Topics
3. Relations & Functions
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United States of AmericaVirginia
Grade 8

3.07 Problem solving with linear functions

Lesson
Practice

Problem solving with linear functions

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.

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\text{independent variable}
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\text{dependent variable}

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

Examples

Example 1

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)}01234
\text{Depth (yds)}0-1.4-2.8-4.2-5.6
a

Identify the independent and dependent variables.

Worked Solution
Create a strategy

The value of the dependent value changes based on the value of the independent variable.

Apply the idea

How far the diver has descended depends on the amount of time passed.

The independent variable is time passed in minutes, and the dependent variable is the depth in yards.

Reflect and check

The independent variable and dependent variable are often associated with consistent variables across different representations if specific variables are not listed.

The independent variable is an input, so it is associated with x. The dependent variable is an output, so it is associated with the variable y.

b

Determine the slope and y-intercept of the situation.

Worked Solution
Create a strategy

The slope, or rate of change, is represented by the ratio \dfrac{\text{change in y}}{\text{change in x}}. The y-intercept is represented by the pair \left(0,\,y\right) in a table.

Apply the idea

To find the slope, we see that the x-values change by 1, so we will need to find the change in y-values by subtracting one value by the previous, checking all pairs to make sure the rate is constant.

\displaystyle -1.4\displaystyle =\displaystyle -1.4-0Subtract 0 from -1.4
\displaystyle -1.4\displaystyle =\displaystyle -2.8-(-1.4)Subtract -1.4 from -2.8
\displaystyle -1.4\displaystyle =\displaystyle -4.2-(-2.8)Subtract -2.8 from -4.2
\displaystyle -1.4\displaystyle =\displaystyle -5.6-(-4.2)Subtract -4.2 from -5.6

The rate of change is \dfrac{-1.4}{1}. The pair \left(0,0\right) is in the table, so this is the y-intercept.

The slope is -1.4, and the y-intercept is 0.

c

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

Worked Solution
Create a strategy

We can use slope-intercept form, y=mx+b, where m is the change in depth per minute and b is the initial depth of the diver.

Apply the idea
\displaystyle y\displaystyle =\displaystyle -1.4x+0Substitute the value of m and b
\displaystyle y\displaystyle =\displaystyle -1.4xEvaluate
d

Graph the relationship and determine the depth after 6 minutes.

Worked Solution
Create a strategy

Graph using the original table or the slope-intercept form from part c. Once graphed, find the point on the line with an x-value of 6.

Apply the idea
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\text{time (min)}
-8.4
-7
-5.6
-4.2
-2.8
-1.4
\text{depth (yds)}

The point \left(6,\,-8.4\right) is on the graph. This means that after 6 minutes, the depth is -8.4 yards.

Reflect and check

If the table is used to graph, the graph could be used to determine the slope and y-intercept. The slope can be found using slope triangles, and the y-intercept is the point \left(0,\,y\right) where the function crosses the y-axis.

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\text{time (min)}
-8.4
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-5.6
-4.2
-2.8
-1.4
\text{depth (yds)}

Example 2

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
Create a strategy

The slope, m, represents a repeated change as x increases. The y-intercept, b, represents an initial value or constant value that is added once.

The slope-intercept form of a linear equation, y=mx+b, will use this rate of change, m, and constant, b, to write the equation.

Apply the idea

A callout fee is a one-time fee that is added to the charge. This value of \$ 150 is b, the y-intercept. A charge of \$ 45 per hour is repeated as the number of hours increases. This value is our slope, m.

\displaystyle y\displaystyle =\displaystyle mx+bSlope-intercept form
\displaystyle y\displaystyle =\displaystyle 45x+150Substitute m=45 and b=150

The equation is y=45x+150.

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}02468
\text{total amount charged} \, (\$)
Worked Solution
Create a strategy

We can use the equation found from part (a), y=45x+150, and substitute and evaluate the values of x.

Apply the idea

Each value of x from the table will be used as an input to the equation and evaluted.

\displaystyle \$ 150\displaystyle =\displaystyle \$45 \cdot 0 + \$ 150Substitute x = 0 and evaluate
\displaystyle \$ 240 \displaystyle =\displaystyle \$45 \cdot 2 + \$ 150Substitute x = 2 and evaluate
\displaystyle \$ 330\displaystyle =\displaystyle \$45 \cdot 4 + \$ 150Substitute x = 4 and evaluate
\displaystyle \$ 420\displaystyle =\displaystyle \$45 \cdot 6 + \$ 150Substitute x = 6 and evaluate
\displaystyle \$ 510\displaystyle =\displaystyle \$45 \cdot 8 + \$ 150Substitute x = 8 and evaluate

Pairing each input with its output in the table:

\text{hours}02468
\text{total amount charged} \, (\$)150240330420510

For 6 hours of work, the total charged will be \$ 420.

Reflect and check

We can also fill out the table by graphing the function in the coordinate plane using the equation y = 45x + 150 and finding the y-values paired with the x-values.

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\text{hours}
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510
\text{total amount charged} \$

Example 3

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

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y
Worked Solution
Create a strategy

The slope, m, describes the rate of change, and the y-intercept, b, gives an initial value. Whether the slope is positive or negative will tell you if the rate of change is increasing or decreasing.

Apply the idea

Using the points on the graph to find the slope and y-intercept, we will be determine the specific values needed and create a realistic situation.

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To find the slope of this graph, we must consider that the x-values have a scale of 1, while the y-values have a scale of 5.

The slope triangles each show a vertical change of 15, and a horizontal change of 5. To write this as slope, we will write

\displaystyle \dfrac{\text{vertical change}}{\text{horizontal change}}\displaystyle =\displaystyle \dfrac{+15}{+5}Write as a ratio
\displaystyle \dfrac{15 \div 5}{5 \div 5}\displaystyle =\displaystyle \dfrac{3}{1}Reduce
\displaystyle \dfrac{3}{1}\displaystyle =\displaystyle 3Divide

The slope is 3, and the situation needs to describe increasing 3 vertical units for every 1 horizontal unit. The y-intercept of \left(0,20\right) means the initial value of the situation is 20.

A possible situation could be a person will pay a one-time fee of \$ 20 to enter an amusement park and pay \$3 per ride. They want to determine how much money will be spent by going on different amounts of rides.

Reflect and check

Several other situations could be used to describe an increasing function.

Other common situations for increasing functions include relating money to time, distance traveled to time, or money spent to items purchased.

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.

Outcomes

8.PFA.3

The student will represent and solve problems, including those in context, by using linear functions and analyzing their key characteristics (the value of the y-intercept (b) and the coordinates of the ordered pairs in graphs will be limited to integers).

8.PFA.3b

Describe key characteristics of linear functions including slope (m), y-intercept (b), and independent and dependent variables.

8.PFA.3c

Graph a linear function given a table, equation, or a situation in context.

8.PFA.3d

Create a table of values for a linear function given a graph, equation in the form of y = mx + b, or context.

8.PFA.3e

Write an equation of a linear function in the form y = mx + b, given a graph, table, or a situation in context.

8.PFA.3f

Create a context for a linear function given a graph, table, or equation in the form y = mx + b.

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