Algebra 1 Linear, Quadratic, and Exponential Models

CONFIDENTIAL 1

Algebra 1Algebra 1

Linear, Quadratic, Linear, Quadratic, and Exponential and Exponential

ModelsModels

CONFIDENTIAL 2

Warm UpWarm Up

Solve by using square roots.

1) 4x2 = 100

2) 10 - x2 = 10

3) 16x2 + 5 = 86

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

CONFIDENTIAL 4

CONFIDENTIAL 5

In the real world, people often gather data and then must decide what kind of relationship (if any) they

think best describes their data.

CONFIDENTIAL 6

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.

CONFIDENTIAL 7

B)

Plot the data points and connect them.

The data appear to be linear.

CONFIDENTIAL 8

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)}

CONFIDENTIAL 9

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

describes a dataset is to use patterns.

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)

CONFIDENTIAL 11

B)

For every constant change in age of +1 year, there is a constant ratio of 0.85.

The data appear to be exponential.

CONFIDENTIAL 12

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.

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

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.

CONFIDENTIAL 15

Step 1: Describe the situation in words.

Each month, the ladybug population is multiplied by 3.In other words, the population triples each month.

CONFIDENTIAL 16

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.

CONFIDENTIAL 17

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

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.

CONFIDENTIAL 19

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)}

CONFIDENTIAL 20

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)}

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.

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

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.

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.

CONFIDENTIAL 25

B)

Plot the data points and connect them.

The data appear to be linear.

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)

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

CONFIDENTIAL 28

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