# Graphs and Functions

## 3.3 Linear Functions

**3.3.1 Graph Linear Functions **

A linear function is a function of the form

f(x) = ax + b

The graph of a linear function is a straight line. Also, for linear
functions, the domain is the set of

all real numbers R . Recall: When graphing y = f(x).

**3.3.2 Intercepts and Standard Form**

The standard form of a linear equation is

ax + by = c

where a, b, and c are real numbers and a and b are not both 0.

In this form it is frequently easier to graph the equation using the intercepts.
The x-intercept

is the point where the graph crosses the x-axis. The y-intercept is the point
where the graph

crosses the y-axis.

•To find the y-intercept, set x = 0 and solve for y .

•To find the x-intercept, set y = 0 and solve for x.

**EX 12.** 1. Graph 3x = 6y + 12 using the x- and y-intercepts.

2. Graph f(x) = 1/2x + 2 using the x- and y-intercepts.

3. Graph -2x + y = 0

**3.3.3 Vertical and Horizontal Lines**

Horizontal Lines

Any equation of the form y = b will always be a horizontal line.

**EX 13.** Graph the equation y = 4 (or written f(x) = 4)

**Vertical Lines
**Any equation of the form x = a will always be a vertical line.

**EX 14.**Graph the equation x = 4

**3.3.4 An Application**

**EX 15. **Suppose a store owner sells widgets for $30 each. If her monthly
expenses are $3,000,

answer the following:

1. Construct a function that relates the number of widgets sold to the pro
fits.

2. How many widgets must she sell to break even?

3. Graph the profit function.

## 3.4 Slope-Intercept Form of a Linear Equation

Our goal in this section will be to completely describe a line using two
numbers which reveal certain

characteristics of the line. The characteristics we will use are the y-intercept
and the slope.

**3.4.1 Understand Translations**

Consider the graph of the function y = 1/2x. What happens if we add 2 to the
right hand side? How

about if I subract 2? Let's graph the following functions on the same coordinate
system .

What are the y-intercepts? Each line is parallel to the other, but the new
lines are shifted, or

translated, up or down by two.

**3.4.2 Slope**

As was mentioned we wish to describe lines using two numerical
characteristics . One of those is the

slope.

**Definition 6.** The slope of a line is the ration of the vertical change
(or rise) to the horizontal

change (or run).

**EX 16.** We examine how to find slope:

1. Look at the graphs from the previous example , find the slope of the lines .

2. Graph the equations y = 2x and and y = 2/3x and find their slopes.

The slope of the line through the distinct points (x_{1}, y_{1}) and (x_{2}, y_{2}) is

provided that We usually use the lowercase letter m to denote the slope.

**EX 17.** Calculate the slope for the following lines:

**REMARK 3.** From the example we notice the following:

•Lines with positive slope increase as we go from left to right.

•Lines with negative slope decrease as we go from left to right.

•Any horizontal line has zero slope .

•What would the slope of a vertical line be?

**3.4.3 Slope-Intercept Form**

The slope-intercept form of a linear equation is

y = mx + b

where m is the slope of the line and (0, b) is the y-intercept of the line.

To write an equation in slope-intercept form, solve the equation for y.

**EX 18. -**

1. Consider the equation y = 2/3x + 2 and determine the slope and y-intercept.

2. Write the equation -3x + 4y = 8 in slope-intercept form and determine the
slope and y-

intercept.

**3.4.4 Graphing Linear Equations Using Slope and y-Intercept**

**EX 19.** Graph the following equations using the slope and y-intercept:

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