Math 142 Final Exam Review: College Algebra Study Guide

Graphs, Functions, and Models

Distance and Midpoint Between Points

Understanding the geometric relationship between points is fundamental in algebra and analytic geometry.

• 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 $(4,6)$. Distance: $d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ Midpoint: $M = \left( \frac{1+4}{2}, \frac{2+6}{2} \right) = (2.5, 4)$

Slope and Equations of Lines

The slope of a line measures its steepness and direction.

• Slope Formula: For points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

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

• Slope-Intercept Form: $y = mx + b$

• Example: Find the equation of the line passing through $(2,3)$ and $(5,11)$. Slope: $m = \frac{11-3}{5-2} = \frac{8}{3}$ Equation: $y - 3 = \frac{8}{3}(x - 2)$

Functions and Their Properties

Linear, Quadratic, and Polynomial Functions

Functions describe relationships between variables. Linear and quadratic functions are foundational in algebra.

• Linear Function: $f(x) = mx + b$

• Quadratic Function: $f(x) = ax^2 + bx + c$

• Polynomial Function: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0$

• Applications: Used to model real-world phenomena such as projectile motion (quadratic) or population growth (polynomial).

Composition of Functions

The composition of functions involves applying one function to the result of another.

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

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

Inverse Functions

An inverse function reverses the effect of the original function.

• Definition: $f^{-1}(f(x)) = x$

• Finding the Inverse: Solve $y = f(x)$ for $x$ in terms of $y$, then interchange $x$ and $y$.

• Example: $f(x) = 3x + 2$ $y = 3x + 2 \implies x = \frac{y - 2}{3}$ So, $f^{-1}(x) = \frac{x - 2}{3}$

Quadratic Equations and Inequalities

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

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

• Example: Solve $x^2 - 5x + 6 = 0$ Factoring: $(x-2)(x-3) = 0 \implies x = 2, 3$

Solving Inequalities

Inequalities can be solved using algebraic manipulation and by analyzing sign charts.

• Example: Solve $x^2 - 4 < 0$ $x^2 < 4 \implies -2 < x < 2$

Polynomial and Rational Functions

Domain and Range

The domain is the set of all possible input values; the range is the set of possible output values.

• Finding Domain: Exclude values that make the denominator zero or result in even roots of negative numbers.

• Example: $f(x) = \frac{1}{x-3}$ has domain $x \neq 3$

Asymptotes

Asymptotes are lines that a graph approaches but never touches.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is nonzero.

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

• Example: $f(x) = \frac{2x}{x-1}$ has a vertical asymptote at $x=1$ and a horizontal asymptote at $y=2$

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions model growth and decay processes.

• General Form: $f(x) = a b^x$

• Example: $f(x) = 2^x$ models doubling growth.

Logarithmic Functions

Logarithms are the inverses of exponential functions.

• Definition: $y = \log_b x \iff b^y = x$

• Properties:

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

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

  • $\log_b (x^r) = r \log_b x$

• Example: $\log_2 8 = 3$ because $2^3 = 8$

Solving Exponential and Logarithmic Equations

• Example: Solve $2^x = 16$ $x = 4$ because $2^4 = 16$

• Example: Solve $\log_3 x = 2$ $x = 9$ because $3^2 = 9$

Systems of Equations and Matrices

Solving Systems of Equations

Systems of equations can be solved using substitution, elimination, or matrix methods.

• Two Variables: Solve $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ Add: $2x = 6 \implies x = 3$, then $y = 2$

• Three Variables: Use elimination or matrices to solve systems with three unknowns.

Applications of Systems

• Applied Problems: Systems can model mixtures, investments, and other real-world scenarios.

Additional Topics

Growth and Decay Applications

Exponential models are used for population growth, radioactive decay, and half-life problems.

• Half-Life Formula: $N(t) = N_0 \left( \frac{1}{2} \right)^{t/h}$ where $h$ is the half-life.

• Example: If $N_0 = 100$ grams and the half-life is 5 years, after 10 years: $N(10) = 100 \left( \frac{1}{2} \right)^{10/5} = 100 \left( \frac{1}{2} \right)^2 = 25$ grams

Summary Table: Key Concepts and Methods

Topic	Key Formula/Method	Example
Distance	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Between $(1,2)$ and $(4,6)$: $5$
Midpoint	$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$	$(2.5, 4)$
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	$x^2 - 5x + 6 = 0 \implies x = 2, 3$
Exponential Growth	$f(x) = ab^x$	$f(x) = 2^x$
Logarithm	$y = \log_b x \iff b^y = x$	$\log_2 8 = 3$
System of Equations	Substitution/Elimination	$x + y = 5, x - y = 1$

Additional info: This guide covers all major topics listed in the exam review outline, providing definitions, formulas, and examples for each. For more advanced applications, consult your textbook or instructor.