We have seen that we can identify characteristics (slope and the y-intercept) of linear functions seen in tables and graphs, and that values can be substituted into equations to create a table needed to plot the function on a coordinate plane. Now we will explore their equations more closely.

Drag the m and b sliders and observe how they change the graph.

- How do different values of m affect the graph?
- How do different values of b affect the graph?

Linear functions can be written in a form that uses the slope as a coefficient of x since it is a rate of change and the y-value of the y-intercept as a constant. The **slope-intercept form** of an equation is

\displaystyle y=mx+b

\bm{m}

is the slope

\bm{b}

is the y-intercept

The value of m affects the steepness of the line, or the slope.

The value of b affects the y-intercept, or where the line crosses the y-axis.

The slope, m, and the y-intercept, b, can be used not only to write equations but also to graph a line. We use the following steps to graph in slope-intercept form:

- Plot b from the equation as the y-intercept.
- Identify the vertical change and horizontal change from m in the equation. If the slope is written as an integer, we can write m =\dfrac{m}{1}.
- Starting at the y-intercept, use the slope to count the vertical and horizontal change and plot a point where you end up.
- Draw a straight line through the points, extending past the points to fill the coordinate plane.

Note that we are able to reverse both directions indicated by the slope in order to graph points to the left of the y-intercept.

Consider the equation y=-4x+5.

a

State the slope and y-intercept of the equation.

Worked Solution

b

Complete the table of values for the given equation:

x | -1 | 0 | 1 | 2 |
---|---|---|---|---|

y |

Worked Solution

Consider the following graph of a line:

a

What is the slope of the line shown in the graph?

Worked Solution

b

What is the y-value of the y-intercept of the line shown in the graph?

Worked Solution

c

Write the equation of the line in slope-intercept form.

Worked Solution

Graph the line y=3x+2 using its slope and y-intercept.

Worked Solution

Given the table of a linear function:

x | -4 | -2 | 0 | 2 | 4 |
---|---|---|---|---|---|

y | -2 | 1 | 4 | 7 | 10 |

a

Find the slope.

Worked Solution

b

Identify the y-intercept of the table and write the equation of the line represented by the table in slope-intercept form.

Worked Solution

The graph of y=-\dfrac{1}{3}x is shown.

a

The graph will be shifted 4 units down. Write the new equation of the line after the translation.

Worked Solution

b

Graph the equation from part (a).

Worked Solution

Idea summary

A linear equation is said to be in** slope-intercept form** when it is expressed as

\displaystyle y=mx+b

\bm{m}

is the slope.

\bm{b}

is the y-intercept.

The y-intercept will shift a line above the x-axis if b is positive and below the x-axis if b is negative.

When given an equation in slope-intercept form, we can graph the line by plotting the y-intercept as the first point. Then we can use the slope to find additional points before drawing our line.