As explored in the previous lesson, transformations on f\left(x\right)=x allow us to shift, dilate, and reflect linear functions. We can write the equations of lines in **slope-intercept form**:

\displaystyle y=mx+b

\bm{m}

slope

\bm{b}

y-intercept

The main advantage of slope-intercept form is that we can easily identify two key features: the **slope** and the **y-intercept** directly from the equation.

Consider the following representations

Table

Graph

Description

Equation

- What do the table, description, graph, and equation representations have in common?
- What is different about each representation?

Given information about a linear function, we can write the equation in slope-intercept form.

Next, identify or solve for the y-intercept, which represents the b in y=mx+b.

In the graph, the y-intercept is at (0, 15), so b=15. From the context, we can see the initial value of the leaked water is 15 cups.

Lastly, rewrite y=mx+b, substituting the solved values of m and b. Since we have found m = 4 and b=15, we can write the equation: y = 4x+15

Slope-intercept form is especially helpful when we want to graph a linear function. The graph of a line represents the set of points that satisfies the equation of a line.

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

d

Find the domain and range of the line.

Worked Solution

Find the equation in slope-intercept form for:

a

A line with a slope of \dfrac{1}{2} and a y-intercept of -3.

Worked Solution

b

A line with a slope of -3 and passes through the point (2,3).

Worked Solution

c

A line that passes through the points (-4, 5) and (8,8).

Worked Solution

Write each of the following equations in slope-intercept form.

a

-4y=12-8x

Worked Solution

b

7x-14y=-28

Worked Solution

c

y+2=-\dfrac{2}{3}(x-6)

Worked Solution

A bathtub has a clogged drain, so it needs to be pumped out. It currently contains 30 gallons of water.

The table of values shows the linear relationship of the amount of water remaining in the tub, y, after x minutes.

\text{Time in minutes } \left(x\right) | 0 | 1 | 2 | 3 |
---|---|---|---|---|

\text{Water remaining in gallons } \left(y\right) | 30 | 28 | 26 | 24 |

a

Determine the linear equation in slope-intercept form that represents this situation.

Worked Solution

b

Draw the graph of this linear relationship with a clearly labeled scale. Only show the viable solutions.

Worked Solution

c

Find the domain and range.

Worked Solution

d

Describe how the graph would change if, instead, there were initially 40 gallons of water in the tub, and it emptied at 2.5 gallons per minute.

Worked Solution

Which of the following has the higher y-intercept?

A

The line with a slope of 4 that crosses the y-axis at (0,\,6).

B

The line given by the equation y=x+4.

Worked Solution

Idea summary

The **slope-intercept form** of a line is:

\displaystyle y=mx+b

\bm{m}

slope

\bm{b}

y-intercept

Slope-intercept form is useful when we know, or want to know the slope of the line and the y-intercept of the line.