Now that we know how to represent linear functions as equations, tables, and graphs we can put this knowledge to use to solve real-world problems.

In each example, we will see that the **slope** represents a constant rate of change, while the **y-intercept ** represents an initial or starting value.

The slope, m, in a situation is related to a value that occurs repeatedly as the independent variable, x, (often time) increases. The y-intercept, b, in a situation is a value that will only occur once and will not be repeated over time. The dependent variable, y, represents the possible outputs that depend on the value of their inputs.

In many contextual situations, specific words or phrases will indicate what part of slope-intercept form they will represent.

Often indicates slope:

- Each
- Per
- For every
- Changes over time

Often indicates the y-intercept:

- Starting value
- One-time fee or cost
- Initial position

A diver starts at the surface of the water and begins to descend below the surface at a constant rate. The table shows the depth of the diver, in yards, over several minutes:

\text{Time passed (min)} | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

\text{Depth (yds)} | 0 | -1.4 | -2.8 | -4.2 | -5.6 |

a

Identify the independent and dependent variables.

Worked Solution

b

Determine the slope and y-intercept of the situation.

Worked Solution

c

Write an equation for the relationship between the number of minutes passed, x, and the depth, y, of the diver.

Worked Solution

d

Graph the relationship and determine the depth after 6 minutes.

Worked Solution

A carpenter charges a callout fee of \$150 plus \$45 per hour.

a

Write a linear equation to represent the total amount, y, charged by the carpenter as a function of the number of hours worked, x.

Worked Solution

b

Use the function to complete the table shown. Use the table to find the total amount charged by the carpenter for 6 hours of work.

\text{hours} | 0 | 2 | 4 | 6 | 8 |
---|---|---|---|---|---|

\text{total amount charged} \, (\$) |

Worked Solution

Create a situation that could be modeled by the graph, describing the slope and y-intercept in context.

Worked Solution

Idea summary

Linear relationships (functions) as equations, tables, and graphs can be used to solve a variety of real-world problems.

The slope, m, will represent a rate of change, and will be connected the the independent variable.

The y-intercept, b, represents a value that occurs once and does not repeat over time.