Function Transformations

Transformations of Quadratic and Absolute Value Functions

Function transformations involve shifting, reflecting, stretching, or compressing the graph of a function. Understanding these transformations is essential for graphing and analyzing functions in algebra.

• Vertical Shifts: Adding or subtracting a constant to a function moves the graph up or down.

• Horizontal Shifts: Adding or subtracting a constant inside the function argument moves the graph left or right.

• Reflections: Multiplying the function by -1 reflects it over the x-axis.

• Examples:

  • Given , the graph of is upside-down (reflection), shifted right 6 units, and up 4 units.

  • Given , the graph of is shifted left 8 units and up 7 units.

  • Given , the graph of is upside-down, shifted left 5 units, and up 3 units.

General Transformation Formula:

• For , the transformed function is where:

  • = vertical stretch/compression/reflection

  • = horizontal shift

  • = vertical shift

Solving Equations

Quadratic, Rational, and Radical Equations

Solving equations is a fundamental skill in algebra. Equations may be quadratic, rational, or involve radicals.

• Quadratic Equations: Equations of the form can be solved by factoring, completing the square, or using the quadratic formula.

• Rational Equations: Equations involving fractions can be solved by finding a common denominator and simplifying.

• Radical Equations: Equations involving roots require isolating the radical and then squaring both sides.

• Examples:

  • Solve

  • Solve

  • Solve

  • Solve

  • Solve

Quadratic Formula:

Function Properties

Intercepts, Zeros, and Multiplicity

Understanding the intercepts and zeros of a function is crucial for graphing and analyzing its behavior.

• x-intercepts: Points where the graph crosses the x-axis ().

• Zeros: Values of for which .

• Multiplicity: The number of times a particular zero occurs. If a zero has even multiplicity, the graph touches the x-axis; if odd, it crosses.

• Example: For , zeros are (multiplicity 2) and (multiplicity 1).

Even, Odd, and Neither Functions

Functions can be classified as even, odd, or neither based on their symmetry.

• Even Function: for all (symmetric about the y-axis).

• Odd Function: for all (symmetric about the origin).

• Example:

Graph Analysis

Increasing, Decreasing, and Constant Intervals

Analyzing a graph involves identifying intervals where the function is increasing, decreasing, or constant.

• Increasing: As increases, increases.

• Decreasing: As increases, decreases.

• Constant: remains the same as increases.

• Example: Use the provided graph to identify intervals of increase, decrease, and constancy.

Inverse Functions

One-to-One Functions and Finding Inverses

A function is one-to-one if each output is produced by exactly one input. The inverse function reverses the roles of inputs and outputs.

• One-to-One Test: A function is one-to-one if implies .

• Finding the Inverse: Solve for in terms of , then replace with .

• Example: For , the inverse is .

Complex Numbers

Operations with Complex Numbers

Complex numbers are numbers of the form , where is the imaginary unit ().

• Addition:

• Example:

HTML Table: Sample Question Distribution

The following table shows the distribution of points across questions in the assignment.

Additional info: The study guide covers all major topics relevant to College Algebra, including function transformations, solving various types of equations, graph analysis, properties of functions, inverse functions, and basic complex number operations. The table above is inferred from the assignment instructions and question layout.