College Algebra Study Guide: Functions, Equations, Graphs, and Applications

Decimals, Fractions, and Percents

Converting Decimals and Fractions to Percent Notation

Understanding how to convert between decimals, fractions, and percents is fundamental in algebra and everyday applications.

• Decimal to Percent: Multiply the decimal by 100 and add the percent symbol (%).

• Fraction to Percent: Divide the numerator by the denominator, multiply by 100, and add the percent symbol (%).

• Example: $0.34 = 34\%$, $\frac{4}{5} = 0.8 = 80\%$

Operations with Fractions

Adding and Subtracting Fractions

To combine fractions, they must have a common denominator. This allows for direct addition or subtraction of numerators.

• Find a common denominator for all fractions involved.

• Rewrite each fraction with the common denominator.

• Add or subtract the numerators and keep the denominator the same.

• Example: $\frac{3}{4} + \frac{5}{8} = \frac{6}{8} + \frac{5}{8} = \frac{11}{8}$

Solving Linear Equations

Solving for the Variable

Linear equations can be solved by isolating the variable using algebraic operations.

• Combine like terms on each side of the equation.

• Move all terms containing the variable to one side and constants to the other.

• Example: Solve $2x - 2 = 3x + 3$ $2x - 2 = 3x + 3$ $2x - 3x = 3 + 2$ $-x = 5$ $x = -5$

Equations of Circles

Standard Form and Application

The equation of a circle with center $(h, k)$ and radius $r$ is:

• $(x - h)^2 + (y - k)^2 = r^2$

• Example: Center $(-2, 3)$, point $(1, 7)$ Find radius: $r = \sqrt{(1 - (-2))^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ Equation: $(x + 2)^2 + (y - 3)^2 = 25$

Functions and Relations

Identifying Functions

A relation is a function if each input (x-value) corresponds to exactly one output (y-value).

• Vertical Line Test: If any vertical line crosses the graph more than once, it is not a function.

• Set of Ordered Pairs: No x-value repeats with different y-values.

• Example: The set $\{(1,2), (2,8), (3,5)\}$ is a function; $\{(1,2), (1,3)\}$ is not.

Intervals of Increase and Decrease

Analyzing Graphs

Functions can be increasing, decreasing, or constant over intervals of their domain.

• Increasing: As x increases, y increases.

• Decreasing: As x increases, y decreases.

• Constant: As x increases, y remains the same.

• Example: From a graph, identify intervals where the function rises or falls.

Slope and Rate of Change

Calculating Slope

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x.

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

• Rate of Change: Slope represents how one variable changes in relation to another.

• Example: If a graph of D has points $(1,2)$ and $(3,6)$, $m = \frac{6-2}{3-1} = 2$

Graphing Linear Equations

Intercepts and Graphing

To graph a line, find its x-intercept (where y=0) and y-intercept (where x=0).

• Equation: $2x + 3y = 6$

• x-intercept: Set $y=0$, solve for $x$.

• y-intercept: Set $x=0$, solve for $y$.

• Plot both intercepts and draw the line through them.

Domain and Range from Graphs

Using Graphs to Find Domain and Range

The domain is the set of all possible x-values; the range is the set of all possible y-values.

• Domain: Read the x-values covered by the graph.

• Range: Read the y-values covered by the graph.

• Example: If the graph extends from $x = -2$ to $x = 4$, domain is $[-2, 4]$.

Piecewise Functions

Graphing and Evaluating Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Example: $f(x) = \begin{cases} x+3 & -3 \leq x < 1 \\ 2x-1 & 1 \leq x \leq 3 \end{cases}$

• Graph each piece over its specified interval.

• Evaluate $f(x)$ for given values by determining which interval $x$ falls into.

Solving Equations and Inequalities Graphically

Graphical Solutions

Equations and inequalities can be solved by finding points of intersection or regions on a graph.

• To solve $f(x) = g(x)$: Find x-values where the graphs of $f$ and $g$ intersect.

• To solve $f(x) < g(x)$: Find intervals where the graph of $f$ is below $g$.

• To solve $f(x) = n$: Find x-values where the graph of $f$ crosses the horizontal line $y = n$.

Solving Linear Inequalities

Interval Notation

Linear inequalities can be solved algebraically and their solutions expressed in interval notation.

• Example: Solve $-5 \leq -2(x+4) - 5 < 5$

• Isolate $x$ and write the solution as an interval, e.g., $[a, b)$

Applications: Hooke's Law

Spring Constant and Stretch

Hooke's Law relates the force applied to a spring to its extension.

• Formula: $F = kx$

• Given: A 20-lb weight stretches a spring 10 inches. Find $k$:

• $k = \frac{F}{x} = \frac{20}{10} = 2$ lb/inch

• How far will a 25-lb weight stretch the spring? $x = \frac{F}{k} = \frac{25}{2} = 12.5$ inches

Reference Formulas and Concepts

Key Algebraic Formulas

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$

• Equation of a Line (Slope-Intercept): $y = mx + b$

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

• Direct Variation: $y = kx$

• Slope Relations: Parallel lines: $m_1 = m_2$ Perpendicular lines: $m_2 = -\frac{1}{m_1}$ or $m_1 m_2 = -1$

Table: Slope Relations

Type of Lines	Slope Relationship
Parallel	$m_1 = m_2$
Perpendicular	$m_1 m_2 = -1$

Table: Key Formulas

Concept	Formula
Distance between points	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint	$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
Circle equation	$(x - h)^2 + (y - k)^2 = r^2$
Slope	$m = \frac{y_2 - y_1}{x_2 - x_1}$
Line (slope-intercept)	$y = mx + b$
Point-slope form	$y - y_1 = m(x - x_1)$
Direct variation	$y = kx$
Hooke's Law	$F = kx$

Additional info:

• Some graphs and tables referenced in the questions are not reproduced here, but the methods for analysis are described.

• Piecewise function evaluation and graphing are included as a bonus topic, as indicated in the test.