Math Analysis - Chapter 3 Mathlete: _______________________________

Date Assigned

Section Homework (due the next day)

✔

Thu 11/3

3.1 Graphs of Quadratic Functions (day 1) You need graph paper!

Fri 11/4

p.225: 7-17odd, 19-26, 27-31odd

Mon 11/7

3.1 Graphs of Quadratic Functions (day 2) You need graph paper!

p.226: 33-39odd, 51-61odd

Tue 11/8

3.1 Graphs of Quadratic Functions (day 3) Applications

p.227: 73, 75, 85, 95a-d

Wed 11/9

3.2 Quadratic Equations - Factoring and CTS (day 1)

p.241: 19-39odd

Thu 11/10

3.2 Quadratic Equations - Quadratic Formula (day 2)

p.242: 41-47odd, 55-63odd, 79, 83, 99

Fri 11/11

Review 3.1 and 3.2 Review Sheets

Mon 11/14

Quiz 3.1 and 3.2

Tue 11/15

Review for Final Exam

Chapter P: Sections 4-8 p.64-65: 53, 55, 63, 65, 67-75odd, 85-93odd, 95, 105

Wed 11/16

Review for Final Exam

Chapter 1: Sections 1-5 p.136-138: 3, 7, 9, 13, 41-47odd, 67, 69

Thu 11/17

Review for Final Exam

Chapter 2: Sections 1-4, 6 p.207: 5-11odd, 17, 27-35odd, 45, 77-80,83, 85, 101, 103, 105-107

Fri 11/18

Review for Final Exam (1st hour Final)

Chapter 3: Sections 1-2 p.280: 3-13odd, 21-39odd

Tue 11/21

Math Final (2nd and 3rd hour Finals)

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3.1 Graphs of Quadratic Functions (Day 1) Warm up: What number must be added to write the expression in the form of (x + b)2 ? x2 - 14x + ________

Definition of a Quadratic Function The graph of a quadratic function is a parabola. A quadratic function can be expressed in two forms:

1. Standard Form The vertex is The axis of symmetry is __________

2. Vertex Form The vertex is __________ The axis of symmetry is __________

Notice that both forms have the coefficient “a”. This number “a” will be the same no matter what form you’re in. If a>0, then the parabola opens ________________, and the vertex is the ________________ point. If a<0, then the parabola opens ________________, and the vertex is the ________________ point. The minimum or maximum is the __________________________

Example 2: Sketching the Graph of f(x)= ax2 Graph , , . Find the domain and range of each function.(x)f = x2 (x) xg = 2 2 (x) −h = x2

x (x)f = x2 (x) xg = 2 2 (x)h = − x2

-2

-1

0

1

2

domain

range

2

Graph each pair of functions on the same set of coordinate axes, and find the domain and range of each function.

10. , (x)f = x2 (x) xg = 3 2

x (x)f = x2 (x) xg = 3 2

-2

-1

0

1

2

domain

range

14. , (x) −f = x2 + 2 (x) −g = x2 − 2

x (x) −f = x2 + 2 (x) −g = x2 − 2

-2

-1

0

1

2

domain

range

18. , (x) − x )f = ( − 4 2 (x) − x )g = ( + 4 2

x (x) − x )f = ( − 4 2 x (x) − x )g = ( + 4 2

2 -6

3 -5

4 -4

5 -3

5 -2

domain domain

range range

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Example 3: Sketching a quadratic function using transformations Use transformations to graph the quadratic function. Find the vertex and state whether it is a minimum or maximum. 1. Let’s do the transformations one at a time:(x) − (x )f = 2 − 1 2 + 3

f(x) = x2

(x) x f = − 2 2

(x) (x )f = − 2 − 1 2

(x) (x )f = − 2 − 1 2 + 3

Conclusion: opens __________, and the ____________ point on the graph is the vertex ______.(x) (x )f = − 2 − 1 2 + 3 Thus, the ____________ . The x and y coordinates of the vertex correspond to the horizontal and vertical shifts. In 2-3, graph the parent function and the given function on the same set of axes.

28. (x) x )f = ( − 3 2 + 2

30. (s) − s )g = ( − 2 2 + 2

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3.1 Graphs of Quadratic Functions (Day 2) Standard Form: ______________________________ Vertex Form: ______________________________ vertex: vertex: axis of symmetry: axis of symmetry:

Note: The lead coefficient “a” is the same in both forms.

Example 4: Converting from Standard form to Vertex Form Write each quadratic function in vertex form. Also find the vertex and determine whether it is a maximum or minimum point. ex. (x) x xf = 2 2 − 4 + 5 38. (x)g = − x2 + x− 7 40. (x) x xf = − 2 2 + 8 + 3

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Example 5: Sketching a Quadratic Function

(x) x xf = 2 2 − 3 − 2 vertex axis of symmetry y-intercept (plug in x=0) symmetry point increasing (state in terms of x) decreasing (state in terms of x) domain range

52. (x) xf = x2 + 6 − 7 vertex axis of symmetry y-intercept (plug in x=0) symmetry point increasing (state in terms of x) decreasing (state in terms of x) domain range

54. (x) − x xg = 4 2 − 8 + 5 vertex axis of symmetry y-intercept (plug in x=0) symmetry point increasing (state in terms of x) decreasing (state in terms of x) domain range

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3.1 Graphs of Quadratic Functions (Day 3) Warm Up: 92. Which of the following points lie on the parabola associated with the function f(s) = -s 2 +6 ? a. (3, -1) b. (0, 6) c. (2, 1) Today we will be given a specific perimeter, and asked to find an area function based on that information. Example 1: Deriving an Expression for Area Traffic authorities have 100 feet of rope to cordon off a rectangular region to form a ticket arena for concert goers who are waiting to purchase tickets.

a. Express the area of this rectangular region as a function of the length of just one of the four side of the region. b. Find the dimensions of the enclosed area that will maximize area. c. Find the maximum area.

72. A rectangular garden is to be enclosed with a fence on three of its sides and a brick wall on the fourth side. If 100 feet of fencing material is available, what dimensions will yield the maximum area?

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74. Physics: The height of a ball that is thrown directly upward from a point 200 feet above the ground with an initial velocity of 40 feet per second is given by , where t is the amount of time elapsed since the ball was thrown.(t) − 6t 0t 00h = 1 2 + 4 + 2 Here, t is in seconds and h(t) is in feet.

a. Sketch the graph of h. b. When will the ball reach its maximum height? c. What is the maximum height?

76. A carpenter wishes to make a rain gutter with a rectangular cross-section by bending up a flat piece of metal that is 18 feet long and 20 inches wide. The top of the rain gutter is open.

a. Write an expression for the cross-sectional area in terms of x, the length of metal that is bent upward. b. How much metal has to be bent upward to maximize the cross sectional area? What is the maximum cross-sectional

area?

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3.2 Quadratic Equations (day 1)

Definition of a Quadratic Equation A quadratic equation is an equation that can be written in the standard form where a, b, and c are real numbers and a≠0.

One of the simplest ways to solve a quadratic equation is by factoring. We need the Zero Product Property to solve equations by factoring.

Zero Product Property If a product of real numbers is zero, then at least one of the factors is zero.

Example 1: Solving an Equation by Factoring 9. 5x2 − 2 = 0 11. x 2x2 − 7 + 1 = 0 13. x 2− 3 2 + 1 = 0 15. x6 2 − x− 2 = 0 17. x4x2 − 4 + 1 = 0

Definition of Real Zeros The real number values of x at which f(x)=0 are called the real zeros of the function f. Graphically, the real zeros are the x-intercepts of a function. To find an x-intercept, set f(x)=0 or y=0, and then solve for x.

A quadratic function may have 1, 2, or no x-intercepts. Sketch these 3 possibilites.

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Find real zeros or x-intercepts: 1. Set the function equal to zero f(x)=0. 2. Factor. 3. Use Zero Product Property: set each factor equal to zero and solve for x. 4. The real zeros are the resulting x-values. The x-intercepts are (x, 0).

Example 2: Relating x-intercepts to Zeros Factor to find the x-intercepts of the parabola described by the quadratic function. Also find the real zeros of the function. 20. (t) − tf = t2 + 4 22. (x) xf = x2 + 4 + 4 24. (x) xf = 6 2 − x− 2 26. (t) − t 0th = 3 2 + 1 − 8 Example 3: Given Zeros, Find the Quadratic Function (Working backwards) Find a possible expression for a quadratic function f(x) with having the given zeros. 1. x=-3 and x=1 28. x=-2 and x=4 30. x=-5 is the only zero 32. x=0.4 and x=0.8

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Not all quadratic equations can be solved by factoring. Another method for solving a quadratic equation is completing the square (CTS). CTS can be used to solve any quadratic equation. We first need the following rule regarding square roots:

Principle of Square Roots If x2 = c, where c≥0, then _________________ or _______________ Example: If x2 = 25, then

Example 5: Solve the Quadratic Equation by Completing the Square x x 3 2 − 6 − 1 = 0

1. Start with the equation.

2. Move the constant to the right side.

3. Divide by “a”, which is 3 in this case,

to get the coefficient of x2 equal to 1.

4. Complete the square by taking . (2b)2

Remember to add this quantity to both sides.

5. Write the left side as a perfect square.

6. Use the principle of square roots.

7. Solve for x.

8. Simplify the radical.

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3.2 Quadratic Equations (day 2) Warm up: Solve the quadratic equation by completing the square. x x 2 2 − 4 − 1 = 0

Today we will learn another method which can be used to solve any quadratic equation.

The Quadratic Formula The solutions of ax2 + bx + c = 0 can be found using the quadratic formula: These solutions are also known as the zeros of ax2 + bx + c = 0.

To solve a quadratic equation, perform these steps:

1. Write the equation in standard form ax2 + bx + c = 0. 2. If possible, factor the left side to find the solution(s). 3. If factoring is not possible, use the quadratic equation or complete the square.

Example 6: Solve the Equation using the Quadratic Formula Solve the equation using the quadratic formula. Find the real zeros and the x-intercepts. a. x x 2 2 − 4 − 1 = 0 b. x x − 4 2 + 3 + 2

1 = 0

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Example 9: Quadratic Model for the Height of an Object in Flight The height of a ball thrown vertically upward from a point 80 feet above the ground with a velocity of 40 feet per second is given by , where t is the time in seconds since the ball was thrown and h(t) is in feet.(t) − 6t 0t 0 h = 1 2 + 4 + 8

a. When will the ball be 50 feet above the ground? b. When will the ball reach the ground? c. For what values of t does this problem make sense (from a physical standpoint)?

84. Construction: A rectangular plot situated along a river is to be fenced in. The side of the plot bordering the river will not need fencing. The builder has 100 feet of fencing available.

a. Draw a picture. Write an equation relating the amount of fencing material available to the lengths of the three sides of the plot that are to be fenced.

b. Use the equation in part a to write an expression for the width of the enclosed region in terms of its length. c. Write an expression for the area of the plot in terms of its length. d. Find the dimensions that will yield the maximum area.

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