Essential Question: How do you find the equation of a trend line?

You can write an equation to model real-world situations

Example #1: Transportation Jacksonville, FL has an elevation of 12 ft

above sea level. A hot-air balloon taking off from Jacksonville rises 50 ft/min. Write an equation to model the balloon’s

elevation as a function of time. Graph the equation. Interpret the intercept at which the graph

intersects the vertical axis.

Jacksonville, FL has an elevation of 12 ft above sea level. A hot-air balloon taking off from Jacksonville rises 50 ft/min.

Balloon’s elevation = rate • time + starting elevation

Let h = the balloon’s current height Let t = time (in minutes) since the balloon lifted off

= • + So an equation that models this data is: We’ll graph it on the next slide

h 50 t 12

h = 50t + 12

h = 50t + 12 Let’s choose two points

If t = 0:h = 50(0) + 12 = 12Use the point (0, 12)

If t = 2h = 50(2) + 12 = 112Use the point (2, 112)

What does the y-intercept (0, 12) represent?

2 4 6 8 t

20

40

60

80

100

120

h

(0,12)

(2,112)

It represents the initial height of the balloon from sea level.It started 12 feet up to begin with.

Your Turn:Suppose a balloon begins descending at a

rate of 20 ft/min from an elevation of 1350 ft.1) Write an equation to model the balloon’s

elevation as a function of time.2) What is true about the slope of the line?3) Graph the equation.4) Interpret the h-intercept.

You can use two data points from a linear relationship (in point-slope form) to write a model.

Example #2/3: ScienceA candle is 6 in tall after burning for 1h.

After 3h, it is 5½ in tall. Write a linear equation to model the height y of

the candle after burning x hours. In how many hours will the candle be 4 in tall?

A candle is 6 in tall after burning for 1 h After 3 h, it is 5½ in tall. Write a linear equation to model the

height y of the candle after burning x hours What are the two data points we have to use?

What is the equation for point slope form?

What do we have from that equation?

What do we need?

(1, 6) and (3, 5½)

y – y1 = m(x – x1)

y1 = 6 and x1 = 1

2 1

2 1

y ym

x x

Points: (1, 6) and (3, 5½)Find the slope:

Find the equation in point-slope form y – y1 = m(x – x1)

1 12 1 2 2

2 1

5 6 1

3 1 2 4

y ym

x x

y – 6 = -¼(x – 1)

y – 6 = -¼ x + ¼y = -¼ x + 6 ¼

y = -¼ x + 6¼ In how many hours will the candle be 4 in

tall?Recall back in our original problem, height

is ySubstitute 4 for y and solve for x

4 = -¼ x + 6¼ The candle will be 4 in tall after 9 hours

-2¼ = -¼ x9 = x

Your Turn: y = -¼ x + 6¼ What does the slope -¼ represent?

What does the y-intercept 6¼ represent?

How tall will the candle be after burning for 11

hours?

When will the candle burn out?

The rate the candle burns down (¼ in per hour)

The original height of the candle

y = -¼(11) + 6¼y = -2¾ + 6¼ = 3½ inches

0 = -¼ x + 6¼-6¼ = -¼ x25 = xAbout 25 hours for the candle to burn out

AssignmentPage 81Problems 1 – 7 (all)Friday: Quiz

Direct Variation (Last Thursday, Section 2-3) Absolute Value Functions/Graphs (Monday,

Section 2-5) Families of Functions (Tuesday, Section 2-6) Using Linear Models (today)

Essential Question: How do you find the equation of a trend line?

Scatter Plot: a graph that relates two different sets of data by plotting the data as ordered pairs.You can use a scatter plot to determine a

relationship between the data sets

weak, positive strong, positivecorrelation correlation

weak, negative strong, negative nocorrelation correlation correlation

Trend line: a line that approximates the relationship between the data sets of a scatter plot. You can use a trend line to make predictions.

See the middle graph above for an example.

Example: AutomobilesA woman is considering buying a 1999 used

car for $4200. She researches prices for various years on the same model and records the data in the table below.

Part A: Let x represent the model year (Use 1 for 2000,

2 for 2001 and so forth.) Let y be the price of the car. Draw a scatter plot. Decide whether a linear model is reasonable.

Model Year

2000 2001 2002 2003 2004

Prices

$5784

$6810

$8237

$9660

$10,948

$5435

$6207

$7751

$9127

$10,455

Example: Automobiles

Is a linear model reasonable?

Draw a trend line. Write the equation of the line and decide whether the asking price is reasonable.

Model Year

2000 2001 2002 2003 2004

Prices

$5784

$6810

$8237

$9660

$10,948

$5435

$6207

$7751

$9127

$10,455

1 2 3 4 5 6 7 x

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

y

Yes, see graph on left

After you’ve drawn your trend line, you can use the slope and y-intercept to determine an equation.

You use the trend line,NOT any of the originalpoints (unless they happento fall on the line)

Equation:1 2 3 4 5 6 7 x

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

y

y-intercept ≈ 4100

slope ≈ 1300

y = 1300x + 4100

Is a 1999 car for $4200 a reasonable price?

The equation is y = 1300x + 4100A 1999 car would represent the year x = 0

Remember, we started by using x = 1 for a 2000 car

So a 1999 car would be fairly priced at: y = 1300(0) + 4100

y = 4100A price of $4200 is a reasonable price

Your Turn:Graph each set of data.

Decide whether a linear model is reasonable. If so, draw a trend line and write its equation.

{(-7.5, 19.75), (-2, 9), (0, 6.5), (1.5, 3), (4, -1.5)}

Done on the whiteboard

Assignment Page 81 Problems 8 – 11

Show your graphs, trend lines, and equations

Tomorrow: Quiz Direct Variation (Last Thursday, Section 2-3) Absolute Value Functions/Graphs (Monday, Section

2-5) Families of Functions (Tuesday, Section 2-6) Using Linear Models (Wednesday, 1st part of 2-4) Today’s material will NOT be on the quiz

Next Week Monday: Chapter 2 Preview Tuesday: Chapter 2 Review Wednesday: Chapter 2 Test