College Algebra Final Exam Review: Key Concepts and Practice Problems

Functions, Graphs, and Models

Understanding Functions and Their Graphs

Functions are mathematical relationships where each input has exactly one output. Graphs visually represent these relationships, helping to identify key features such as domain, range, intercepts, and behavior.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) the function can produce.

• Piecewise-Defined Functions: Functions defined by different expressions over different intervals of the domain.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Example: For the piecewise function $f(x) = \begin{cases} -4 & \text{if } x \leq -2 \\ x^2 & \text{if } -2 < x \leq 1 \\ 3 & \text{if } x > 1 \end{cases}$, evaluate $f(-2)$, $f(0)$, and $f(5)$.

Linear Functions and Models

Equations of Lines

Linear functions have the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope measures the steepness of the line, and the y-intercept is where the line crosses the y-axis.

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals.

Example: Find the equation of the line passing through $(-6,4)$ with slope $\frac{1}{2}$.

Quadratic and Higher-Degree Polynomial Functions

Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$. Their graphs are parabolas, which open upwards if $a > 0$ and downwards if $a < 0$.

• Vertex: The highest or lowest point on the graph, found at $x = -\frac{b}{2a}$.

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$.

• Y-intercept: The point where $x = 0$.

• Factoring: Expressing the quadratic as a product of two binomials.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve $2x^2 + 5x + 3 = 0$ using the quadratic formula.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = ab^x$, where $a$ is the initial value and $b$ is the base (growth or decay factor).

• Compound Interest Formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• Continuous Compounding: $A = Pe^{rt}$

Example: If $P = 4000$, $r = 0.0275$, $n = 2$, $t = 5$, find $A$.

Logarithmic Functions

Logarithms are the inverses of exponential functions. The logarithm $\log_b a$ answers the question: "To what power must $b$ be raised to get $a$?"

• Definition: $y = \log_b x$ if and only if $b^y = x$

• Properties:

  • $\log_b (xy) = \log_b x + \log_b y$

  • $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

  • $\log_b (x^k) = k \log_b x$

Example: Write $\log_2 2x - \log_2 y + 3\log_2 x$ as a single logarithm.

Solving Equations and Inequalities

Linear, Quadratic, and Rational Equations

Solving equations involves finding all values of the variable that make the equation true.

• Linear Equations: $ax + b = 0$

• Quadratic Equations: $ax^2 + bx + c = 0$

• Rational Equations: Equations involving fractions with polynomials in the numerator and denominator.

Example: Solve $-8[2x + 6] - 3[2x - 7] + 4 = -2$.

Systems of Equations and Applications

Systems of Linear Equations

Systems of equations involve finding values that satisfy all equations simultaneously. Methods include substitution, elimination, and using matrices.

• Application: Compound interest, population growth, and demand functions can be modeled using algebraic equations.

Example: If the demand function is $p = \frac{80}{q + 1}$, find $q$ when $p = 56.71$.

Parent Functions and Transformations

Identifying Parent Functions

Parent functions are the simplest forms of common functions, such as linear, quadratic, cubic, absolute value, and square root functions. Transformations include shifts, reflections, stretches, and compressions.

• Absolute Value Function: $f(x) = |x|$

• Quadratic Function: $f(x) = x^2$

• Cubic Function: $f(x) = x^3$

• Square Root Function: $f(x) = \sqrt{x}$

Example: Identify the parent function and domain for $f(x) = \sqrt{x}$.

Summary Table: Key Parent Functions

Function	Equation	Domain	Range
Linear	$f(x) = x$	$(-\infty, \infty)$	$(-\infty, \infty)$
Quadratic	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Absolute Value	$f(x) = |x|$	$(-\infty, \infty)$	$[0, \infty)$
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$
Cubic	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$

Additional info:

• Some questions involve interpreting the slope of a linear model in context (e.g., economic data).

• Students are expected to show all steps for open-ended questions and understand the meaning of logarithms and exponential growth.

• Practice includes evaluating functions, solving equations, graphing, and applying algebraic models to real-world scenarios.