UNIT 5 • QUADRATIC FUNCTIONS Lesson 6: Analyzing Quadratic ...lilliepad.pbworks.com/w/file/fetch/74659820/AG Walch U5 L6Pt3 Exa… · Through comparison shopping, you have obtained

UNIT 5 • QUADRATIC FUNCTIONSLesson 6: Analyzing Quadratic Functions

Instruction

CCGPS Analytic Geometry Teacher Resource U5-330

© Walch Education

IntroductionRelationships can be presented through different forms: tables, equations, graphs, and words. This section explores how to compare functions presented in all four forms.

Key Concepts

• Recall that for a quadratic function in standard form, f(x) = ax2 + bx + c, the value of a determines whether the quadratic has a maximum or a minimum.

• If a is negative, the quadratic function will have a maximum.

• If a is positive, the quadratic function will have a minimum.

• Recall that linear functions have a constant slope.

• When x increases by 1, y increases by a constant value.

• A linear function will either increase, decrease, or remain constant.

• The change in y when x increases by 1 is called a first difference.

• Recall that exponential functions either increase or decrease.

• There is a constant multiple in the rate of change between the y-values.

• A growing exponential function will eventually exceed a linear or quadratic function.

• A quadratic function changes from increasing to decreasing, or vice versa.

• There is neither a constant rate of change nor a constant multiple in a quadratic function.

• In a quadratic model, the change in the first differences is constant.

• The change in first differences is called a second difference.

Prerequisite Skills

This lesson requires the use of the following skills:

• finding the vertex of a parabola

• finding the x-intercepts of a parabola

• graphing a parabola

• writing an equation for a simple exponential function


Instruction

CCGPS Analytic Geometry Teacher Resource © Walch EducationU5-331

• Compare the y-values of the vertices of quadratic functions to determine the greatest maximum or least minimum values between two or more quadratic functions.

Common Errors/Misconceptions

• misidentifying the appropriate type of function (linear, quadratic, or exponential)

• thinking that an equation with a higher y-intercept has a higher maximum value


Instruction


© Walch Education

Example 1

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 the youngest drivers?

• Wreck-for-Less Insurance: R(x) = 0.87x2 – 91.48x + 3185.16, 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 change in age, younger or older, the customer is charged an additional $300, then $500, then $700, and so on.

• Fender and Bender Insurance:

Age of customer Under 24 25–29 30–39 40–49 50–59 Over 59

Insurance rate ($) 1,750 1,500 1,125 916 875 916

1. Verify that the vertex for each insurance company rate is the minimum value.

For Wreck-for-Less, the value of a is positive, implying that it is an upward-opening parabola.

The rates for Careless Insurance increase as the customer’s age moves farther away from age 51, so the value at age 51 must be a minimum value. Therefore, the parabola opens upward.

The table for Fender and Bender Insurance decreases then increases, so it is also an upward-opening parabola.

Guided Practice 5.6.3


Instruction


2. Find the vertex for each insurance company.

Find the vertex of the given Wreck-for-Less function using b

af

b

a2,

2

− −

.

xb

a2=−

Formula to find the x-value of the vertex

x( 91.48)

2(0.87)=− −

Substitute 0.87 for a and –91.48 for b.

x =91 48

1 74

.

. Simplify.

x ≈ 52.57

Use the given function to find fb

a2

−

.

R(x) = 0.87x2 – 91.48x + 3185.16 Original equation

R(52.57) = 0.87(52.57)2 – 91.48(52.57) + 3185.16Substitute 52.57 for x.

R(52.57) ≈ 780.39 Simplify.

The y-value of the vertex is about 780.39.

The vertex for Wreck-for-Less is approximately (52.57, 780.39).

The table for Fender and Bender shows that the vertex is found within the 50–59 age group at a rate of $875.

The minimum value for Careless Insurance is (51, 900), since the rate increases as the age moves away from 51 in either direction.

3. Determine the company with the cheapest rate.

Compare the y-values of each vertex.

780.39 < 875 < 900

The minimum premium for Wreck-for-Less is the cheapest, at about $780.39.


Instruction


© Walch Education

Example 2

Three students are shooting wads of paper with a rubber band, aiming for a trash can in the front of the room. The height of each student’s paper wad in feet is given as a function of the time in seconds. Which student’s paper wad flies the highest?

• The path of Alejandro’s paper wad is modeled by the equation f(x) = –x2 + 2x + 7.

• Melissa’s paper wad is estimated to reach the heights shown in the table below.

x 0 2 3 4y 3 6 7 6

• After 3 seconds, Connor’s paper wad achieves a maximum height of 6.5 feet above the floor.

1. Determine if each function represents a quadratic.

Alejandro’s path is represented by a quadratic function written in standard form.

Melissa’s table has symmetry about x = 3, which implies a quadratic relationship.

In general, a projectile follows a parabolic path, gaining height until reaching a maximum, then descending. We can assume Connor’s paper wad will follow a parabolic path.

2. Verify that the extremum for each function is the maximum value of each function.

The value of a in the equation for Alejandro’s paper wad is a negative value. Therefore, the function has a maximum value.

The table for Melissa’s paper wad increases then decreases, so its path will have a maximum value.

The maximum height of the path of Connor’s paper wad is given.


Instruction


3. Determine the vertex of each function.

Find the vertex of Alejandro’s function using b

af

b

a2,

2

− −

.

x = b

a2

−

Formula to find the x-coordinate of the vertex

x = (2)

2( 1)

−−

Substitute –1 for a and 2 for b.

x = 1 Simplify.

Use the given function to find fb

a2

−

= f(1).

f(x) = –x2 + 2x + 7 Original equation

f(1) = –(1)2 + 2(1) + 7 Substitute 1 for x.

f(1) = –1 + 2 + 7 Simplify.

f(1) = 8

The vertex is at (1, 8).

Find the vertex for Melissa’s paper wad using symmetry. Her paper wad’s function is symmetric about the vertex at (3, 7).

The vertex for Connor’s paper wad is given as (3, 6.5).

4. Use the vertices to determine whose paper wad goes the highest.

Compare the y-values of the vertices.

8 > 7 > 6.5

Alejandro’s paper wad flies the highest, at 8 feet.


Instruction


© Walch Education

Example 3

Which of the following quadratic functions has a vertex with a larger y-value: f(x) = 2x2 – 12x + 25, or g(x) as presented in the table?

x –4 –3 –2 0 2

g(x) 7 8 7 –1 –17

1. Verify for each function whether the vertex is a minimum or maximum.

For f(x), a is positive, which means that the vertex is a minimum.

For g(x), as x increases, the y-values first increase and then decrease. This indicates the vertex is a maximum.

2. Find the vertex for each function.

For f(x), complete the square of f(x) = 2x2 – 12x + 25.

f(x) = 2(x2 – 6x) + 25Factor 2 out of each term containing an x.

f(x) = 2(x2 – 6x + 9) + 25 – 2(9) Add and subtract 2(9).

f(x) = 2(x – 3)2 + 25 – 2(9)Rewrite the perfect square trinomial as a squared binomial.

f(x) = 2(x – 3)2 + 7 Simplify.

From this form, we can see that the vertex of f(x) is (3, 7).

For g(x), the points are symmetric about x = –3. Therefore, the line of symmetry contains (–3, 8), which must be the vertex.

3. Determine which vertex has a larger y-value.

Compare the y-values of each function.

7 < 8

The function g(x) has a vertex with a larger y-value.


Instruction


Example 4

You are considering investing $5,000 in one of two mutual funds. The first fund will pay $500 each year. The second fund is predicted to have end-of-year balances as shown in the following table. Which fund should you choose if you want to withdraw your money after 5 years? Which fund should you choose if you want to invest the money for 10 years?

x (year) 0 1 2 3 4 5I(x) ($) 5,000 5,200 5,500 5,900 6,400 7,000

1. Determine if the first fund is quadratic.

The first fund has a constant rate of change of $500 each year; therefore, it is linear.

2. Determine if the second fund is quadratic.

There is no constant rate of change; therefore, the function is not linear.

In the first year, the fund increases by $100, then $200, then $300, and so on.

There is no constant multiple from one end-of-year balance to the next.

This means it is not exponential.

If we find the second differences by subtracting $100 from $200, $200 from $300, and so on, we find that the second differences are constant.

Therefore, the function is quadratic.

3. Determine which fund you should choose if you want to withdraw your money after 5 years.

At the end of the fifth year, the first fund would contain $5000 + $500(5) = $7500.

The second fund contains only $7,000 after 5 years.

If you want to withdraw your money after 5 years, you should choose the first fund.


Instruction


© Walch Education

4. Determine which fund should you choose if you want to invest the money for 10 years.

The first fund increases by $500 each year, but the second fund increases by more and more each year.

At the end of the fifth year, the second account holds only $500 less than the first fund, but the balance will increase by $600 to exceed the first fund by the end of the sixth year.

Based on this information, the second fund would have more money after 10 years. This can be verified. At the end of the tenth year, the first fund would contain $5000 + $500(10) = $10,000, and the second fund would continue to increase following the same pattern and contain $11,500.

Example 5

Suppose that you have been offered a position at a prestigious company. You may choose how your salary is paid. Option 1 is described by the quadratic equation S(x) = 2500x2 + 2500x + 60,000, where x is the number of years you are with the company and S(x) is the yearly salary in dollars. Option 2 has a starting yearly salary of $35,000, but you will get a 25% raise each year. Make a table of values for each salary and graph both functions on a coordinate plane. If you plan to work for this company for 5 years, which option should you choose? If you plan to work for this company until you retire at age 70, which option should you choose?

1. Write an equation for Option 2.

Since you have been given the equation for Option 1 already, start by determining the equation for Option 2. Option 2 is an exponential function because it increases by the same factor (125%) each year. We can write the function by multiplying the initial salary by 1.25 raised to the number of years.

T(x) = 35,000(1.25)x


Instruction


2. Make a table of values for each salary.

Salaries listed in the table that follows are in thousands of dollars.

Year 1 2 3 4 5 6 7 8 9 10 11Option 1 65 75 90 110 135 165 200 240 285 335 390Option 2 43.8 54.6 68.4 85.4 106.8 133.5 166.9 208.6 260.8 326.0 407.5

3. Graph both functions on the same coordinate plane.

Plot the points from the table and connect them with a smooth curve. Label each function clearly.

Years of employment

20 4 6 8 10 12 14 16 18

Sala

ry in

dol

lars

50,000

100,000

150,000

200,000

250,000

300,000

350,000

400,000

450,000

500,000

550,000

y = 35,000

y = 60,000

Option 1

Option 2

(10.4, 356,410)

y

x

4. Determine which option you should choose if you plan to work for this company for only 5 years.

The table of values shows us that Option 1 yields a salary of $135,000 in the fifth year, compared to $106,800 for Option 2.

If you plan to work for this company for only 5 years, you should select Option 1.


Instruction


© Walch Education

5. Determine which option you should choose if you plan to work for this company until retirement at age 70.

We can see from the graph that although Option 2 pays less for the first 10 years, from year 11 on it is the better choice.

Exponential functions eventually exceed quadratic functions.

If you plan to work for this company until retirement, you should select Option 2.

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UNIT 5 • QUADRATIC FUNCTIONS Lesson 6: Analyzing Quadratic ...lilliepad.pbworks.com/w/file/fetch/74659820/AG Walch U5 L6Pt3 Exa… · Through comparison shopping, you have obtained