Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

Linear models can also be found by hand in some situations when the data follows a linear pattern since the slope remains the same between data points.

Linear Models from Tables

For 1 - 4, use the scenario and table below.

The table below shows the number of songs Marcus will have in his collection as he adds new songs each month.

Number of months, $t$ 0 1 2 3

Number of songs, $N$ 200 215 230 245

1.

What is the initial amount of songs, or the $y$ -intercept of this situation?

200

2.

What is the slope, or rate of change?

15

3.

Write the equation of the linear model that represents the situation in slope-intercept form.

Compare your answer:

$N = 15 t + 200$

4.

How many songs will Marcus have in 8 months?

320

Compare your answer: $N = 15 (8) + 200 = 320$

For 5 - 8, use the situation and table below.

A new plant food is introduced to a young tree to test its effect on the height of the tree. The table shows the height of the tree, $H$ , in feet $x$ months since measurements began.

Number of months, $x$ 2 4 6 8

Height in feet, $H$ 13.5 14.5 15.5 16.5

5.

What is the initial height of the tree, or the $y$ -intercept, when measurements began?

(Hint: Work each row backwards to where $x = 0$ .)

12.5

6.

What is the slope, or rate of change?

$\frac{1}{2}$

7.

Write the equation of the linear model that represents the situation in slope-intercept form.

Compare your answer:

$H = \frac{1}{2} x + 12.5$

8.

What will the height of the tree be in 14 months?

19.5

Compare your answer: $H = \frac{1}{2} (14) + 12.5 = 19.5$

Linear Models from Graphs

For 9 - 12, use the situation and graph below.

The graph models the cost in dollars, $C$ , of renting a tent at a campground for $n$ nights.

9.

What is the $y$ -intercept of the graph?

30

10.

What is the slope, or rate of change?

10

11.

Write the equation of the linear model that represents the situation in slope-intercept form.

Compare your answer:

$C = 10 n + 30$

12.

How much does it cost to rent a tent for 5 nights?

Compare your answer: $C = 10 n + 30$ $C = 10 (5) + 30 = 80$

Self Check

The table below shows the revenue for the number of pizzas sold at a restaurant.

Number of Pizzas	20	30	40	50
Revenue	110	160	210	260

Which linear model could be used to predict the revenue if 4000 pizzas were sold?

$y = 80 (4000)$

$y = 5 (4000) + 10$

$y = 4000 (10)$

I am not sure.

$y = \frac{1}{5} (4000) + 10$

Additional Resources

Writing Linear Equations From Tables and Graphs

Linear Equations from Tables

The number of texts a teen sends, $T$ , in days, $d$ , is shown in the table below.

Days, $d$	1	2	3	4
Number of texts, $T$	65	130	195	260

Write an equation in slope-intercept form to represent the situation then predict the number of texts sent in 8 days.

Step 1 - Find the $y$ -intercept.

Since the value when $d = 0$ is not present, work backwards to $d = 0$ .

If $d = 0$ was present, then $T = 0$ since as each $x$ -value increases by 1, the $y$ -values increase by 65.

Step 2 - Find the slope of the situation.

Find the rate of change. The change in $y$ is 65 and the change in $x$ is 1, so the slope is 65

Step 3 - Write the equation in slope-intercept form.

$y = m x + b$

$T = 65 d + 0$

$T = 65 d$

Step 4 - Make a prediction.

Substitute $d = 8$ into the equation.

$T = 65 (8) = 520$

After 8 days, the teen would have sent 520 texts.

Linear Equations from Graphs

The cost, $C$ , for the number of days, $d$ , a dog spends at doggie daycare is shown in the graph below.

A scatterplot graph showing the cost in dollars graphed on the y-axis versus the number of days represented on the x-axis. Points are plotted and labeled at: (1, 35), (2, 70), (3, 105), and (4, 140). The cost increases steadily as the number of days increases.

Write an equation in slope-intercept form to represent the situation then predict the cost of a dog staying at doggie daycare after 7 days.

Step 1 - Find the $y$ -intercept.

Since the value when $d = 0$ is not present, work backwards to $d = 0$ .

If $d = 0$ was present, then $C = 0$ since as each $x$ -value increases by 1, the $y$ -values increase by 35.

Step 2 - Find the slope of the situation.

Find the rate of change. The change in $y$ is 35 and the change in $x$ is 1, so the slope is 35

Step 3 -Write the equation in slope-intercept form.

$y = m x + b$

$C = 35 d + 0$

$C = 35 d$

Step 4 -Make a prediction.

Substitute $d = 7$ into the equation.

$C = 35 (7) = 245$

After 7 days, the cost of doggie daycare would be $245.

Try it

Try It-Writing Linear Equations From Tables and Graphs

Naomi is a professional painter. The table below shows how many square feet, $F$ , she can paint in $h$ hours.

Time in hours, $h$	1	2	3	4
Number of square feet, $F$	120	240	360	480

How many square feet can Naomi paint in 8 hours?

Compare your answer:

960

Step 1 - Find the $y$ -intercept.

Since the value when $h = 0$ is not present, work backwards to $h = 0$ .

If $h = 0$ was present, then $F = 0$ since as each $x$ -value increases by 1, the $y$ -values increase by 120.

Step 2 - Find the slope of the situation.

Find the rate of change. The change in $y$ is 120 and the change in $x$ is 1, so the slope is 120

Step 3 -Write the equation in slope-intercept form.

$y = m x + b$

$F = 120 h + 0$

$F = 120 h$

Step 4 -Make a prediction.

Substitute $h = 8$ into the equation.

$F = 120 (8) = 960$

Naomi can paint 960 square feet in 8 hours.

Number of months, $t$	0	1	2	3
Number of songs, $N$	200	215	230	245

Number of months, $x$	2	4	6	8
Height in feet, $H$	13.5	14.5	15.5	16.5

3.2.3 Writing Linear Models without Technology

Activity

Additional Resources

Writing Linear Equations From Tables and Graphs

Try It-Writing Linear Equations From Tables and Graphs