College Algebra Practice Test 2: Functions, Graphs, Transformations, and Polynomial Analysis

Functions and Their Graphs

Vertical Line Test and Function Properties

The Vertical Line Test is a graphical method used to determine if a curve represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.

• Function: A relation in which each input (x-value) has exactly one output (y-value).

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Example: The graph of $y = x^2$ passes the vertical line test and is a function. Its domain is $(-\infty, \infty)$ and its range is $[0, \infty)$.

Transformations of Functions

Types of Transformations

Transformations change the position or shape of a graph. Common transformations include translations, reflections, stretches, and compressions.

• Translation: Shifts the graph horizontally or vertically.

• Reflection: Flips the graph over a line (usually the x-axis or y-axis).

• Stretch/Compression: Changes the steepness or width of the graph.

Transformation	Equation	Description
Horizontal Shift	$y = f(x + c)$	Shift left by $c$ units
Vertical Shift	$y = f(x) + c$	Shift up by $c$ units
Reflection over y-axis	$y = f(-x)$	Flip horizontally
Reflection over x-axis	$y = -f(x)$	Flip vertically
Vertical Stretch	$y = a f(x)$	Stretch by factor $a$
Horizontal Stretch	$y = f(ax)$	Compress horizontally by $a$

Example: The graph of $y = f(x - 2)$ is the graph of $f(x)$ shifted right by 2 units.

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions over different intervals of the domain.

• Example: $h(x) = \begin{cases} x^2 & \text{if } x \leq 2 \\ 6 - x & \text{if } x > 2 \end{cases}$

• To graph, plot each piece over its specified interval.

• Identify intervals where the function is increasing or decreasing by analyzing the slope or derivative.

Function Operations and Composition

Operations on Functions

Functions can be added, subtracted, multiplied, divided, or composed.

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Quotient: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

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

Domain: The domain of the resulting function depends on the domains of $f$ and $g$ and the operation performed.

Inverse Functions

Definition and Verification

An inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. A function has an inverse if and only if it is one-to-one (passes the Horizontal Line Test).

• Verification by Composition: $f(g(x)) = x$ and $g(f(x)) = x$

• Example: If $f(x) = 2x - 1$ and $g(x) = \frac{x + 1}{2}$, then $f(g(x)) = 2\left(\frac{x + 1}{2}\right) - 1 = x$

Quadratic Functions and Graphing

Vertex, Intercepts, and Axis of Symmetry

A quadratic function has the form $f(x) = ax^2 + bx + c$. Its graph is a parabola.

• Vertex: The point $(h, k)$ where $h = -\frac{b}{2a}$ and $k = f(h)$

• Axis of Symmetry: The vertical line $x = h$

• Intercepts: Find $f(0)$ for the y-intercept and solve $f(x) = 0$ for x-intercepts.

Example: For $f(x) = \frac{1}{2}(x - 1)^2 - 3$, the vertex is $(1, -3)$ and the axis of symmetry is $x = 1$.

Polynomial Functions

End Behavior, Zeros, and Factorization

Polynomial functions are expressions of the form $P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0$.

• End Behavior: Determined by the leading term $a_nx^n$.

• Zeros: Values of $x$ where $P(x) = 0$.

• Factor Theorem: If $P(a) = 0$, then $(x - a)$ is a factor of $P(x)$.

• Rational Root Theorem: Possible rational zeros are $\pm \frac{\text{factors of constant}}{\text{factors of leading coefficient}}$.

Example: For $P(x) = 2x^3 - 13x^2 + 11x + 25$, if $x = 5$ is a zero, then $(x - 5)$ is a factor.

Rational Functions and Asymptotes

Finding Asymptotes

Rational functions are quotients of polynomials. Asymptotes describe the behavior of the graph as $x$ approaches certain values.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is not zero.

• Horizontal Asymptotes: Determined by the degrees of numerator and denominator.

• Oblique (Slant) Asymptotes: Occur when the degree of the numerator is one more than the degree of the denominator.

Function	Vertical Asymptote	Horizontal Asymptote
$f(x) = \frac{2x^2 - 5}{3x^2 + 2x - 12}$	Set $3x^2 + 2x - 12 = 0$	Compare degrees: $y = \frac{2}{3}$
$g(x) = \frac{2x^2 + 1}{3x^2 - 4x - 4}$	Set $3x^2 - 4x - 4 = 0$	$y = \frac{2}{3}$
$h(x) = \frac{2x^4 - 4x^3 + 5}{2x}$	Set $2x = 0$	No horizontal asymptote (degree numerator > denominator)

Optimization Problems

Maximizing Area with Constraints

Optimization involves finding the maximum or minimum value of a function given certain constraints.

• Example: A rancher has 600 yards of fencing to enclose a rectangular corral with a partition. Let $x$ and $y$ be the dimensions. The constraint is $3x + 2y = 600$.

• Express area $A = x \cdot y$ in terms of one variable using the constraint, then maximize $A$.

Additional info:

• Questions cover topics from Ch. 3 (Functions and Their Graphs), Ch. 4 (Linear and Quadratic Functions), Ch. 5 (Polynomial and Rational Functions), and Ch. 7 (Analytic Geometry).

• Some questions require graphing, analysis of piecewise functions, and application of the Rational Root Theorem.