Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions

Июль 21, 2022

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Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions

Содержание

2. Recognize characteristics of parabolas. Graph parabolas. Determine a quadratic function’s minimum or maximum value. Solve problems
3. The Standard Form of a Quadratic Function The quadratic function is in standard form. The graph
4. Graphing Quadratic Functions with Equations in Standard Form To graph 1. Determine whether the parabola opens
5. Graphing Quadratic Functions with Equations in Standard Form (continued) To graph 4. Find the y-intercept by
6. Example: Graphing a Quadratic Function in Standard Form Graph the quadratic function Step 1 Determine how
7. Example: Graphing a Quadratic Function in Standard Form (continued) Graph: Step 3 Find the x-intercepts by
8. Example: Graphing a Quadratic Function in Standard Form (continued) Graph: Step 4 Find the y-intercept by
9. Example: Graphing a Quadratic Function in Standard Form Graph: The parabola opens downward. The x-intercepts are
10. The Vertex of a Parabola Whose Equation is Consider the parabola defined by the quadratic function
11. Graphing Quadratic Functions with Equations in the Form To graph 1. Determine whether the parabola opens
12. Graphing Quadratic Functions with Equations in the Form (continued) To graph 4. Find the y-intercept by
13. Example: Graphing a Quadratic Function in the Form Graph the quadratic function Step 1 Determine how
14. Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 2 (continued) find the vertex.
15. Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 3 Find the x-intercepts by
16. Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 4 Find the y-intercept by
17. Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 5 Graph the parabola. The
18. Minimum and Maximum: Quadratic Functions Consider the quadratic function 1. If a > 0, then f
19. Minimum and Maximum: Quadratic Functions (continued) Consider the quadratic function In each case, the value of
20. Example: Obtaining Information about a Quadratic Function from Its Equation Consider the quadratic function Determine, without
21. Example: Obtaining Information about a Quadratic Function from Its Equation (continued) Consider the quadratic function Find
22. Example: Obtaining Information about a Quadratic Function from Its Equation Consider the quadratic function Identify the
23. Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions 1. Read the problem carefully and
24. Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions (continued) 4. Calculate If a >
25. Example: Maximizing Area You have 120 feet of fencing to enclose a rectangular region. Find the
26. Example: Maximizing Area (continued) Step 2 Express this quantity as a function in one variable. We
27. Example: Maximizing Area (continued) Step 3 Write the function in the form Step 4 Calculate a
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Слайд 2

Recognize characteristics of parabolas.
Graph parabolas.
Determine a quadratic function’s minimum or maximum

value.
Solve problems involving a quadratic function’s minimum or maximum value.

Objectives:

Слайд 3

The Standard Form of a Quadratic Function
The quadratic function
is in standard

form. The graph of f is a parabola whose vertex is the point (h, k). The parabola is symmetric with respect to the line x = h. If a > 0, the parabola opens upward; if a < 0, the parabola opens downward.

Слайд 4

Graphing Quadratic Functions with Equations in Standard Form
To graph
1. Determine whether

the parabola opens upward or downward. If a > 0, it opens upward. If a < 0, it opens downward.
2. Determine the vertex of the parabola. The vertex is (h, k).
3. Find any x-intercepts by solving f(x) = 0. The function’s real zeros are the x-intercepts.

Слайд 5

Graphing Quadratic Functions with Equations in Standard Form (continued)
To graph
4. Find

the y-intercept by computing f(0).
5. Plot the intercepts, the vertex, and additional points as necessary. Connect these points with a smooth curve that is shaped like a bowl or an inverted bowl.

Слайд 6

Example: Graphing a Quadratic Function in Standard Form
Graph the quadratic function
Step

1 Determine how the parabola opens.
a = –1, a < 0; the parabola opens downward.
Step 2 Find the vertex.
The vertex is at (h, k). Because h = 1 and k = 4, the parabola has its vertex at (1, 4)

Слайд 7

Example: Graphing a Quadratic Function in Standard Form (continued)
Graph:
Step 3 Find

the x-intercepts by solving f(x) = 0.
The x-intercepts are (3, 0) and (–1, 0)

Слайд 8

Example: Graphing a Quadratic Function in Standard Form (continued)
Graph:
Step 4 Find

the y-intercept by computing f(0).
The y-intercept is (0, 3).

Слайд 9

Example: Graphing a Quadratic Function in Standard Form
Graph:
The parabola opens

downward.
The x-intercepts are
(3, 0) and (–1, 0).
The y-intercept is (0, 3).
The vertex is (1, 4).

Слайд 10

The Vertex of a Parabola Whose Equation is
Consider the parabola

defined by the quadratic function
The parabola’s vertex is
The x-coordinate is
The y-coordinate is found by substituting the
x-coordinate into the parabola’s equation and evaluating the function at this value of x.

Слайд 11

Graphing Quadratic Functions with Equations in the Form
To graph
1. Determine

whether the parabola opens upward or downward. If a > 0, it opens upward. If a < 0, it opens downward.
2. Determine the vertex of the parabola. The vertex is
3. Find any x-intercepts by solving f(x) = 0. The real solutions of are the x-intercepts.

Слайд 12

Graphing Quadratic Functions with Equations in the Form (continued)
To graph
4. Find

the y-intercept by computing f(0). Because
f(0) = c (the constant term in the function’s equation), the y-intercept is c and the parabola passes through
(0, c).
5. Plot the intercepts, the vertex, and additional points as necessary. Connect these points with a smooth curve.

Слайд 13

Example: Graphing a Quadratic Function in the Form
Graph the quadratic function
Step

1 Determine how the parabola opens.
a = –1, a < 0, the parabola opens downward.
Step 2 Find the vertex.
The x-coordinate of the vertex is
a = –1, b = 4, and c = 1

Слайд 14

Example: Graphing a Quadratic Function in the Form (continued)
Graph:
Step 2 (continued) find

the vertex.
The coordinates of the vertex are
We found that x = 2 at the vertex.
The coordinates of the vertex are (2, 5).

Слайд 15

Example: Graphing a Quadratic Function in the Form (continued)
Graph:
Step 3 Find the

x-intercepts by solving f(x) = 0.
The x-intercepts are (–0.2, 0) and (4.2, 0).

Слайд 16

Example: Graphing a Quadratic Function in the Form (continued)
Graph:
Step 4 Find the

y-intercept by computing f(0).
The y-intercept is (0, 1).

Слайд 17

Example: Graphing a Quadratic Function in the Form (continued)
Graph:
Step 5 Graph

the parabola.
The x-intercepts are
(–0.2, 0) and (4.2, 0).
The y-intercept is (0, 1).
The vertex is (2, 5).
The axis of symmetry is x = 2.

Слайд 18

Minimum and Maximum: Quadratic Functions
Consider the quadratic function
1. If a >

0, then f has a minimum that occurs at
This minimum value is
2. If a < 0, then f has a maximum that occurs at
This maximum value is

Слайд 19

Minimum and Maximum: Quadratic Functions (continued)
Consider the quadratic function
In each case,

the value of gives the location
of the minimum or maximum value.
The value of y, or gives that minimum or maximum value.

Слайд 20

Example: Obtaining Information about a Quadratic Function from Its Equation
Consider the

quadratic function
Determine, without graphing, whether the function has a minimum value or a maximum value.
a = 4; a > 0.
The function has a minimum value.

Слайд 21

Example: Obtaining Information about a Quadratic Function from Its Equation (continued)
Consider

the quadratic function
Find the minimum or maximum value and determine where it occurs.
a = 4, b = –16, c = 1000
The minimum value of f is 984.
This value occurs at x = 2.

Слайд 22

Example: Obtaining Information about a Quadratic Function from Its Equation
Consider the

quadratic function
Identify the function’s domain and range (without graphing).
Like all quadratic functions, the domain is
We found that the vertex is at (2, 984).
a > 0, the function has a minimum value at the vertex.
The range of the function is

Слайд 23

Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions
1. Read

the problem carefully and decide which quantity is to be maximized or minimized.
2. Use the conditions of the problem to express the quantity as a function in one variable.
3. Rewrite the function in the form

Слайд 24

Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions (continued)
4.

Calculate If a > 0, f has a minimum at
This minimum value is If a < 0, f has a
maximum at This maximum value is
5. Answer the question posed in the problem.

Слайд 25

Example: Maximizing Area
You have 120 feet of fencing to enclose a

rectangular region. Find the dimensions of the rectangle that maximize the enclosed area. What is the maximum area?
Step 1 Decide what must be maximized or minimized
We must maximize area.

Слайд 26

Example: Maximizing Area (continued)
Step 2 Express this quantity as a function

in one variable.
We must maximize the area of the rectangle, A = xy.
We have 120 feet of fencing, the perimeter of the rectangle is 120. 2x + 2y = 120
Solve this equation for y:

Слайд 27

Example: Maximizing Area (continued)
Step 3 Write the function in the form
Step

4 Calculate
a < 0, so the function has a maximum at this value.
This means that the area, A(x), of a rectangle with perimeter 120 feet is a maximum when one of the rectangle’s dimensions, x, is 30 feet.

Chapter 3. Polynomial and Rational Functions. 3.1 Quadratic Functions

Содержание

Recognize characteristics of parabolas.Graph parabolas.Determine a quadratic function’s minimum or maximum

The Standard Form of a Quadratic FunctionThe quadratic functionis in standard

Graphing Quadratic Functions with Equations in Standard FormTo graph1. Determine whether

Graphing Quadratic Functions with Equations in Standard Form (continued)To graph4. Find

Example: Graphing a Quadratic Function in Standard FormGraph the quadratic functionStep

Example: Graphing a Quadratic Function in Standard Form (continued)Graph:Step 3 Find

Example: Graphing a Quadratic Function in Standard Form (continued)Graph:Step 4 Find

Example: Graphing a Quadratic Function in Standard FormGraph: The parabola opens

The Vertex of a Parabola Whose Equation is Consider the parabola

Graphing Quadratic Functions with Equations in the Form To graph1. Determine

Graphing Quadratic Functions with Equations in the Form (continued)To graph4. Find

Example: Graphing a Quadratic Function in the Form Graph the quadratic functionStep

Example: Graphing a Quadratic Function in the Form (continued) Graph:Step 2 (continued) find

Example: Graphing a Quadratic Function in the Form (continued) Graph:Step 3 Find the

Example: Graphing a Quadratic Function in the Form (continued) Graph:Step 4 Find the

Example: Graphing a Quadratic Function in the Form (continued) Graph: Step 5 Graph

Minimum and Maximum: Quadratic FunctionsConsider the quadratic function1. If a >

Minimum and Maximum: Quadratic Functions (continued)Consider the quadratic functionIn each case,

Example: Obtaining Information about a Quadratic Function from Its EquationConsider the

Example: Obtaining Information about a Quadratic Function from Its Equation (continued)Consider

Example: Obtaining Information about a Quadratic Function from Its EquationConsider the

Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions1. Read

Strategy for Solving Problems Involving Maximizing or Minimizing Quadratic Functions (continued)4.

Example: Maximizing AreaYou have 120 feet of fencing to enclose a

Example: Maximizing Area (continued)Step 2 Express this quantity as a function

Example: Maximizing Area (continued)Step 3 Write the function in the formStep

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