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.

• 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's argument moves the graph left or right.

• Reflections: Multiplying the function by -1 reflects it across the x-axis (vertical reflection).

• Examples:

  • Upside-down and shifted right 6 units, up 4 units (for $y = x^2$): $y = - (x - 6)^2 + 4$

  • Shifted left 8 units, up 7 units (for $y = |x|$): $y = |x + 8| + 7$

  • Upside-down, shifted left 5 units, up 3 units (for $y = x^2$): $y = - (x + 5)^2 + 3$

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 $ax^2 + bx + c = 0$. Solutions can be found by factoring, completing the square, or using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Rational Equations: Equations involving fractions with polynomials in the numerator and denominator. Clear denominators by multiplying both sides by the least common denominator (LCD).

• Radical Equations: Equations involving roots. Isolate the radical and then square both sides to eliminate it, checking for extraneous solutions.

• Examples:

  • $x^2 - \frac{3}{3} = 2x$

  • $\frac{1}{x} + \frac{1}{x+1} = \frac{5}{6}$

  • $\sqrt{x+4} + 5 = 0$

  • $\sqrt{4x+1} - 6 = 2$

  • $\frac{3x}{x+5} + \frac{15}{x} = \frac{75}{x^2+5x}$

Graph Analysis

Intercepts, Zeros, and Graph Behavior

Analyzing the graph of a function involves identifying key features such as intercepts, zeros, and intervals of increase or decrease.

• x-intercepts (Zeros): Points where the graph crosses the x-axis. Found by solving $f(x) = 0$.

• y-intercept: The point where the graph crosses the y-axis, found by evaluating $f(0)$.

• Intervals of Increase/Decrease: Where the function is rising or falling as x increases.

• Constant Intervals: Where the function remains unchanged as x increases.

• Example: Given a graph, identify the x-intercepts and zeros by finding where the curve meets the x-axis.

Polynomial and Radical Equations

Solving Higher-Degree and Radical Equations

Equations of degree three or higher, and those involving radicals, require specialized techniques for solution.

• Cubic Equations: $x^3 - 4 = 4x - x^2$

• Radical Equations: $\sqrt{x+4} + 2 = x$

• Absolute Value Equations: $7 - |4x + 3| = 2$

• Strategy: Isolate the variable, use factoring, or apply the quadratic formula as needed. For absolute value, consider both positive and negative cases.

Function Properties

One-to-One Functions and Inverses

A function is one-to-one if each output is produced by exactly one input. Such functions have inverses.

• Test for One-to-One: A function $f(x)$ is one-to-one if $f(a) = f(b)$ implies $a = b$.

• Finding the Inverse: Swap $x$ and $y$ in $y = f(x)$ and solve for $y$.

• Example: For $f(x) = 8x - 1$, the inverse is found as follows:

  • Let $y = 8x - 1$

  • Swap $x$ and $y$: $x = 8y - 1$

  • Solve for $y$: $y = \frac{x + 1}{8}$

Even and Odd Functions

Classification of Functions

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

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain. Graph is symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain. Graph is symmetric about the origin.

• Neither: If neither condition holds.

• Example: $f(x) = x^4 - 4x^2 + 6$

Complex Numbers

Operations with Complex Numbers

Complex numbers are numbers of the form $a + bi$, where $i = \sqrt{-1}$.

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Multiplication: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

• Example: Compute $(-8 + 9i)^2$ and $(-4 + 3i)(8 - 7i)$

Zeros and Multiplicity

Finding Zeros and Their Multiplicities

The zeros of a function are the values of $x$ for which $f(x) = 0$. The multiplicity of a zero is the number of times it appears as a root.

• Example: For $f(x) = (x + 2)^2(x - 1)$:

  • Zero at $x = -2$ with multiplicity 2

  • Zero at $x = 1$ with multiplicity 1

Summary Table: Types of Equations and Solution Methods