Let's have a look...
So far, you have learned about and practiced constructing a linear function using both a table of values and a graph. Regardless of whether you are using a table or graph, you construct the function by calculating the rate of change and solving for the initial value. Then, you substitute those values into the slope-intercept form of a line, \( y = mx + b \).
Now, you will see how to construct linear functions from descriptions of real-world scenarios. For example:
One weekend you decide to mow the neighbors' lawn for some extra money. The lawn has an area of 10,000 square feet. After 20 minutes of work, you still have 7,500 square feet left to mow.
If you mow the lawn at a constant rate of change, what is the linear function that models this scenario?
The steps for constructing a linear function from a description are shown in the table below. Click each step to see it applied to the example.
|
Recognize that you were given the ordered pairs (0, 10000) and (20, 7500). Use these values to calculate the rate of change. \( m = \frac{\left( 10000 \right) - \left( 7500 \right)}{\left( 20 \right) - \left( 0 \right)} \) \( m = - \frac{2500 \textsf{ square feet}}{20 \textsf{ minutes}} \) \( m = - \frac{125 \textsf{ square feet}}{1 \textsf{ minute}} \) |
|
|
You can substitute either ordered pair, along with the rate of change \( m = - 125 \), and solve for \( b \) in \( y = mx + b \). You can also recognize that one of the ordered pairs (0, 10000), is the initial value. |
|
|
The value of \( m = - 125 \) and \( b = 10000. \) \( y = - 125x + 10000 \) The equation indicates that you started with 10,000 square feet of lawn to mow, and you complete 125 square feet for every minute you work. |
Show your skills!
How well can you write a linear function from a description? Use the activity below to practice. Read the scenario on each tab, then construct the linear function and check your answer.
A streaming video service membership is $45 plus rental fees. If you rent 2 movies, the total cost of this service is $48.10. If you rent 5 movies, the total cost of this service is $52.75.
Construct the linear function that models this scenario.
\( y = \) 1.55\( x + \) 45
If you need help arriving at this answer, click the Solution button.
|
Step 1: Determine the constant rate of change. |
Recognize that you were given the ordered pairs (0, 45), (2, 48.10) and (5, 52.75). Use any two of these ordered pairs to calculate the rate of change. \( m = \frac{\left( 48.10 \right) - \left( 45 \right)}{\left( 2 \right) - \left( 0 \right)} \) \( m = \frac{\$ 3.10}{2 \textsf{ movies}} \) \( m = \frac{\$ 1.55}{1 \textsf{ movie}} \) |
|
Step 2. Solve for the initial value, \( b \). |
You can substitute any of the ordered pairs along with the rate of change \( m = 1.55 \) and solve for \( b \) in \( y = mx + b \). You can also recognize that one of the ordered pairs (0, 45), is the initial value. |
|
Step 3: Write the function in \( y = mx + b \) form. |
The value of \( m = 1.55 \) and \( b = 45. \) \( y = 1.55x + 45 \) The equation indicates that the streaming service costs $45 and each movie rental costs $1.55. |
A scientist discovered a new species of underwater plant. She is studying its growth. After 24 days, the plant grew to 16 cm in height. If the scientist determined the underwater plant is growing at a constant rate of 3 cm every 8 days, construct the linear function that models this situation. The input variable is the number of days, and the output is the height of the underwater plant.
\( y = \) \( \frac{3}{8}x + 7 \)
If you need help arriving at this answer, click the Solution button.
|
Step 1: Determine the constant rate of change. |
The rate of change is given in the problem: 3 cm every 8 days. \( m = \frac{3 \textsf{ cm}}{8 \textsf{ days}} \) |
|
Step 2. Solve for the initial value, \( b \). |
You know one ordered pair that lies along the line; it is (24, 16). Use this ordered pair along with the rate of change, \( m = \frac{3}{8}, \) to solve for the value of \( b \) in \( y = mx + b \). \( \left( 16 \right) = \left( \frac{3}{8} \right)\left( 24 \right) + b \) \( 16 = 9 + b \) \( 7 = b \) |
|
Step 3: Write the function in \( y = mx + b \) form. |
The value of \( m = \frac{3}{8} \) and \( b = 7. \) \( y = \frac{3}{8}x + 7 \) The equation indicates that the plant grows 3 cm every 8 days, and its initial height was 7 cm. |
Are you planning on continuing your education after high school? There are many options to continue your education, each with different benefits and costs. You can go to a two-year community college, a four-year private or public college, or many other options. How long you go, the amount you pay, and how you pay depends on the career you want, the type of college you choose, and the financial aid available. Financial aid is money given or loaned to you to help you pay for college. Financial aid may be awarded to you based on your financial need or based on merit such as academic or athletic ability. Most full-time college students receive some form of financial aid.
The table below shows the total estimated costs for different colleges. There is an extra charge of $4,500 for each for room and board.
Construct a linear function that models the scenario of the cost of college for 1 year including room and board. Complete a linear function for both the two-year and four-year college options.
| Room and Board | Two-Year Community College Tuition |
Four-Year Public College Tuition |
|---|---|---|
| $4,500 | $18,000 | $80,000 |
Two Year:
\(y = \frac{(18,000)}{(2)}\)
\(y = $9,000\)
\(y = $9,000 + $4,500\)
\(y = $13,500\)
Four Year:
\(y= \frac{(80,000)}{(4)}\)
\(y = \frac{$20,000}{1}\)
\(y = $20,000 + $4,500\)
\(y = $24,500\)
Alex has decided to go to a two-year community college and needs $9,000 to pay tuition for his first year. His family is giving him $5,000 to put towards that amount. Alex has determined he can save $500 each month.
How many months will it take Alex to save the $9,000 for his first year of college? Construct a linear function that models the scenario of saving for college for 1 year.
$9,000 = $500x + $5,000
