Exponents and Quadratic Equations

Simplifying Exponential Expressions Involving Integer Exponents

Exponential expressions with integer exponents can be simplified using the laws of exponents. These rules help in rewriting expressions in a more manageable form.

• Product Rule:

• Quotient Rule:

• Power Rule:

• Zero Exponent: (for )

• Negative Exponent:

Example: Simplify

Solving Quadratic Equations Using the Quadratic Formula

Quadratic equations are equations of the form . The quadratic formula provides a method for finding their solutions.

• Quadratic Formula:

• The expression under the square root, , is called the discriminant.

• If the discriminant is positive, there are two real solutions; if zero, one real solution; if negative, two complex solutions.

Example: Solve using the quadratic formula.

Linear Equations and Graphs

Finding the Equation of a Line Using the Slope-Intercept Form

The slope-intercept form of a line is , where is the slope and is the y-intercept.

• Given a point and slope , the equation is (point-slope form), which can be rearranged to slope-intercept form.

Example: Find the equation of a line with slope $2(1, 3)y - 3 = 2(x - 1) \implies y = 2x + 1$

Determining the Slope of a Line

The slope of a line measures its steepness and is calculated as the ratio of the change in to the change in between two points.

• Slope Formula:

Example: For points and ,

Finding the Equations of Parallel and Perpendicular Lines

• Parallel lines have the same slope:

• Perpendicular lines have slopes that are negative reciprocals:

Example: A line perpendicular to has slope

Functions and Their Properties

Using Function Notation; Evaluating Functions

A function assigns each input to exactly one output . Function notation is used to express this relationship.

• To evaluate, substitute the input value into the function.

Example: If , then

Determining the Domain of a Function Given the Equation

The domain of a function is the set of all input values for which the function is defined.

• For rational functions, exclude values that make the denominator zero.

• For even roots, exclude values that make the radicand negative.

Example: The domain of is all real numbers except .

Determining Whether a Function Is Even, Odd, or Neither

Functions can be classified based on their symmetry:

• Even function: (symmetric about the y-axis)

• Odd function: (symmetric about the origin)

Example: is even; is odd.

Determining Information about a Function from a Graph

Graphs can reveal properties such as intercepts, intervals of increase/decrease, and symmetry.

• Identify - and -intercepts.

• Check for symmetry and behavior at endpoints.

Analyzing Piecewise-Defined Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Evaluate by determining which interval the input belongs to.

Example:

Using Combinations of Transformations to Graph Functions

Functions can be transformed by shifting, reflecting, stretching, or compressing their graphs.

• Vertical shift:

• Horizontal shift:

• Reflection: (over x-axis), (over y-axis)

• Vertical stretch/compression:

Evaluating a Combined Function

Combined functions involve operations such as addition, subtraction, multiplication, or division of two functions.

• ,

Finding the Intersection of Intervals

The intersection of intervals is the set of numbers common to both intervals.

• For example,

Forming and Evaluating Composite Functions

The composite function is defined as .

• Evaluate first, then substitute into .

Example: If and , then

Determining the Domain of Composite Functions

The domain of consists of all in the domain of such that is in the domain of .

Understanding the Definition of a One-to-One Function

A function is one-to-one if each output is paired with exactly one input.

• No horizontal line intersects the graph more than once.

Determining Whether a Function Is One-to-One Using the Horizontal Line Test

If every horizontal line crosses the graph at most once, the function is one-to-one.

Understanding and Verifying Inverse Functions

The inverse function reverses the effect of .

• and

Sketching the Graphs of Inverse Functions

The graph of is the reflection of the graph of across the line .

Finding the Inverse of a One-to-One Function

To find the inverse, solve for and interchange and .

• Example: , so

Quadratic and Polynomial Functions

Determining the Equation of a Quadratic Function Given Its Graph

A quadratic function has the form , where is the vertex.

• Use the vertex and another point to solve for .

Maximizing Projectile Motion Functions

Projectile motion is modeled by (in feet, with in seconds).

• The maximum height occurs at for .

Determining the Real Zeros of Polynomial Functions and Their Multiplicities

The real zeros of a polynomial are the -values where the function equals zero. The multiplicity of a zero is the number of times it occurs as a factor.

• If a zero has even multiplicity, the graph touches the -axis and turns around.

• If a zero has odd multiplicity, the graph crosses the -axis.

Example: has a zero at (multiplicity 2) and (multiplicity 1).