Unit 5: Comparing Linear, Quadratic, & Exponential …...Algebra 1 Unit 5: Comparing Linear, Quadratic, and Exponential Functions Notes 3 Identifying Types of Functions from a Table

Algebra 1 Unit 5: Comparing Linear, Quadratic, and Exponential Functions Notes

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Unit 5: Comparing Linear, Quadratic, & Exponential Functions

Comparing Multiple Representations

After completion of this unit, you will be able to…

Learning Target: Comparing Functions in Multiple Representations

Compare and contrast characteristics of linear, quadratic, and exponential models

Recognize that exponential and quadratic functions have variable rates of changes whereas linear functions have

constant rates of change

Observe that graphs and tables of exponential functions eventually exceed linear and quadratic functions

Find and interpret domain and range of linear, quadratic, and exponential functions

Interpret parameters of linear, quadratic, and exponential functions

Calculate and interpret average rate of change over a given interval

Write a function that describes a linear, quadratic, or exponential relationship

Solve problems in different representations using linear, quadratic, and exponential models

Construct and interpret arithmetic and geometric sequences

Timeline for Unit 5

Monday Tuesday Wednesday Thursday Friday

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Day 1:

Distinguishing between

Linear, Quadratic, and

Exponential Equations,

Tables, and Graphs

16

Day 2:

Domain, Range, &

Parameters in a Real

World Context

17

Day 3:

Comparing Graphs &

Tables of Functions

18

Day 4:

Average Rate of

Change

Day 5:

Comparing Multiple

Representations of

Functions


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Day 1 – Distinguishing Between Linear, Quadratic, & Exponential Functions

You should refer to your All about Linear, Quadratic, and Exponential Functions graphic organizers throughout

this entire unit.

Identifying Types of Functions from an Equation

Classify each equation as linear, quadratic, or exponential:

a. f(x) = 3x + 2 b. y = 5x c. f(x) = 2

d. f(x) = 4(2)x + 1 e. y = 7(.25)3x f. y = 4x2 + 2x - 1

Identifying Types of Functions from a Graph

Determine if the following graphs represent an exponential function, linear function, quadratic function, or

neither.

a. b. c. d.

e. f. g. h.


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Identifying Types of Functions from a Table

Remember with linear functions, they have constant (same) first differences (add same number over and over).

Quadratic Functions have constant second differences.

Exponential functions have constant ratios (multiply by same number over and over).

Linear

Function Quadratic Function Exponential Function

Determine if the following tables represent linear, quadratic, exponential, or neither and explain why.

a. b. c.

d. e. f.


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Writing Equations from a Graph or Table

Linear Functions

y = mx + b

y = (slope)x + y-intercept

slope = # you add/sub each time

y-intercept: starting amount or y-value

when x = 0

Quadratic Functions

y = a(x – h)2 + k

y = opens(x – x-value)2 + y-value

(h, k) is vertex

y = a(x – p)(x – q)

y = opens(x – zero)(x – zero)

You then have to multiply your equation out

to get to standard form.

Exponential Functions

y = abx

y = y-intercept(constant ratio)x

y-intercept: starting amount or y-value when

x = 0

constant ratio = # you multiply by each time

For each table or graph below, identify if it is linear, quadratic, or exponential. Then write an equation that

represents it.

a. Type: ______________________ b. Type: ______________________

Equation: ____________________ Equation: ____________________

c. Type: ______________________ d. Type: ______________________



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e. Type: ______________________ f. Type: ______________________


g. Type: ______________________ h. Type: ______________________


i. Type: ______________________ j. Type: ______________________


k. Type: ______________________ l. Type: ______________________



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Day 2 –Domain, Range, & Parameters in Real World Contexts

1. Determine the Domain and Range of each of the following graphed functions (using Interval and Set Notations).

A.

B.

C.

Domain (INTERVAL): _________

Domain (SET):

Range (INTERVAL):

Range (SET):


Domain (SET):

Range (INTERVAL):

Range (SET):


Domain (SET):

Range (INTERVAL):

Range (SET):

2. Determine the Domain and Range of each of the following graphed functions (using Interval and Set Notations).

A. 𝑚(𝑥) = 2(𝑥 − 1)2 − 3 B. 𝑝(𝑥) = 2𝑥 + 1 C. 𝑞(𝑥) = 2𝑥 − 4

Domain (INTERVAL):

Domain (SET):

Range (INTERVAL):

Range (SET):

Domain (INTERVAL):

Domain (SET):

Range (INTERVAL):

Range (SET):

Domain (INTERVAL):

Domain (SET):

Range (INTERVAL):

Range (SET):

3. If we only considered the functions LINEAR, QUADRATIC, and EXPONENTIAL, which is the only one that could

have a range of [–∞, ∞) ?

4. If we only considered the functions LINEAR, QUADRATIC, and EXPONENTIAL, which is the only one that could

have a range of (2, ∞)?

5. If we only considered the functions LINEAR, QUADRATIC, and EXPONENTIAL, which is the only one that

could have a range of [– 5, ∞)?


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Linear Characteristics

Practice Example 1

Practice Example 2

Domain:

Range:

Domain:

Range:

Y-intercept:

X-intercept:

Zero:

Y-intercept:

X-intercept:

Zero:

Interval of Increase:

Interval of Decrease:

Interval of Constant:

Interval of Increase:

Interval of Decrease:

Interval of Constant:

Maximum:

Minimum:

Maximum:

Minimum:

Positive:

Negative:

Positive:

Negative:

End Behavior:

As x -∞, f(x) _____

As x ∞, f(x) _____

End Behavior:

As x -∞, f(x) _____

As x ∞, f(x) _____


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Quadratic Characteristics

Domain: ___________________ Range: ______________________

Vertex: ____________________ Axis of Sym.__________________

Y-Intercept: ________________ Zeroes: ______________________

Extrema: ___________________ Max/Min Value: _____________

Int of Inc: __________________ Int of Dec: ___________________

Positive: ____________________ Negative: ____________________

End Behavior: As x -∞, f(x) ______. As x ∞, f(x) ______

Domain: ___________________ Range: ______________________







Domain: ___________________ Range: ______________________








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Exponential Characteristics

Domain: _____________________ Range: _________________________

X-intercept: __________________ y-intercept: ____________________

Interval of Increase: __________ Interval of Decrease: ___________

Maximum(s): ________________ Minimum(s):____________________

Asymptote: _________________

End Behavior: as x , f(x) ______

as x , f(x) ______

Domain: _____________________ Range: _________________________




Asymptote: _________________


as x , f(x) ______

Domain: _____________________ Range: _________________________




Asymptote: _________________


as x , f(x) ______


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When we apply domain and range to real world situations, we need to consider what types of numbers are

suitable for a domain and range. Typically, we describe domain and range using one of the types of number

classifications.

Types of Numbers Example

Counting Numbers 1, 2, 3, 4… (Zero is not included)

Whole Numbers 0, 1, 2, 3… (Also called non-negative integers)

Integers …-3, -2, -1, 0, 1, 2, 3, …

Rational Numbers Everything above plus terminating/repeating decimals

& fractions

Real Numbers Everything above plus irrational numbers

Most of the real world applications of domain and range do not include rational numbers (you can’t have a

fractional piece of an item or person) or negative numbers (such as time).

1. Which domain would be the most appropriate to use for a function that

predicts the number of household online-devices in terms of the number of

people in the household?

A. Integers

B. Whole Numbers

C. Rational Numbers

D. Irrational Numbers

2. A construction company uses the function f(p), where p is the number of

people working on a project, to model the amount of money it spends to

complete a project. A reasonable domain would be?

A. Positive Integers

B. Positive Real Numbers

C. Integers

D. Real Numbers

3. A store sells self-serve frozen yogurt sundaes. The function C(w) represents

the cost, in dollars, of a sundae weighing w ounces. An appropriate domain

would be?

A. Integers

B. Rational Numbers

C. Nonnegative Integers

D. Nonnegative Rational Numbers


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4. The function h(t) = -16t2 + 144 represents the height, h(t), in feet, of an object

from the ground t seconds after it dropped. A realistic domain for this function

is?

A. -3 ≤ t ≤ 3

B. 0 ≤ t ≤ 3

C. 0 ≤ t ≤ 144

D. All real numbers

5. A sample of 1,000 bacteria becomes infected with a virus. Each day, one

fourth of the bacteria sample dies due to the virus. A biologist studying the

bacteria models the population of the bacteria with the function P(t) =

1,000(0.75)t, where t is the time, in days. What is the range of this function in this

context?

A. Any real number such that t ≥ 0.

B. Any whole number such that t ≥ 0.

C. Any real number such that 0 ≤ P(t)≤ 1,000.

D. Any whole number such that 0 ≤ P(t)≤ 1,000.

6. Using Question 5 above, what would be a reason domain?

A. Any real number such that t ≥ 0.

B. Any whole number such that t ≥ 0.

C. Any real number such that 0 ≤ P(t)≤ 1,000.

D. Any whole number such that 0 ≤ P(t)≤ 1,000.

7. A flying disk is thrown into the air from a height of 25 feet at time t = 0. The

function hat models this situation is h(t) = -16tt + 75t + 25, where t is measured in

seconds and h is the height in feet. What values of t best describe the times when

the disk is flying in the air?

A. 0 < t < 5

B. 0 < t < 25

C. All real numbers

D. All positive integers


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Day 3 – Comparing Graphs and Tables of Functions

For the following functions, create a table and graph each function in a different color.

x y = 2x x y = x2 x y = 2x

0 0 0

1 1 1

2 2 2

3 3 3

4 4 4

5 5 5

Looking at the graphs above:

a) Which function shows a constant rate of change in its y values? How is this displayed on your graph?

b) Which function is largest between1 2x ? How is this displayed on your graph?

c) Eventually, which type of function shows the most rapid rate of growth in its y values? How is this

displayed on your graph?


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1. Consider the following graphed functions; fill in the blank comparing the functions with an inequality

symbol.

2. For which values of x is f(x) > g(x)? (Write the interval using set notation and interval notation.)

3. Write an inequality statement for all x of which function is greater.

A. f(x) = – x2 – 3 B. p(x) = 3x + 2

g(x) = ½ x + 2 q(x) = ½ x – 2

For all values of x,

f(x) g(x)

Determine the appropriate inequality symbol

(< or >) to put between the two functions.

For all values of x,

h(x) k(x)

Determine the appropriate inequality symbol

(< or >) to put between the two functions.

A. B.

A. B. C.

f(x) g(x) p(x) q(x)


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4. Consider the following functions.

5. A partial set of values is provided for two functions in each problem below. For all x≥0, which function

would most likely be greater. If the greater function changes determine the appropriate intervals.

M. Winking Unit 5-2 page 127

As x → , which function becomes the largest?



A. B. C.

A. B.

C. D.


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6. Consider the function and given the transformation determine which statements are true and which

are false.

a. p(x) < h(x) + 3 for all values of x.

b. p(x) < h(x – 3) for all values of x.

c. p(x – 4) < h(x) for all values of x.

d. – f(x) < g(x) for all values of x.

e. f(– x) < g(x) for all values of x.

f. f(x) > – g(x) for all values of x.

TRUE FALSE

Model (circle one):

TRUE FALSE

Model (circle one):

TRUE FALSE

Model (circle one):

TRUE FALSE

Model (circle one):

TRUE FALSE

Model (circle one):

TRUE FALSE

Model (circle one):



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Day 4 – Comparing Average Rate of Change

If you are asked to calculate the average rate of change on an interval without a graph, you will have to

come up with two points to calculate the slope.

You will get your two points by taking the bounds of your interval and substitute those x-values into your

equation to find the y-values. Then use the slope formula to calculate the slope.

Remember slope is: rise

run or 2 1

2 1

y y

x x

1. Find the average rate of change from x = – 1 to x = 2 for each of the functions below.

a. 𝑎(𝑥) = 2𝑥 + 3 b. 𝑏(𝑥) = 𝑥2 − 1 c. 𝑐(𝑥) = 2𝑥 + 1

d. Which function has the greatest average rate of change over the interval [ – 1, 2]?

2. Find the average rate of change on the interval [ 2, 5] for each of the functions below.

a. 𝑎(𝑥) = 2𝑥 + 1 b. 𝑏(𝑥) = 𝑥2 + 2 c. 𝑐(𝑥) = 2𝑥 − 1

d. Which function has the greatest average rate of change over the interval x = 2 to x = 5?

3. In general as x→, which function eventually grows at the fastest rate?

a. 𝑎(𝑥) = 2𝑥 b. 𝑏(𝑥) = 𝑥2 c. 𝑐(𝑥) = 2𝑥


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4. Find the average rate of change from x = – 1 to x = 2 for each of the continuous functions below based on

the partial set of values provided.

a. b. c.

d. Which function has the greatest average rate of change over the interval [ – 1, 2]?

5. Consider the table below that shows a partial set of values of two continuous functions. Based on any interval

of x provided in the table which function always has a larger average rate of change?

6. Find the average rate of change from x = 1 to x = 3 for each of the functions graphed below.

a. b. c.

d. Find an interval of x over which all three graphed functions above have the same average rate of change.


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Day 5 – Comparing Multiple Representations

When comparing different functions, it is helpful to think about what characteristics apply most to each type of

function. Use the chart below to help you compare the most important features of each function.

Linear Quadratic Exponential

Slope (Rate of Change)

Y-intercept (Starting Point)

Vertex

X-intercepts

Y-intercepts

Growth/Decay Rate

Y-intercept (Starting Point)

Scenario 1:

Two airplanes are in flight. The function f(x) = 400x + 1200 represents the altitude, f(x) of one airplane after x

minutes. The graph below represents the altitude of the second airplane.

a. Compare the starting altitudes of both airplanes.

b. Compare the rates of changes of both airplanes.

Scenario 2: Compare the two functions below:

Function A

A local newspaper began with a circulation of

1,300 readers in its first year. Since then, its

circulation has increased by 150 readers per

year.

Function B

The function g(x) = 225x + 950 represents the

circulation of another newspaper where g(x)

represents total subscriptions and x represents

the number of years since its first year.


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Scenario 3: Compare the two functions below:

Scenario 4:

Through comparison shopping, you have obtained the yearly insurance rate quotes from three car insurance

companies. The insurance rate quote is given as a function of the age of the customer. Using the information

presented below, which company has the lowest rate for a person in their fifties?

Wreck-for-Less Insurance: R(x) = 0.87x2 – 91.48x + 3185, where R(x) is the yearly premium in dollars and x is

the age in years?

Careless Insurance: A 51 year old customer pays a yearly premium of $900. For every 5 year increase or

decrease in age, the customer is charged an additional $300, then $500, then $700, and so on.

Fender & Bender Insurance:

Function A

The value of a car in dollars, f(x), depreciates

after each year, x. The following table shows

the value of a car for each of the first 4 years

after it was purchased.

Function B

The value of a second car is modeled by the

equation g(x) = 19,374(1-0.16)x, where g(x)

represents the value of the car x years after the

date it was purchased.


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Which of the following would be best modeled by a LINEAR, QUADRATIC, or EXPONENTIAL function.

a. A window cleaner is cleaning windows about half way up the Peachtree Plaza Hotel

in Downtown Atlanta. The cleaner didn’t properly clip one of his squeegee tools and it

fell from 400 feet up in the air. The height of the squeegee t seconds after it fell is given

below. What type of function (Linear, Quadratic, or Exponential) would best describe

the squeegee’s height as a function of time?

b. Joey’s shower fixture began leaking water. So, he called a plumber to help him fix the problem. The

plumber said he would charge $100 for making a house call and $60 for every hour he is there working on

the problem. What type of function (Linear, Quadratic, or Exponential) would best describe the amount

the plumber charges the customer? Can you write the function?

c. A pot of tea was brewed such that its temperature was 200˚F and then the stove was turned off. The pot

of tea slowly cooled back to the room temperature of 80˚F over a few hours. The data is shown below in

the table. What type of function (Linear, Quadratic, or Exponential) would best describe the

temperature?

d. A company named ‘Fone Faze Foundary’ designs the hardware for new smart phones and is offering a

new employee an initial salary of $40,000 a year and will get a raise of an additional $6,000 for each year

she works for the company. What type of function (Linear, Quadratic, or Exponential)

would best describe the employee’s salary based on the number of years that she works for

the company? Can you determine the function?

e. A company named ‘Phone Program Ring’ designs the software for new smartphones is offering a new

employee an initial salary of $40,000 a year and will get a raise of an increase of 12% each

year he works for the company. What type of function (Linear, Quadratic, or

Exponential)would best describe the employee’s salary based on the number of years that

he works for the company? Can you determine the function?



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Explain what each of the parameter of each model mean.

f. A student was marking the growth of a corn stalk plant and at least for the first several weeks the plant’s

height in inches could be described by the following model (where t is the time in weeks).

h(t) = 5(1.40) t

i) What does the 5 represent? ii) What does the 1.40 represent?

g. A person purchased a new car which depreciates in value. The car owner determined the value of the

car in dollars could be modeled by the following function (where t is in years after the car was

purchased).

v(t) = 23000(0.93) t

i) What does the 23000 represent? ii) What does the 0.93 represent?

h. A physical therapist charges an initial fee and then charges another amount per hour of therapy. The

charges could be described by the following function model (where t is in hours of therapy).

c(t) = 140 + 40t

i) What does the 40 represent? ii) What does the 140 represent?

i. A baseball is struck by a bat. The height in feet of the ball can described by the following function (where

t is in seconds after the ball was struck).

h(t) = – 16(t – 2.4) 2 + 70

i) What does the 2.4 represent? ii) What does the 70 represent?


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Create a function model for each of the following:

a. Some kids are selling lemonade for $1.50 per cup at a high school baseball game. They spent $14 on

all of the items needed for the lemonade stand (cups, lemonade, tablecloth, sign, etc.). Create a

function that would represent their profit based on the number of cups of lemonade they sold.

b. A first year teacher is paid $38,000. Each year she is paid an additional 5% over

the previous year. Create function that would represent the teacher’s salary

based on the number of years that the teacher worked.

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Unit 5: Comparing Linear, Quadratic, & Exponential …...Algebra 1 Unit 5: Comparing Linear, Quadratic, and Exponential Functions Notes 3 Identifying Types of Functions from a Table