Pre cal R review
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Questions and Answers

What is one method to solve the equation (𝑥 − 1)(3𝑥 + 2) = 0?

  • Set each factor to zero separately (correct)
  • Use the Quadratic Formula directly
  • Combine the factors on the left side
  • Multiply both sides by zero

Which equation represents the product of complex conjugates 𝑎 + 𝑏𝑖 and 𝑎 − 𝑏𝑖?

  • 𝑎^2 + 𝑏^2 (correct)
  • 𝑎^2 + 2𝑏𝑖
  • 𝑎^2 - 𝑏^2
  • 𝑎^2 - 2𝑏𝑖

What is the first step to solve a quadratic equation using the Square Root Method?

  • Apply the Quadratic Formula
  • Take the derivative of the equation
  • Get the squared term on one side (correct)
  • Factor the equation

What form does a complex number take?

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

What does the √𝑏² − 4𝑎𝑐 represent in the Quadratic Formula?

<p>The discriminant of the equation (A)</p> Signup and view all the answers

What is the standard equation of a line?

<p>Ax + By = C (D)</p> Signup and view all the answers

How do you find the x-intercept of a graph?

<p>Let y = 0 and solve for x (D)</p> Signup and view all the answers

What does the slope of a line represent?

<p>The steepness of the line (A)</p> Signup and view all the answers

Which of these slopes corresponds to a horizontal line?

<p>Slope equal to 0 (A)</p> Signup and view all the answers

Under what condition are two lines considered parallel?

<p>If their slopes are the same (B)</p> Signup and view all the answers

What rule describes the operation order in mathematical expressions?

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

What is the result of $a^0$ for any real number a?

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

If the slopes of two lines are m1 and m2, how are the lines classified as perpendicular?

<p>If m1 = -1/m2 (B)</p> Signup and view all the answers

What type of transformation is represented by the function $y = √x - 2$?

<p>A vertical shift downwards by 2 units (C)</p> Signup and view all the answers

How does the function $y = |x - 2|$ differ from the basic absolute value function $y = |x|$?

<p>It shifts the graph 2 units to the right. (C)</p> Signup and view all the answers

What feature would you find in the graph of $y = -x^2$?

<p>A downward-opening parabola (D)</p> Signup and view all the answers

Which transformation is present in the function $y = √−x$?

<p>A reflection across the y-axis (A)</p> Signup and view all the answers

What is the domain of the function $y = √x - 2$?

<p>$[0, ∞)$ (B)</p> Signup and view all the answers

What is the effect of the coefficient in the function $y = 2|x|$ compared to $y = |x|$?

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

How does changing the function to $y = |x + 3|$ impact the graph of $y = |x|$?

<p>Shifts the graph 3 units to the left (D)</p> Signup and view all the answers

Which of these functions contains a transformation that reflects the graph across the x-axis?

<p>$y = -|x - 2|$ (C)</p> Signup and view all the answers

What is the result of $(ab)^n$ in terms of $a$ and $b$?

<p>$a^n b^n$ (B)</p> Signup and view all the answers

Which of the following correctly represents the domain of the function $f(x) = \sqrt{x - 1}$?

<p>[1, ∞) (D)</p> Signup and view all the answers

What must be done to the inequality if both sides are multiplied by a negative number?

<p>It is reversed. (D)</p> Signup and view all the answers

In the function notation $f(x) = 3x - 5$, what is the value of $f(2)$?

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

What does the Vertical Line Test determine about a graph?

<p>Whether the graph represents a function. (A)</p> Signup and view all the answers

Using the Zero Product Property, if $xy = 0$, what can be concluded?

<p>At least one of $x$ or $y$ must be zero. (C)</p> Signup and view all the answers

If $x^2 + bx + c = 0$ is a quadratic equation, what strategy can be used to factor it when $c$ and $b$ are integers?

<p>Find two numbers that add to $b$ and multiply to $ac$. (C)</p> Signup and view all the answers

Which expression represents the range of the inequality $x eq 2$?

<p>(-∞, 2) ∪ (2, ∞) (C)</p> Signup and view all the answers

What is the sum of the functions 𝑓(𝑥) and 𝑔(𝑥) if 𝑓(𝑥) = 3𝑥^2 - 1 and 𝑔(𝑥) = 𝑥^3?

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

What is the correct expression for the difference of the functions 𝑓(𝑥) and 𝑔(𝑥) if 𝑓(𝑥) = 3𝑥 - 1 and 𝑔(𝑥) = √𝑥?

<p>3𝑥 - 1 - √𝑥 (D)</p> Signup and view all the answers

What is the product of the functions 𝑓(𝑥) and 𝑔(𝑥) for 𝑓(𝑥) = 3𝑥 - 2 and 𝑔(𝑥) = 2𝑥 + 1?

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

What can be concluded about the domain of 1/(𝑔(𝑥)) if 𝑔(𝑥) is a function defined for all 𝑥, except when it equals zero?

<p>The domain excludes zero. (B)</p> Signup and view all the answers

What is the difference quotient formula for a function 𝑓(𝑥)?

<p>(𝑓(𝑥 + 𝑕) - 𝑓(𝑥))/𝑕 (A)</p> Signup and view all the answers

For 𝑔(𝑥) = −3|𝑥 + 1| − 2, what is the general behavior of the graph?

<p>It opens downwards. (C)</p> Signup and view all the answers

In the expression 𝑓(𝑥) = 3𝑥^3, what is the degree of the polynomial?

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

How is the function 𝐻(𝑥) = (𝑥 − 2)^2 + 1 transformed from its parent function?

<p>Shifted right 2 units and up 1 unit. (C)</p> Signup and view all the answers

Study Notes

Lines and Slopes

  • Standard equation of a line: ( Ax + By = C ) where ( A, B, C ) are real numbers; ( A ) and ( B ) cannot both be zero.
  • X-intercept: Found by setting ( y = 0 ) and solving for ( x ); point where the graph crosses the x-axis.
  • Y-intercept: Found by setting ( x = 0 ) and solving for ( y ); point where the graph crosses the y-axis.
  • Slope (m): Calculated through points ( (x_1, y_1) ) and ( (x_2, y_2) ): [ m = \frac{y_2 - y_1}{x_2 - x_1} ]
  • Types of lines:
    • Vertical line: ( x = \text{constant} ), slope is undefined.
    • Horizontal line: ( y = \text{constant} ), slope is zero.

Line Equations

  • Slope-intercept form: ( y = mx + b ), where ( m ) is slope and ( b ) is the y-intercept.
  • Point-slope form: [ y - y_1 = m(x - x_1) ]
  • Parallel lines: Lines ( L_1 ) and ( L_2 ) are parallel if ( m_1 = m_2 ).
  • Perpendicular lines: Lines ( L_1 ) and ( L_2 ) are perpendicular if ( m_1 = -\frac{1}{m_2} ).

Order of Operations and Exponent Rules

  • Order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • Key exponent rules:
    • ( a^0 = 1 ), ( a^1 = a )
    • ( a^m \cdot a^n = a^{m+n} )
    • ( \frac{a^m}{a^n} = a^{m-n} )
    • ( (a^m)^n = a^{mn} )
    • ( (ab)^n = a^n b^n )

Inequalities and Functions

  • When multiplying or dividing inequalities by a negative number, flip the inequality sign.
  • Interval notation: e.g., ( x < -3 ) is ( (-\infty, -3) ); ( x \leq 5 ) is ( (-\infty, 5] ); ( x \geq 0.63 ) is ( [0.63, \infty) ).
  • A function corresponds each x-value to exactly one y-value.
  • The domain of a function includes all x-values that yield a real number; the range includes all resulting y-values.

Factoring Techniques

  • Steps in factoring:
    • Check for a GCF (Greatest Common Factor) and factor out.
    • Use grouping for 4 terms or apply specific methods for 2 or 3 terms.
    • Zero Product Property states if ( ab = 0 ), then ( a = 0 ) or ( b = 0 ).

Complex Numbers

  • Imaginary unit: ( i = \sqrt{-1} ) and ( i^2 = -1 ).
  • A complex number has the form ( a + bi ) with real ( a ) and ( b ).
  • Complex conjugates ( a + bi ) and ( a - bi ) yield a positive real number when multiplied.

Quadratic and Difference Quotient

  • Quadratic Formula: Solutions to ( ax^2 + bx + c = 0 ): [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
  • Difference Quotient: [ \frac{f(x+h) - f(x)}{h} ]

Graphs and Transformations

  • Changes that affect y-values occur on the outside of a function, while changes that affect x-values occur on the inside.
  • Examples of functions to graph include squares, absolute values, and cube functions.

These notes encompass essential definitions, rules, and examples necessary for understanding lines, functions, factoring, and other algebraic concepts.

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Chapter R Notes

Description

This quiz covers the standard equation of a line, x-intercept, y-intercept, and slope calculations. Learn to identify and apply these concepts through examples and problem-solving. Test your understanding of these fundamental algebraic principles.

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