Pre cal R review

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?

𝑎 + 𝑏𝑖 Signup and view all the answers

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

The discriminant of the equation Signup and view all the answers

What is the standard equation of a line?

Ax + By = C Signup and view all the answers

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

Let y = 0 and solve for x Signup and view all the answers

What does the slope of a line represent?

The steepness of the line Signup and view all the answers

Which of these slopes corresponds to a horizontal line?

Slope equal to 0 Signup and view all the answers

Under what condition are two lines considered parallel?

If their slopes are the same Signup and view all the answers

What rule describes the operation order in mathematical expressions?

PEMDAS Signup and view all the answers

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

1 Signup and view all the answers

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

If m1 = -1/m2 Signup and view all the answers

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

A vertical shift downwards by 2 units Signup and view all the answers

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

It shifts the graph 2 units to the right. Signup and view all the answers

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

A downward-opening parabola Signup and view all the answers

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

A reflection across the y-axis Signup and view all the answers

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

$[0, ∞)$ Signup and view all the answers

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

It results in a vertical stretch Signup and view all the answers

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

Shifts the graph 3 units to the left Signup and view all the answers

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

$y = -|x - 2|$ Signup and view all the answers

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

$a^n b^n$ Signup and view all the answers

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

[1, ∞) Signup and view all the answers

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

It is reversed. Signup and view all the answers

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

1 Signup and view all the answers

What does the Vertical Line Test determine about a graph?

Whether the graph represents a function. Signup and view all the answers

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

At least one of $x$ or $y$ must be zero. 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?

Find two numbers that add to $b$ and multiply to $ac$. Signup and view all the answers

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

(-∞, 2) ∪ (2, ∞) 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?

3𝑥^2 + 𝑥^3 - 1 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?

3𝑥 - 1 - √𝑥 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?

6𝑥^2 + 4𝑥 - 2 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?

The domain excludes zero. Signup and view all the answers

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

(𝑓(𝑥 + 𝑕) - 𝑓(𝑥))/𝑕 Signup and view all the answers

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

It opens downwards. Signup and view all the answers

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

3 Signup and view all the answers

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

Shifted right 2 units and up 1 unit. 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.

Studying That Suits You

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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.