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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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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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Another way to decide which kind of relationship (if any) best

describes a dataset is to use patterns.

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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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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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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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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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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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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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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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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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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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You did a great job You did a great job today!today!