Introduction Relationships can be presented through different forms: tables, equations, graphs, and...

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

1

5.6.3: Comparing Properties of Functions Given in Different 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. 2


Key Concepts, continued• 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.

3


Key Concepts, continued• 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.• 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.

4


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

5


Guided Practice

Example 2Three 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 functions for each of the three students are described on the next slide.

6


Guided Practice: Example 2, continued• 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.

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

7


x 0 2 3 4

y 3 6 7 6

Guided Practice: Example 2, continued

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.

8



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.

9



3. Determine the vertex of each function. Find the vertex of Alejandro’s function using

10


Formula to find the x-coordinate of the vertex

Substitute –1 for a and 2 for b.

x = 1 Simplify.


Use the given function to find

The vertex is at (1, 8).11


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

Guided Practice: Example 2, continuedFind 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).

12



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.

13


✔


14


http://www.walch.com/ei/00267

Guided Practice

Example 3Which 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?

15


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.

16



2. Find the vertex for each function. For f(x), complete the square of f(x) = 2x2 – 12x + 25.

17


f(x) 2(x2 – 6x) + 25 Factor 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.

Guided Practice: Example 3, continuedFrom 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.

18



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.

19


✔


20


http://www.walch.com/ei/00268

Introduction Relationships can be presented through different forms: tables, equations, graphs, and...

Documents

Transcript of Introduction Relationships can be presented through different forms: tables, equations, graphs, and...