College Algebra Study Guide: Functions, Equations, and Graphs

Basic Algebraic Operations

Order of Operations and Arithmetic

Algebraic expressions often require careful application of the order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) to simplify correctly.

• Order of Operations: Always evaluate expressions inside parentheses first, then exponents, followed by multiplication/division, and finally addition/subtraction.

• Example: Simplify $-5 \cdot 2^3 + 2 - (-2)^2$.

• Multiplying Fractions: Multiply numerators and denominators directly.

• Dividing Fractions: Multiply by the reciprocal of the divisor.

• Zero in Fractions: $\frac{0}{a} = 0$ for $a \neq 0$; $\frac{a}{0}$ is undefined.

Algebraic Expressions and Exponents

Exponent Rules

Exponents follow specific rules for multiplication, division, and powers.

• Product of Powers: $a^m \cdot a^n = a^{m+n}$

• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$, $a \neq 0$

• Power of a Power: $(a^m)^n = a^{mn}$

• Example: $(4x^2)(2x^3) = 8x^{5}$

Linear Equations and Graphs

Slope and Intercept

The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Equation of a Line: $y - y_1 = m(x - x_1)$

• Parallel Lines: Have the same slope.

• Perpendicular Lines: Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$

• Example: Find the slope and equation of the line through $(2,4)$ and $(-8,7)$.

Graphing Linear Equations

To graph a line, identify the slope and y-intercept, plot the intercept, and use the slope to find another point.

• Example: Graph $y = -\frac{1}{2}x - 2$ by plotting $(0, -2)$ and using the slope $-\frac{1}{2}$.

Solving Linear Equations and Inequalities

Solving Equations

To solve linear equations, isolate the variable using inverse operations.

• Example: $3(2x + 1) - 2(x + 5) = 4(x - 3)$

• Distribute, combine like terms, and solve for $x$.

Solving Inequalities

Similar to equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

• Example: $1 - 2(3x - 1) \leq 2x + 5$

Functions and Their Domains

Definition of a Function

A function is a relation that assigns exactly one output for each input.

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

• Example: For $f(x) = \sqrt{3x + 9}$, the domain is all $x$ such that $3x + 9 \geq 0$.

• Example: For $h(x) = \frac{2}{\sqrt{3x + 11}}$, the denominator must be positive and nonzero.

Piecewise Functions

Evaluating Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} 3x - 2 & x \leq 0 \\ x^2 & 0 < x < 3 \\ 2x + 3 & x \geq 3 \end{cases}$

• To evaluate $f(a)$, determine which interval $a$ falls into and use the corresponding formula.

Function Operations

Sum, Difference, Product, and Quotient of Functions

Functions can be combined using addition, subtraction, multiplication, and division.

• Sum: $(f + g)(x) = f(x) + g(x)$

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

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

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

• Example: If $f(x) = x^2 + 2$ and $g(x) = 3x - 2$, then $(f + g)(2) = f(2) + g(2)$.

Profit, Revenue, and Cost Functions

Business Applications

In applied algebra, functions can represent revenue, cost, and profit.

• Revenue Function: $R(x)$, total income from selling $x$ units.

• Cost Function: $C(x)$, total cost to produce $x$ units.

• Profit Function: $P(x) = R(x) - C(x)$

• Example: If $R(x) = 20x - 2x^2$ and $C(x) = 5x + 2$, then $P(x) = (20x - 2x^2) - (5x + 2)$.

Intervals of Increase and Decrease; Relative Extrema

Analyzing Graphs

Functions can be increasing, decreasing, or constant over intervals. Relative maxima and minima are the highest or lowest points in a local region.

• Increasing Interval: Where $f(x)$ rises as $x$ increases.

• Decreasing Interval: Where $f(x)$ falls as $x$ increases.

• Constant Interval: Where $f(x)$ remains unchanged as $x$ increases.

• Relative Maximum: A point higher than nearby points.

• Relative Minimum: A point lower than nearby points.

• Example: Use the graph to identify intervals and extrema.

Summary Table: Key Formulas and Concepts

Concept	Formula	Description
Slope	$m = \frac{y_2 - y_1}{x_2 - x_1}$	Rate of change between two points
Line Equation	$y - y_1 = m(x - x_1)$	Point-slope form of a line
Profit	$P(x) = R(x) - C(x)$	Profit equals revenue minus cost
Function Domain	Set of $x$ for which $f(x)$ is defined	Restrictions depend on the function's formula
Function Operations	$(f \pm g)(x), (fg)(x), \left(\frac{f}{g}\right)(x)$	Combine functions by addition, subtraction, multiplication, division

Additional info:

• Questions cover core College Algebra topics: arithmetic, exponents, linear equations, graphing, functions, domains, and applications.

• Piecewise functions and function operations are included, which are standard in College Algebra.

• Business applications (profit, revenue, cost) are typical in applied algebra sections.