College Algebra Study Guide: Functions, Graphs, and Linear Equations (Sections 1.3–2.4)

Introduction to Functions and Graphs

Relations and Functions

A relation is a set of ordered pairs. A function is a special type of relation where each input (x-value) has exactly one output (y-value).

• Determining if a Relation is a Function: Each x-value must correspond to only one y-value. If any x-value is paired with more than one y-value, the relation is not a function.

• Vertical Line Test: If a vertical line crosses a graph more than once, the graph does not represent a function.

• Example: The set {(1,2), (2,3), (3,4)} is a function. The set {(1,2), (1,3), (2,4)} is not a function.

Evaluating Functions

To evaluate a function, substitute the given value for x into the function's formula.

• Symbolic Evaluation: For $f(x) = 2x + 3$, find $f(4)$: $f(4) = 2 \times 4 + 3 = 11$.

• Graphical Evaluation: Locate the x-value on the graph and read the corresponding y-value.

Domain and Range

The domain of a function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

• Finding the Domain: Identify all x-values for which the function is defined (no division by zero, no square roots of negative numbers, etc.).

• Interval Notation: Example: $(-\infty, 2]$ means all real numbers less than or equal to 2.

• Set Builder Notation: Example: $\{x \mid x \leq 2\}$ means all x such that x is less than or equal to 2.

• Example: For $f(x) = \sqrt{x}$, domain is $[0, \infty)$.

Increasing and Decreasing Intervals

A function is increasing on intervals where its graph rises as x increases, and decreasing where it falls.

• Finding Intervals: Examine the graph or use calculus (if known) to determine where the function rises or falls.

• Example: For $f(x) = x^2$, the function decreases on $(-\infty, 0)$ and increases on $(0, \infty)$.

Intercepts and Zeros

• x-intercept: The point(s) where the graph crosses the x-axis ($y=0$).

• y-intercept: The point where the graph crosses the y-axis ($x=0$).

• Zeros of a Function: The x-values for which $f(x) = 0$.

• Example: For $f(x) = 2x - 4$, the zero is at $x = 2$.

Linear Functions and Equations

Slope and Rate of Change

The slope of a line measures its steepness and direction. The average rate of change describes how a quantity changes, on average, between two points.

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

• Average Rate of Change: Same as the slope between two points on a function.

• Example: For points (1,2) and (3,6): $m = \frac{6-2}{3-1} = 2$.

Equations of Lines

• Slope-Intercept Form: $y = mx + b$

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

• Standard Form: $Ax + By = C$ (not explicitly listed but commonly used)

• Finding Slope and y-Intercept: In $y = mx + b$, m is the slope, b is the y-intercept.

• Example: For $y = 3x - 5$, slope is 3, y-intercept is -5.

Graphing Lines

• Plot the y-intercept (b), then use the slope (rise/run) to find another point.

• Draw a straight line through the points.

Parallel and Perpendicular Lines

• Parallel Lines: Have the same slope ($m_1 = m_2$).

• Perpendicular Lines: Slopes are negative reciprocals ($m_1 \times m_2 = -1$).

• Example: A line parallel to $y = 2x + 1$ is $y = 2x + c$. A line perpendicular is $y = -\frac{1}{2}x + c$.

Solving Linear Equations and Inequalities

• Linear Equation: An equation of the form $ax + b = 0$.

• Solving: Isolate x using algebraic operations.

• Inequality: Similar to equations, but with $<$, $>$, $\leq$, or $\geq$.

• Expressing Solutions: Use interval or set builder notation.

• Example: Solve $2x - 4 > 0$: $x > 2$; interval notation: $(2, \infty)$.

Word Problems

• Distance Problems: $\text{Distance} = \text{Rate} \times \text{Time}$

• Direct Variation: $y = kx$, where k is the constant of variation.

• Finance, Population, Cost Trends: Set up equations based on the context and solve for the unknown.

• Example: If $y$ varies directly with $x$ and $y = 10$ when $x = 2$, then $k = 5$ and $y = 5x$.

Piecewise Functions

Graphing and Evaluating Piecewise Functions

A piecewise function is defined by different expressions for different intervals of the domain.

• Graphing: Graph each piece on its specified interval.

• Evaluating: Determine which piece applies for the given x-value, then substitute x into that expression.

• Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x + 1 & x \geq 0 \end{cases}$

• For $x = -2$, $f(-2) = (-2)^2 = 4$; for $x = 3$, $f(3) = 3 + 1 = 4$.

Common Function Types and Their Graphs

Function	Equation	General Shape
Line	$f(x) = mx + b$	Straight line
Absolute Value	$f(x) = |x|$	V-shaped
Parabola	$f(x) = x^2$	U-shaped
Cubic	$f(x) = x^3$	S-shaped
Square Root	$f(x) = \sqrt{x}$	Starts at (0,0), increases slowly

Formulas to Memorize

• Slope-Intercept Form: $y = mx + b$

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

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

• Distance Formula: $\text{Distance} = \text{Rate} \times \text{Time}$

• Direct Variation: $y = kx$

Additional info: Practice problems are referenced by section and page number for further study. The above notes cover the foundational concepts and skills needed for Test #1 in a college algebra course, focusing on functions, their properties, and linear equations.