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