Linear Functions, Equations, and Inequalities

Linear Functions

A linear function is a function of the form , where a and b are real numbers. The graph of a linear function is a straight line. The solution to a linear equation is an ordered pair that satisfies the equation.

• Standard form:

• Graph: A straight line in the Cartesian plane

• Intercepts: Points where the graph crosses the axes

Locating x- and y-Intercepts

To find the intercepts of the graph of (assuming ):

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

Zero of a Function and x-Intercepts

A zero of a function is any number in the domain of for which . The point is an x-intercept of the graph of .

Graphing a Linear Function

To graph :

• Find the y-intercept:

• Find the x-intercept (zero): Set

• Plot both intercepts and draw the line through them.

A graphing calculator can also be used to verify the intercepts and the zero of the function.

Modeling with Linear Functions

Example: A 100-gallon tank is drained at 5 gallons per minute.

• Function: models the gallons left after minutes.

• y-intercept: (initial amount)

• x-intercept: (time to empty the tank)

Constant Functions

A constant function is defined by , where is a real number. Its graph is a horizontal line at . For , it has no x-intercept.

• Example: is a horizontal line through .

Comprehensive Graphs of Lines

To obtain a comprehensive graph of a line, adjust the viewing window so that both intercepts are visible. For :

• A window of by may not show both intercepts.

• Expanding to by reveals both intercepts.

Slope of a Line

The slope of a line passing through points and is given by:

Using the Slope Formula

Example: Find the slope of the line through and .

Using Slope and a Point to Graph a Line

Given a point and a slope, plot the point and use the slope (rise over run) to find another point. Draw the line through both points.

Geometric Orientation Based on Slope

• If , the line rises from left to right.

• If , the line falls from left to right.

• If , the line is horizontal.

• A vertical line has an undefined slope and equation .

Interpreting Slope as Average Rate of Change

The slope of a line segment connecting two points and represents the average rate of change of with respect to .

• Example: Online sales increased from $205 million in 2011.

• Average rate of change: million per year.

Slope-Intercept Form

The slope-intercept form of a line is , where is the slope and is the y-intercept.

Matching Graphs with Equations

Given several lines, match each to its equation by analyzing the sign of the slope and the y-intercept:

• If , the line rises; if , it falls.

• If , the y-intercept is above the x-axis; if , below.

• : B (rises, y-intercept positive)

• : D (falls, y-intercept positive)

• : A (rises, y-intercept negative)

• : C (falls, y-intercept negative)

Interpreting Slope–Intercept Form in Applications

Example: In 2006, 60 billion text messages were sent in the U.S. The number increased by 407 billion per year for the next 6 years.

• Let correspond to 2006, so

• Rate of change

• Model:

• For 2012 (): billion messages

Finding an Equation from a Graph

Given a graph, determine:

• Slope: Calculate rise over run between two points.

• y-intercept: Where the line crosses the y-axis.

• x-intercept: Where the line crosses the x-axis.

• Equation: Use the slope and y-intercept in .

• Zero: The x-coordinate of the x-intercept.

Example: Slope is , y-intercept is , x-intercept is . Equation: .