College Algebra: Functions, Graphs, and Applications Study Guide

Functions and Their Graphs

Intervals of Increase, Decrease, and Constancy

Understanding where a function increases, decreases, or remains constant is fundamental in analyzing its behavior. These intervals are typically identified by examining the graph of the function.

• Increasing Interval: A function is increasing on an interval if, as x increases, f(x) also increases.

• Decreasing Interval: A function is decreasing on an interval if, as x increases, f(x) decreases.

• Constant Interval: A function is constant on an interval if, as x increases, f(x) remains the same.

• Interval Notation: Use parentheses for open intervals and brackets for closed intervals, e.g., (a, b) or [a, b].

Example: If a graph rises from left to right between x = -2 and x = 1, the function is increasing on (-2, 1).

Relative Maximum and Minimum

Relative extrema are points where a function reaches a local highest or lowest value.

• Relative Maximum: The highest point in a particular section of a graph.

• Relative Minimum: The lowest point in a particular section of a graph.

• To find these, look for points where the graph changes from increasing to decreasing (maximum) or decreasing to increasing (minimum).

Example: If f(x) has a peak at x = 2, then f(2) is a relative maximum.

Applications of Functions

Area Problems with Constraints

Many real-world problems involve maximizing or minimizing a quantity, such as area, given certain constraints.

• Rectangular Area with Fixed Perimeter: If a rectangle has a fixed amount of fencing, express the area as a function of one side's length.

• Formula: For a rectangle with length l and width w, and total fencing P: $2l + 2w = P$; Area $A = l \times w$.

Example: If 200 ft of fencing is available and the width is x, then length is $l = (200 - 2x)/2$, so $A(x) = x \cdot (100 - x)$.

Piecewise Functions and Graphing

Piecewise-Defined Functions

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

• Definition: A function defined by multiple sub-functions, each applying to a certain interval.

• Graphing: Plot each piece over its specified interval, paying attention to open and closed endpoints.

Example: $f(x) = \begin{cases} 2x+1 & x < 1 \\ x^2 & x \geq 1 \end{cases}$

Transformations of Functions

Shifts, Reflections, and Stretches

Transformations alter the position or shape of a function's graph.

• Vertical Shift: $f(x) + k$ shifts up by k units.

• Horizontal Shift: $f(x - h)$ shifts right by h units.

• Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

Example: $g(x) = -2f(x+3) + 1$ reflects f(x) over the x-axis, stretches vertically by 2, shifts left by 3, and up by 1.

Difference Quotient

Definition and Computation

The difference quotient is a formula that gives the average rate of change of a function over an interval.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• Used as the foundation for the derivative in calculus.

Example: For $f(x) = x^2$, the difference quotient is $\frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$.

Function Composition and Inverses

Composition of Functions

Composing functions involves applying one function to the result of another.

• Notation: $(f \circ g)(x) = f(g(x))$

• Evaluate the inner function first, then the outer function.

Example: If $f(x) = 2x$ and $g(x) = x+3$, then $(f \circ g)(x) = f(x+3) = 2(x+3) = 2x+6$.

Inverse Functions

An inverse function reverses the effect of the original function.

• Definition: $f^{-1}(f(x)) = x$ for all x in the domain of f.

• To find the inverse, solve $y = f(x)$ for x in terms of y, then swap x and y.

Example: If $f(x) = 3x + 2$, then $y = 3x + 2 \implies x = (y-2)/3$, so $f^{-1}(x) = (x-2)/3$.

Even and Odd Functions

Definitions and Tests

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

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

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

• Neither: If neither condition holds.

Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Trigonometric Functions and Applications

Basic Trigonometric Equations

Solving trigonometric equations often involves using inverse trigonometric functions and understanding the unit circle.

• Example: To solve $\sin \theta = 1/2$, find all angles where the sine value is 1/2.

• General Solution: $\theta = \arcsin(1/2) + 2\pi n$ or $\theta = \pi - \arcsin(1/2) + 2\pi n$ for integer n.

Amplitude, Period, and Frequency

For sinusoidal functions, amplitude, period, and frequency describe the shape and repetition of the wave.

• Amplitude: The maximum distance from the midline to the peak.

• Period: The length of one complete cycle, $T = \frac{2\pi}{|b|}$ for $y = a \sin(bx)$.

• Frequency: The number of cycles per unit interval, $f = \frac{1}{T}$.

Example: For $y = 3 \sin(2x)$, amplitude is 3, period is $\pi$.

Tables: Function Properties and Transformations

Summary Table: Types of Function Symmetry

Type	Algebraic Test	Graphical Symmetry
Even	$f(-x) = f(x)$	y-axis
Odd	$f(-x) = -f(x)$	Origin
Neither	Neither condition holds	No symmetry

Summary Table: Common Function Transformations

Transformation	Equation	Effect
Vertical Shift	$f(x) + k$	Up by k units
Horizontal Shift	$f(x - h)$	Right by h units
Reflection (x-axis)	$-f(x)$	Flip over x-axis
Reflection (y-axis)	$f(-x)$	Flip over y-axis
Vertical Stretch	$a f(x)$, $|a| > 1$	Stretches vertically
Vertical Compression	$a f(x)$, $0 < |a| < 1$	Compresses vertically

Additional info:

• Some questions involve graph identification, which requires understanding of how algebraic changes affect the graph's appearance.

• Real-world applications (e.g., fencing, sound waves) connect algebraic concepts to practical problems.