$\frac{a^2}{b^{4/3}}$ (A)</p> Signup and view all the answers

If $\frac{2^{3/n}}{2^{1/n}}$ is simplified, what is the result?

<p>$2^{2/n}$ (C)</p> Signup and view all the answers

Which equation correctly represents the law of exponents for the expression $(x^2y^{-3})^{3}$?

<p>$x^6 y^{-9}$ (A)</p> Signup and view all the answers

What is the result of simplifying the expression $a^0 \cdot b^0$?

<p>$1$ (A)</p> Signup and view all the answers

When using logarithms to solve the equation $2^x = 16$, which step should be taken first?

<p>Express both sides with the same base. (B)</p> Signup and view all the answers

What is the outcome of simplifying the expression $\frac{(x^4y^{-2})^2}{x^{-2}y}$?

<p>$x^{8}y^{-1}$ (C)</p> Signup and view all the answers

What describes the turning point of the graph when the coefficient a is less than zero?

<p>The graph has a maximum turning point at (0, q) (B)</p> Signup and view all the answers

For the function of the form y = ax^2 + q, what determines the direction of the parabolic graph?

<p>The value of a (C)</p> Signup and view all the answers

What is the behavior of y = (a/x) + q as x approaches zero?

<p>It becomes undefined (C)</p> Signup and view all the answers

How is the range of the function y = ax^2 + q determined if a is greater than zero?

<p>[q, infinity) (C)</p> Signup and view all the answers

When a hyperbolic function y = (a/x) + q has q greater than zero, what is the vertical shift of the graph?

<p>It shifts upwards by q units (D)</p> Signup and view all the answers

What conditions must be met for a hyperbolic function to have no x-intercept?

<p>When y cannot equal q (C)</p> Signup and view all the answers

In an exponential function represented as y = ab^x + q, which factor affects the growth direction of the graph?

<p>Only a affects the base b (D)</p> Signup and view all the answers

What is the horizontal asymptote for a hyperbolic function of the form y = (a/x) + q?

<p>y = q (B)</p> Signup and view all the answers

The axis of symmetry for the function of the form f(x) = ax^2 + q is defined as:

<p>The line x = 0 (A)</p> Signup and view all the answers

How does changing the sign of the coefficient 'a' in the equation of a parabola affect its orientation?

<p>The parabola opens downwards when a &lt; 0. (D)</p> Signup and view all the answers

What is the significance of the y-intercept 'c' in the linear equation y = mx + c?

<p>It shifts the entire graph vertically. (A)</p> Signup and view all the answers

What can be inferred about a parabola when 'q' equals zero in the equation y = ax^2 + q?

<p>The vertex is located at the origin. (D)</p> Signup and view all the answers

For a linear equation with a gradient of m = 0, what can be said about the graph?

<p>It is a horizontal line. (A)</p> Signup and view all the answers

Which statement accurately describes the domain of the function f(x) = mx + c?

<p>It can take any real number. (B)</p> Signup and view all the answers

What impact does a negative y-intercept have on the graph of a linear function?

<p>It shifts the graph down, crossing the x-axis. (C)</p> Signup and view all the answers

How does increasing the absolute value of 'a' in the equation y = ax^2 + q affect the graph of the parabola?

<p>The graph becomes narrower and steeper. (A)</p> Signup and view all the answers

Which of the following accurately describes the range of a quadratic function when 'a' is less than zero?

<p>All real numbers less than or equal to q. (A)</p> Signup and view all the answers

What is the effect of 'q' when it is a positive value in the function y = ax^2 + q?

<p>The parabola's vertex is above the x-axis. (D)</p> Signup and view all the answers

What principle defines the steepness of the graph in relation to the gradient 'm' for a linear function?

<p>The absolute value of m represents the ratio of vertical change to horizontal change. (A)</p> Signup and view all the answers

What is the significance of checking for extraneous solutions after solving an equation?

<p>To verify that the solution satisfies the original equation. (B)</p> Signup and view all the answers

In the context of solving quadratic equations, what format must be achieved for the equation before applying factorization?

<p>In the form of $ax^2 + bx + c = 0$. (B)</p> Signup and view all the answers

What must be ensured when applying operations to both sides of an equation?

<p>The equation remains balanced throughout. (C)</p> Signup and view all the answers

When solving simultaneous equations using substitution, what is typically the first step?

<p>Use the simplest equation to express one variable in terms of the other. (A)</p> Signup and view all the answers

How many solutions can a quadratic equation have under certain conditions?

<p>At most two solutions. (B)</p> Signup and view all the answers

What is the purpose of factoring out common terms when solving linear equations?

<p>To simplify the equation and reduce the number of variables. (A)</p> Signup and view all the answers

What common mistake should be avoided when rearranging terms in an equation?

<p>Displacing constant terms without balancing the equation. (D)</p> Signup and view all the answers

Which characteristic differentiates linear equations from quadratic equations?

<p>Linear equations have the highest exponent of 1; quadratics have at most 2. (C)</p> Signup and view all the answers

What is typically the last step after finding solutions for simultaneous equations?

<p>Check the solutions against the original equations. (A)</p> Signup and view all the answers

What unique feature applies to quadratic equations when considering their solutions?

<p>They can have no solutions depending on the discriminant. (C)</p> Signup and view all the answers

What happens to the graph of the function when the parameter q is set to a negative value?

<p>The horizontal asymptote moves downwards. (B)</p> Signup and view all the answers

If a function is defined as y = 3(2^x) - 5, what is the y-intercept of the graph?

<p>-5 (C)</p> Signup and view all the answers

How is the amplitude of the sine function affected when a > 1?

<p>It results in vertical stretch. (D)</p> Signup and view all the answers

For a cosine function of the form y = -2 ext{cos}( heta) + 3, what is the range of the function?

<p>[1, 5] (A)</p> Signup and view all the answers

In the equation of the tangent function y = 5 ext{tan}( heta) + 2, what effect does the factor of 5 have on the graph?

<p>It results in vertical stretch. (C)</p> Signup and view all the answers

What is the period of the function y = 4 ext{sin}(2 heta) + 1?

<p>180° (D)</p> Signup and view all the answers

How does the value of b in the exponential function y = ab^x influence the graph if 0 < b < 1?

<p>It represents exponential decay. (C)</p> Signup and view all the answers

In the context of the sine function, which of the following statements is true when a < 0?

<p>The maximum and minimum points are inverted. (A)</p> Signup and view all the answers

What vertical shift occurs in the function y = ext{sin}( heta) + 4?

<p>The graph is shifted upwards by 4 units. (B)</p> Signup and view all the answers

In an exponential function of the form y = ab^x + q, which conditions on a and b will result in a graph curving downwards?

<p>a &lt; 0 and b &gt; 1 (D)</p> Signup and view all the answers

What does the parameter 'q' do in the equations for parabolas and hyperbolas?

<p>Adjusts the vertical shift of the graph (A)</p> Signup and view all the answers

Which of the following statements accurately describes the asymptotes of a hyperbola?

<p>The graph has vertical asymptotes at specific angle values (C)</p> Signup and view all the answers

In determining the equation of a sine function, what indicates the amplitude?

<p>The coefficient 'a' (A)</p> Signup and view all the answers

How is the range defined for trigonometric functions like sine and cosine?

<p>It consists of all possible real values (B)</p> Signup and view all the answers

What is the correct process to find the value of 'a' in the equation of a hyperbola?

<p>Use another point on the curve to create an equation from 'y = a/x + q' (C)</p> Signup and view all the answers

What does the sign of 'a' in the equation of a parabola indicate?

<p>It reveals whether the parabola opens upwards or downwards (B)</p> Signup and view all the answers

How would you describe the domains of trigonometric functions like tangent?

<p>All real numbers except where the function is undefined (A)</p> Signup and view all the answers

When determining the characteristics of the graph of a tangent function, which feature is crucial?

<p>The behavior at the asymptotes (B)</p> Signup and view all the answers

What characteristic is critical in differentiating between the functions y = ax^2 + q and y = a/x + q?

<p>The quadratic versus hyperbolic nature of the functions (A)</p> Signup and view all the answers

What impact does increasing the absolute value of 'a' have on the graph of a tangent function?

<p>It increases the steepness of the graph branches (B)</p> Signup and view all the answers

Which number set includes all numbers that can be expressed as a ratio of two integers?

<p>Rational Numbers (C)</p> Signup and view all the answers

Which of the following options describes all possible values in the real number system?

<p>Rational and Irrational Numbers (D)</p> Signup and view all the answers

What distinguishes irrational numbers from rational numbers?

<p>Irrational numbers have non-repeating, non-terminating decimal expansions. (B)</p> Signup and view all the answers

Which set of numbers does NOT include negative values?

<p>Whole Numbers (B), Natural Numbers (D)</p> Signup and view all the answers

Which of the following symbols represents the set of integers?

<p>Z (C)</p> Signup and view all the answers

Which statement best describes imaginary numbers?

<p>They have a negative square root. (A)</p> Signup and view all the answers

Which of the following best describes a rational number?

<p>It can be expressed as the ratio of two integers. (D)</p> Signup and view all the answers

What differentiates an irrational number from a rational number?

<p>Irrational numbers cannot be expressed as a fraction. (D)</p> Signup and view all the answers

In converting a recurring decimal into a rational number, what is the primary operation performed?

<p>Subtraction of the original equation from a new equation. (B)</p> Signup and view all the answers

How does rounding off a number change its value?

<p>It can either increase or decrease the original number depending on the next digit. (A)</p> Signup and view all the answers

Which of the following statements accurately describes perfect squares in the context of surds?

<p>Perfect squares are obtained from squaring integers. (D)</p> Signup and view all the answers

When estimating a surd, what should you primarily identify?

<p>The nearest perfect power that surrounds the surd. (B)</p> Signup and view all the answers

In rounding off a decimal number, what does a digit of 9 result in during the rounding process?

<p>The last digit becomes 0 and the preceding digit increases by one. (A)</p> Signup and view all the answers

Which of the following types of decimal numbers are classified as rational numbers?

<p>Terminating and repeating decimals. (D)</p> Signup and view all the answers

What component of a mathematical expression is defined as a numerical factor?

<p>Coefficient (C)</p> Signup and view all the answers

What is the interpretation of the value of 'm' in the equation y = mx + c?

<p>The slope or gradient of the line (D)</p> Signup and view all the answers

If the y-intercept 'c' is negative in the equation y = mx + c, how does this affect the graph?

<p>The graph will still intersect the y-axis but below the origin (A)</p> Signup and view all the answers

What does an upward opening parabola indicate about the value of 'a' in the equation y = ax^2 + q?

<p>a is greater than zero (D)</p> Signup and view all the answers

In the function y = ax^2 + q, how does adjusting 'q' impact the graph?

<p>It causes a vertical shift of the graph (D)</p> Signup and view all the answers

What characterizes the turning point of a parabola when 'a' is negative?

<p>It is the highest point on the graph (D)</p> Signup and view all the answers

Which statement about the domain of the function y = ax^2 + q is correct?

<p>The domain is all real numbers, ∈ ℝ (D)</p> Signup and view all the answers

How does a value of 'a' between 0 and 1 affect the graph of y = ax^2 + q?

<p>It makes the parabola wider (A)</p> Signup and view all the answers

What happens to the graph of y = mx + c if 'm' is zero?

<p>The graph becomes a horizontal line (C)</p> Signup and view all the answers

What is the effect of a positive value of 'q' in the function y = ax^2 + q?

<p>Shifts the graph upwards (C)</p> Signup and view all the answers

What is the result of simplifying the expression $\frac{a^5}{a^2} \times a^3$?

<p>$a^6$ (A)</p> Signup and view all the answers

If $a^x = a^y$, what can be concluded when $a > 0$ and $a \neq 1$?

<p>$x = y$ (A)</p> Signup and view all the answers

Which property applies to the expression $(ab)^{m/n}$?

<p>$a^{m/n} b^{m/n}$ (D)</p> Signup and view all the answers

What happens when simplifying the expression $ rac{a^{m/n}}{a^{p/q}}$?

<p>$a^{m/n - p/q}$ (A)</p> Signup and view all the answers

Which expression corresponds to the zero exponent rule?

<p>$a^{0} = 1$ (C)</p> Signup and view all the answers

What is the first step to solve the equation $a^{x} = b^{y}$ using logarithms?

<p>Take the logarithm of both sides. (D)</p> Signup and view all the answers

How can the expression $ rac{a^{1/n}}{b^{1/n}}$ be rewritten?

<p>$\left(\frac{a}{b}\right)^{1/n}$ (B)</p> Signup and view all the answers

When factorizing an expression of the form $x^2 - 9$, what is the correct factorization?

<p>$(x - 3)(x + 3)$ (C)</p> Signup and view all the answers

Which of the following describes how to simplify the expression $(a^2b^3)^{3/2}$?

<p>$a^{3/2}b^{9/2}$ (B)</p> Signup and view all the answers

What is the maximum number of solutions a linear equation can have?

<p>One solution (C)</p> Signup and view all the answers

What is the result of multiplying the monomial 5 and the binomial (3x + 2)?

<p>15x + 10 (D)</p> Signup and view all the answers

Which step is essential after factoring a quadratic equation to ensure the solution is valid?

<p>Check the solution (C)</p> Signup and view all the answers

In the elimination method for solving simultaneous equations, what operation is commonly used to eliminate a variable?

<p>Adding or subtracting equations (A)</p> Signup and view all the answers

Which of the following correctly expands the binomials (x + 3) and (2x - 5)?

<p>2x^2 + 6x - 15 (D)</p> Signup and view all the answers

What is the first step in factoring the quadratic trinomial 3x^2 + 12x + 12?

<p>Identify common factors. (B)</p> Signup and view all the answers

What type of equation allows only one solution as opposed to potentially two solutions?

<p>Linear equations (D)</p> Signup and view all the answers

When simplifying the fraction ( \frac{x^2 - 1}{x^2 + x - 2} ), what expression is factored in the denominator?

<p>(x + 1)(x - 2) (B)</p> Signup and view all the answers

When solving a quadratic equation using factoring, which form should the equation be in before applying the method?

<p>ax^2 + bx + c = 0 (D)</p> Signup and view all the answers

What is a necessary condition for any operation performed on one side of an equation?

<p>It must be performed on both sides (D)</p> Signup and view all the answers

What identity can be used to factor the expression x^3 - 8?

<p>x^3 - y^3 = (x - y)(x^2 + xy + y^2) (A)</p> Signup and view all the answers

Which method involves pairing and factoring terms in the expression x^3 + 3x^2 + 2x?

<p>Factoring by grouping (C)</p> Signup and view all the answers

Which method is efficient for simplifying the number of equations in simultaneous equations?

<p>Substituting variables (B)</p> Signup and view all the answers

In solving simultaneous equations, how many equations are required to find the values of two unknowns?

<p>Two independent equations (A)</p> Signup and view all the answers

In the expression ( \frac{3x^2 + 6x}{3x} ), after canceling common factors, what remains?

<p>x + 2 (D)</p> Signup and view all the answers

What is often the first step when expanding expressions while solving linear equations?

<p>Expand all brackets (B)</p> Signup and view all the answers

Factoring the expression 4x^2 - 16 involves recognizing which type of factorization?

<p>Difference of squares (D)</p> Signup and view all the answers

When a quadratic equation has no real solutions, what can typically be inferred about its discriminant?

<p>It is less than zero (A)</p> Signup and view all the answers

What is the result of simplifying the expression ( \frac{x^3 - 1}{x - 1} )?

<p>x^2 + x + 1 (A)</p> Signup and view all the answers

What is the correct method used to multiply a binomial by a trinomial?

<p>Distribute each term of the trinomial to each term of the binomial. (C)</p> Signup and view all the answers

What does the solution of a system of simultaneous equations represent when solved graphically?

<p>The coordinates of the intersection point (C)</p> Signup and view all the answers

Which step is crucial when translating words into algebraic expressions for word problems?

<p>Identify the unknown quantity (C)</p> Signup and view all the answers

When solving literal equations, which principle is important to isolate the unknown variable?

<p>Perform the same operation to both sides (A)</p> Signup and view all the answers

What happens to the inequality symbol when both sides of a linear inequality are divided by a negative number?

<p>The signs are flipped (B)</p> Signup and view all the answers

In the equation of a line, what does the value of c represent?

<p>The point where the line intersects the y-axis (D)</p> Signup and view all the answers

What must be done if the unknown variable in a literal equation is part of multiple terms?

<p>Factor it out (C)</p> Signup and view all the answers

What approach should be taken when solving a word problem that involves multiple steps?

<p>Translate the problem into equations systematically (D)</p> Signup and view all the answers

Which of the following statements holds true regarding the value of m in a linear function?

<p>It indicates the direction of the slope (A)</p> Signup and view all the answers

When solving a linear inequality such as $2x + 2 < 1$, what is the first step?

<p>Subtract 2 from both sides (B)</p> Signup and view all the answers

What is necessary to check once a set of equations from a word problem is solved?

<p>Ensure the solution makes sense in the context of the problem (B)</p> Signup and view all the answers

What effect does the parameter $q$ have in the equations of the hyperbola and tangent functions?

<p>It specifies the vertical shift of the graph. (C)</p> Signup and view all the answers

How does the sign of $a$ influence the shape of a parabola?

<p>A positive $a$ results in a 'smile' shaped parabola. (C)</p> Signup and view all the answers

What can be inferred about the range of the function $y = a an heta + q$?

<p>It covers all real numbers without any restrictions. (D)</p> Signup and view all the answers

Which characteristics can be determined from the asymptotes of the tangent function?

<p>They indicate where the function is not defined. (D)</p> Signup and view all the answers

When solving for $a$ in the equation of a hyperbola $y = rac{a}{x} + q$, what method must be used?

<p>Use given points to substitute and create a system of equations. (C)</p> Signup and view all the answers

What is the primary goal when determining the equation of a parabola using its sketch?

<p>To confirm the direction of the parabola. (C)</p> Signup and view all the answers

In which scenario is it necessary to set $y = 0$ to find the x-intercept for parabolic graphs?

<p>For any standard form quadratic equations. (B)</p> Signup and view all the answers

Which equation represents the graph of a sine function modified by vertical shifts and amplitude?

<p>$y = a ext{sin} heta + q$ (D)</p> Signup and view all the answers

What defines the domain of the tangent function based on its behavior?

<p>It excludes specific $ heta$ values that cause undefined outputs. (C)</p> Signup and view all the answers

What determines the direction in which an exponential graph curves when both $a$ and $b$ are greater than 1?

<p>The sign of $a$ (A)</p> Signup and view all the answers

Which of the following accurately describes the behavior of the sine function at its minimum turning point?

<p>It occurs at $270°$ (D)</p> Signup and view all the answers

What effect does a negative value of $a$ have on the graph of a sine function?

<p>It reflects the graph about the x-axis (C)</p> Signup and view all the answers

For a cosine function $y = a , cos , heta + q$, which condition indicates a vertical compression?

<p>$|a| &lt; 1$ (C)</p> Signup and view all the answers

What happens to the horizontal asymptote of an exponential function when $q < 0$?

<p>It moves downwards by $q$ units (C)</p> Signup and view all the answers

Which of the following describes the domain of the tangent function?

<p>$ heta : 0° ext{ to } 360°, heta eq 90°, 270°$ (C)</p> Signup and view all the answers

How does the value of $b$ influence an exponential function of the form $y = ab^x + q$?

<p>It determines whether the function grows or decays (A)</p> Signup and view all the answers

What is the period of the sine function expressed as $y = ext{sin} heta$?

<p>360° (C)</p> Signup and view all the answers

Which point represents the y-intercept of the function $y = an heta$?

<p>(0°, 0) (B)</p> Signup and view all the answers

What is the range of a function of the form $f(x) = ax^2 + q$ when $a < 0$?

<p>$(- ext{infinity}; q]$ (C)</p> Signup and view all the answers

At what point does the turning point occur for the graph of $f(x) = ax^2 + q$?

<p>At $(0, q)$ (D)</p> Signup and view all the answers

What is the effect of setting $x = 0$ on the hyperbolic function $y = rac{a}{x} + q$?

<p>It causes the function to be undefined. (B)</p> Signup and view all the answers

Which of the following statements accurately describes the horizontal asymptote of the function $y = rac{a}{x} + q$?

<p>It is at $y = q$. (B)</p> Signup and view all the answers

What is the domain of the hyperbolic function $y = rac{a}{x} + q$?

<p>$ ext{R} ackslash ext{0}$ (C)</p> Signup and view all the answers

In the equation of an exponential function $y = ab^x + q$, what does a positive value of $a$ indicate?

<p>The graph will lie above the line $y = q$. (D)</p> Signup and view all the answers

For the function $y = ax^2 + q$, if $a$ is negative, which of the following is true?

<p>The graph has a maximum turning point. (A)</p> Signup and view all the answers

What describes the axes of symmetry for the function $y = rac{a}{x} + q$?

<p>Symmetrical about the line $y = x + q$. (A)</p> Signup and view all the answers

When designing a graph for $f(x) = ax^2 + q$, which characteristic must be established first?

<p>The sign of $a$. (C)</p> Signup and view all the answers

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