Graphs, Functions, and Systems of Equations: Study Guide

Graphs and Functions

Relations, Domain, and Range

A relation is a set of ordered pairs, typically written as (x, y). The domain is the set of all possible x-values, and the range is the set of all possible y-values.

• Domain: All x-values in the relation.

• Range: All y-values in the relation.

• Example: For R = {(1,2), (3,4), (5,6)}, domain = {1,3,5}, range = {2,4,6}.

Functions and the Vertical Line Test

A function is a relation in which each input (x-value) corresponds to exactly one output (y-value). The vertical line test is used to determine if a graph represents a function.

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

• Example: The graph of y = x2 passes the vertical line test; y2 = x does not.

Equations of Lines

Slope and Slope Formula

The slope of a line measures its steepness and is calculated using two points on the line.

• Slope Formula:

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

• Example: Given points (2,3) and (4,7): $m = \frac{7-3}{4-2} = \frac{4}{2} = 2$

Graphing Lines: Slope-Intercept and General Form

Lines can be written in different forms. The slope-intercept form is useful for graphing.

• Slope-Intercept Form: $y = mx + b$

• General (Standard) Form: $ax + by = c$

• To convert: Solve for y in terms of x.

Example: Convert $2x + 3y = 6$ to slope-intercept form:

$3y = -2x + 6$ $y = -\frac{2}{3}x + 2$

Special Lines: Horizontal, Vertical, and Through the Origin

• Horizontal Line: $y = c$ (slope = 0)

• Vertical Line: $x = k$ (undefined slope)

• Line Through Origin: $y = mx$ (intercept b = 0)

Parallel and Perpendicular Lines

• Parallel Lines: Have equal slopes ($m_1 = m_2$)

• Perpendicular Lines: Product of slopes is -1 ($m_1 \cdot m_2 = -1$)

• Example: If one line has slope 2, a perpendicular line has slope $-\frac{1}{2}$

Inequalities and Graphing

Graphing Linear Inequalities

Linear inequalities are graphed by first drawing the boundary line, then shading the appropriate region.

• Boundary Line: Use the equation (e.g., $y = mx + b$)

• Solid Line: For $\leq$ or $\geq$

• Dotted Line: For $<$ or $>$

• Example: Graph $y > 2x + 1$: Draw dotted line $y = 2x + 1$, shade above.

Systems of Linear Equations

Solving Systems: Addition (Elimination) and Substitution Methods

A system of linear equations consists of two or more equations. Solutions are found where the equations intersect.

• Addition (Elimination) Method: Add or subtract equations to eliminate a variable.

• Substitution Method: Solve one equation for a variable, substitute into the other.

• Check Solution: Substitute values into both original equations to verify.

• Example:

Given: $x + y = 5$ $x - y = 1$ Add equations: $(x + y) + (x - y) = 5 + 1$ $2x = 6$ $x = 3$ Substitute $x$ into first equation: $3 + y = 5$ $y = 2$

Summary Table: Line Types and Properties

Line Equation	Type	Slope	Intercept
$y = c$	Horizontal	0	y-intercept: c
$x = k$	Vertical	Undefined	x-intercept: k
$y = mx$	Through Origin	m	Origin (0,0)

Additional info: This guide covers key concepts from intermediate algebra chapters on graphs, functions, equations of lines, inequalities, and systems of equations. Practice with representative questions from homework to reinforce understanding.