College Algebra Final Exam Study Guide

Equations & Inequalities

Solving Equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations involves finding the value(s) of the variable(s) that make the equation true.

• Linear Equations: Equations of the form $ax + b = 0$.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$.

• Logarithmic Equations: Equations involving logarithms, such as $\log_a x = b$.

Example: Solve $2x^2 - 4 = 0$.

Solution: $2x^2 = 4 \implies x^2 = 2 \implies x = \pm \sqrt{2}$

Functions

Domain of Functions

The domain of a function is the set of all possible input values (usually $x$) for which the function is defined.

• For rational functions $f(x) = \frac{1}{x-7}$, the domain excludes values that make the denominator zero: $x \neq 7$.

• For square root functions $f(x) = \sqrt{x+1}$, the domain is $x + 1 \geq 0 \implies x \geq -1$.

Example: Find the domain of $f(x) = \frac{1}{x-7}$.

Domain: $x \in \mathbb{R}, x \neq 7$

Function Notation and Evaluation

Functions can be expressed in various forms, such as $f(x) = x^2 + 1$. Evaluating a function means substituting a value for $x$.

• Example: If $f(x) = x^2 + 1$, then $f(2) = 2^2 + 1 = 5$.

Polynomial & Rational Functions

Simplifying Expressions

Simplifying algebraic expressions involves combining like terms, factoring, and reducing fractions.

• Example: Simplify $\sqrt{9} - \sqrt{25}$.

• Solution: $\sqrt{9} = 3$, $\sqrt{25} = 5$, so $3 - 5 = -2$.

• Rational Expressions: Simplify $\frac{x^2 - 6x + 7}{x^2 - x - 2}$ by factoring numerator and denominator.

Exponential & Logarithmic Functions

Properties of Logarithms

Logarithms are the inverses of exponential functions. Key properties include:

• $\log_a (xy) = \log_a x + \log_a y$

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

• $\log_a (x^k) = k \log_a x$

Example: Simplify $\log_2 8$.

Solution: $\log_2 8 = 3$ because $2^3 = 8$.

Solving Exponential and Logarithmic Equations

• To solve $\log_x y = k$, rewrite as $x^k = y$.

• To solve $a^x = b$, take logarithms of both sides: $x = \frac{\log b}{\log a}$.

Graphs of Equations & Functions

Graphing Exponential Functions

Exponential functions have the form $f(x) = a^x$. Their graphs show rapid growth or decay depending on the base $a$.

• If $a > 1$, the function increases as $x$ increases.

• If $0 < a < 1$, the function decreases as $x$ increases.

Example: Graph $f(x) = e^{x^2}$.

The graph is always above the $x$-axis and increases rapidly for large $|x|$.

Systems of Equations

Word Problems and Applications

Systems of equations can be used to solve real-world problems, such as rates and investments.

• Example: If David labels 144 tins in 6 days, how many tins in 15 days?

• Set up a proportion: $\frac{144}{6} = \frac{x}{15}$, solve for $x$.

Exponential Growth & Compound Interest

Compound Interest Formula

Compound interest is calculated using the formula:

$A(t) = P(1 + r/n)^{nt}$

• $A(t)$: Amount after time $t$

• $P$: Principal (initial amount)

• $r$: Annual interest rate

• $n$: Number of compounding periods per year

• $t$: Number of years

Example: $A(t) = 10000(1.01)^{12t}$ for monthly compounding at 12% annual rate.

Quadratic Functions

Vertex, Axis of Symmetry, and Minimum/Maximum

A quadratic function $f(x) = ax^2 + bx + c$ has a vertex, axis of symmetry, and minimum or maximum value.

• Vertex: $x = -\frac{b}{2a}$

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

• Minimum/Maximum Value: Substitute $x$ into $f(x)$

Example: For $f(x) = x^2 - 1x + 6$, vertex at $x = \frac{1}{2}$.

Summary Table: Properties of Logarithms

Property	Formula	Example
Product Rule	$\log_a (xy) = \log_a x + \log_a y$	$\log_2 (8 \cdot 4) = \log_2 8 + \log_2 4 = 3 + 2 = 5$
Quotient Rule	$\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$	$\log_2 \left(\frac{8}{4}\right) = \log_2 8 - \log_2 4 = 3 - 2 = 1$
Power Rule	$\log_a (x^k) = k \log_a x$	$\log_2 (8^2) = 2 \log_2 8 = 2 \times 3 = 6$

Additional info:

• Some questions involve graph selection and matching, which tests understanding of function behavior.

• Word problems and applications are included to assess real-world algebraic modeling.