Graphing and Interpreting Applications of Slope–intercept
Graphing and Interpreting Applications of Slope–intercept
Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real-world situations.
Usually when a linear equation models a real-world situation, different letters are used for the variables, instead of x and y. The variable names remind us of what quantities are being measured.
Example
The equation \(F=\frac{9}{5}C+32\) is used to convert temperatures, \(C\), on the Celsius scale to temperatures, \(F\), on the Fahrenheit scale.
a. Find the Fahrenheit temperature for a Celsius temperature of 0.
b. Find the Fahrenheit temperature for a Celsius temperature of 20.
d. Graph the equation.
Solution
(a)
\(\begin{array}{cccccc}\text{Find the Fahrenheit temperature for a Celsius temperature of 0.}\hfill & & & & & F=\frac{9}{5}C+32\hfill \\ \text{Find}\phantom{\rule{0.2em}{0ex}}F\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}C=0.\hfill & & & & & F=\frac{9}{5}\left(0\right)+32\hfill \\ \text{Simplify.}\hfill & & & & & F=32\hfill \end{array}\)
(b)
\(\begin{array}{cccccc}\text{Find the Fahrenheit temperature for a Celsius temperature of 20.}\hfill & & & & & F=\frac{9}{5}C+32\hfill \\ \text{Find}\phantom{\rule{0.2em}{0ex}}F\phantom{\rule{0.2em}{0ex}}\text{when}\phantom{\rule{0.2em}{0ex}}C=20.\hfill & & & & & F=\frac{9}{5}\left(20\right)+32\hfill \\ \text{Simplify.}\hfill & & & & & F=36+32\hfill \\ \text{Simplify.}\hfill & & & & & F=68\hfill \end{array}\)
(c)
Interpret the slope and F-intercept of the equation.
Even though this equation uses \(F\)and \(C\), it is still in slope–intercept form.
The slope, \(\frac{9}{5}\), means that the temperature Fahrenheit (F) increases 9 degrees when the temperature Celsius (C) increases 5 degrees.
The F-intercept means that when the temperature is \(0\text{°}\) on the Celsius scale, it is \(32\text{°}\) on the Fahrenheit scale.
(d)
Graph the equation.
We’ll need to use a larger scale than our usual. Start at the F-intercept \(\left(0,32\right)\) then count out the rise of 9 and the run of 5 to get a second point. See the figure below.
The cost of running some types business has two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.
Example
Stella has a home business selling gourmet pizzas. The equation \(C=4p+25\) models the relation between her weekly cost, C, in dollars and the number of pizzas, p, that she sells.
a. Find Stella’s cost for a week when she sells no pizzas.
c. Interpret the slope and C-intercept of the equation.
d. Graph the equation.
Solution
| Find Stella's cost for a week when she sells no pizzas. | |
| Find C when \(p=0\). | |
| Simplify. | |
| Stella's fixed cost is $25 when she sells no pizzas. | |
| Find the cost for a week when she sells 15 pizzas. | |
| Find C when \(p=15\). | |
| Simplify. | |
| Stella's costs are $85 when she sells 15 pizzas. | |
| Interpret the slope and C-intercept of the equation. | |
| The slope, 4, means that the cost increases by $4 for each pizza Stella sells. The C-intercept means that even when Stella sells no pizzas, her costs for the week are $25. | |
| Graph the equation. We'll need to use a larger scale than our usual. Start at the C-intercept (0, 25) then count out the rise of 4 and the run of 1 to get a second point. |
This lesson is part of:
Graphs and Equations