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: $N = 15 (8) + 200 = 320$ $N = 15 (8) + 200 = 320$

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

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

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?

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