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CONFIDENTIAL 1
Algebra 1Algebra 1
Linear, Quadratic, Linear, Quadratic, and Exponential and Exponential
ModelsModels
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Warm UpWarm Up
Solve by using square roots.
1) 4x2 = 100
2) 10 - x2 = 10
3) 16x2 + 5 = 86
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CONFIDENTIAL 3
Look at the tables and graphs below. The data show three ways you have learned that variable quantities can be related. The
relationships shown are linear, quadratic, and exponential.
Linear, Quadratic, and Exponential ModelsLinear, Quadratic, and Exponential Models
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In the real world, people often gather data and then must decide what kind of relationship (if any) they
think best describes their data.
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Graphing Data to Choose a ModelGraphing Data to Choose a Model
Graph each data set. Which kind of model best describes the data?
A) Plot the data points and connect them.
The data appear to be exponential.
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B)
Plot the data points and connect them.
The data appear to be linear.
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Now you try!
Graph each data set. Which kind of model best describes the data?
1) { (-3, 0.30) , (-2, 0.44) , (0, 1) , (1, 1.5) , (2, 2.25) , (3,3.38}
2) {(-3, -14) , (-2, -9) , (-1, -6) , (0, -5) , (1, -6) , (2, -9) , (3, -14)}
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CONFIDENTIAL 9
Another way to decide which kind of relationship (if any) best
describes a dataset is to use patterns.
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CONFIDENTIAL 10
Graphing Data to Choose a ModelGraphing Data to Choose a Model
Look for a pattern in each data set to determine which kind of model best describes the data.
For every constant change in distance of +100 feet, there is a constant second difference of +32.
The data appear to be quadratic.
A)
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B)
For every constant change in age of +1 year, there is a constant ratio of 0.85.
The data appear to be exponential.
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Now you try!
1) Look for a pattern in the data set {(-2, 10) , (-1, 1) , (0, -2) , (1, 1) , (2, 10)} to determine which kind of model best describes the data.
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CONFIDENTIAL 13
General Forms of FunctionsGeneral Forms of Functions
After deciding which model best fits the data, you can write a function. Recall the general forms of
linear, quadratic, and exponential functions.
LINEAR y = mx + b
QUADRATIC y = ax2 + bx + c
EXPONENTIAL y = abx
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CONFIDENTIAL 14
Problem-Solving ApplicationProblem-Solving Application
Use the data in the table to describe how the ladybug population is changing. Then write a function that models
the data. Use your function to predict the ladybug population after one year.
Determine whether the data is linear, quadratic, or exponential. Use the general form to write a function. Then use the function
to find the population after one year.
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Step 1: Describe the situation in words.
Each month, the ladybug population is multiplied by 3.In other words, the population triples each month.
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Step 2: Write the function.
There is a constant ratio of 3. The data appear to be exponential.
y = abx
y = a(3)x
10 = a(3)x
10 = a (1)
10 = a
y = 10(3)x
Write the general form of an exponential function.
Substitute the constant ratio, 3, for b.
Choose an ordered pair from the table, such as (0, 10) . Substitute for x and y.
Simplify. 30 = 1
The value of a is 10.
Substitute 10 for a in y = a (3)x.
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Step 3: Predict the ladybug population after one year.
y = 10(3)x
y = a(3)12
= 5,314,410
Substitute 12 for x (1 year = 12 mo).
Use a calculator.
Write the function.
You chose the ordered pair (0, 10) to write the function. Check that every other ordered pair in the table satisfies your function.
y = 10(3)x
30 10(3)1
30 10(3)
30 30
y = 10(3)x
90 10(3)2
90 10(9)
90 90
y = 10(3)x
270 10(3)3
270 10(27)
270 270
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CONFIDENTIAL 18
Now you try!
1) Use the data in the table to describe how the oven temperature is changing. Then write a function that models the data. Use your function to predict the temperature after 1 hour.
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Assessment
1) {(-1, 4) , (-2, 0.8) , (0, 20) , (1, 100) , (-3, 0.16)}
Graph each data set. Which kind of model best describes the data?
2) {(0, 3) , (1, 9) , (2, 11) , (3, 9) , (4, 3)}
3) {(2, -7) , (-2, -9) , (0, -8) , (4, -6) , (6, -5)}
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Look for a pattern in each data set to determine which kind of model best describes the data.
5) {(-2, 0.75) , (-1, 1.5) , (0, 3) , (1, 6) , (2, 12)}
4) {(-2, 1) , (-1, 2.5) , (0, 3) , (1, 2.5) , (2, 1)}
6) {(-2, 2) , (-1, 4) , (0, 6) , (1, 8) , (2, 10)}
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CONFIDENTIAL 21
7) Use the data in the table to describe the cost of grapes. Then write a function that models the data. Use your function to predict the cost of 6 pounds of grapes.
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CONFIDENTIAL 22
Look at the tables and graphs below. The data show three ways you have learned that variable quantities can be related. The
relationships shown are linear, quadratic, and exponential.
Linear, Quadratic, and Exponential ModelsLinear, Quadratic, and Exponential Models
Let’s review
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CONFIDENTIAL 23
In the real world, people often gather data and then must decide what kind of relationship (if any) they
think best describes their data.
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CONFIDENTIAL 24
Graphing Data to Choose a ModelGraphing Data to Choose a Model
Graph each data set. Which kind of model best describes the data?
A) Plot the data points and connect them.
The data appear to be exponential.
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B)
Plot the data points and connect them.
The data appear to be linear.
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CONFIDENTIAL 26
Graphing Data to Choose a ModelGraphing Data to Choose a Model
Look for a pattern in each data set to determine which kind of model best describes the data.
For every constant change in distance of +100 feet, there is a constant second difference of +32.
The data appear to be quadratic.
A)
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CONFIDENTIAL 27
General Forms of FunctionsGeneral Forms of Functions
After deciding which model best fits the data, you can write a function. Recall the general forms of
linear, quadratic, and exponential functions.
LINEAR y = mx + b
QUADRATIC y = ax2 + bx + c
EXPONENTIAL y = abx
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CONFIDENTIAL 28
You did a great job You did a great job today!today!