Math 142 Final Exam Review: College Algebra Study Guide

Distance and Midpoint Between Points

• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: The midpoint between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Example: Find the distance and midpoint between $(1,2)$ and $(4,6)$.

Slope and Intercept of a Line

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$

• Example: Find the equation of a line passing through $(2,3)$ with slope 4.

Linear Equations, Composition of Functions

• Linear Equation: $y = mx + b$ Composition of Functions: $(f \circ g)(x) = f(g(x))$

• Example: If $f(x) = 2x + 1$ and $g(x) = x^2$, find $(f \circ g)(x)$.

Quadratic Functions & Equations; Inequalities

• Standard Form: $ax^2 + bx + c = 0$ Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Inequalities

Solving inequalities involves finding the set of values that satisfy a given condition.

• Linear Inequality: $ax + b > c$ Quadratic Inequality: $ax^2 + bx + c < 0$

• Example: Solve $x^2 - 4 < 0$.

Polynomial Functions & Rational Functions: Domain & Range

Understanding the domain and range is essential for analyzing functions.

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

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

• Rational Function: $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$.

• Finding Asymptotes: Vertical asymptotes occur where $Q(x) = 0$;

• horizontal asymptotes depend on the degrees of $P(x)$ and $Q(x)$.

• Example: Find the domain and asymptotes of $f(x) = \frac{2x}{x-3}$.

Exponential Functions & Logarithmic Functions Inverse Functions; Exponential & Log Equations

• Exponential Function: $f(x) = a^x$ Logarithmic Function: $f(x) = \log_a x$

• Inverse Function: If $f(x)$ and $g(x)$ are inverses, $f(g(x)) = x$ and $g(f(x)) = x$.

• Example: Find the inverse of $f(x) = 2^x$.

Properties and Applications of Logarithms

Product Rule: $\log_a (xy) = \log_a x + \log_a y$ Quotient Rule: $\log_a \left( \frac{x}{y} \right) = \log_a x - \log_a y$

Power Rule: $\log_a (x^r) = r \log_a x$ Example: Expand $\log_2 (8x)$.

Solving Systems of Equations

• System of Two Equations: $\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$ System of Three Equations: $\begin{cases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{cases}$

• Methods: Substitution, elimination, and matrix methods (such as Gaussian elimination).

• Example: Solve $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$

  Exponential Growth and Decay Applications to Half-Life and Growth Problems

• Exponential Growth: $N(t) = N_0 e^{kt}$ Exponential Decay: $N(t) = N_0 e^{-kt}$

• Half-Life Formula: $t_{1/2} = \frac{\ln 2}{k}$

Topic	Key Formula	Example
Distance Between Points	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	$(1,2)$ and $(4,6)$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 - 5x + 6 = 0$
Exponential Growth	$N(t) = N_0 e^{kt}$	Population growth
Logarithm Properties	$\log_a (xy) = \log_a x + \log_a y$	$\log_2 (8x)$
System of Equations	$\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$	Find $x$ and $y$