[PDF] - Key Terms Solve Quadratic Equations by Factoring Application of PDF Document

SLIDE 1

Slide 1 / 222 Slide 2 / 222

Algebra II

Quadratic Functions

www.njctl.org 2014-10-14

Slide 3 / 222 Table of Contents

Explain Characteristics of Quadratic Functions Graph Quadratic Functions Solve Quadratic Equations by Graphing Solve Quadratic Equations by Factoring Solve Quadratic Equations Using Square Roots

Key Terms Application of Zero Product Property

Solve Quadratic Equations by Completing the Square Solve Quadratic Equations using the Quadratic Formula The Discriminant Combining Transformations (review) Vertex Form More Application Problems using Quadratics

click on the topic to go to that section

Slide 4 / 222

Key Terms

Return to Table of Contents

Slide 5 / 222 Slide 6 / 222

SLIDE 2

Slide 7 / 222

Zero(s) of a Function: An x value that makes the function equal zero. Also called a "root," "solution" or "x-intercept"

Key Terms Slide 8 / 222

Minimum Value: The y-value of the vertex if a > 0 and the parabola opens upward Maximum Value: The y-value of the vertex if a < 0 and the parabola opens downward Vertex: The highest or lowest point on a parabola.

Key Terms Slide 9 / 222

Axis of symmetry: The vertical line that divides a parabola into two symmetrical halves

Key Terms Slide 10 / 222

Explain Characteristics

f Quadratic

Functions

Return to Table of Contents

Slide 11 / 222

Remember: A quadratic equation is any equation that can be written in the form ax2 + bx + c =0 Where a, b, and c are real numbers and a ≠ 0 Question 1: Is a quadratic equation? Question 2: Is a quadratic equation?

Characteristics of Quadratics Slide 12 / 222

The form ax2 + bx + c = 0 is called the standard form of a quadratic equation. The standard form is not unique. For example, x2 - x + 1 = 0 can also be written -x2 + x - 1 = 0. Also, 4x2 - 2x + 2 = 0 can be written 2x2 - x + 1 = 0.

Characteristics of Quadratics

SLIDE 3

Slide 13 / 222

Practice writing quadratic equations in standard form: (Simplify if possible.) Write 2x2 = x + 4 in standard form:

Standard Form Slide 14 / 222

Write 3x = -x2 + 7 in standard form, if possible:

Standard Form Slide 15 / 222

Write 6x2 - 6x = 12 in standard form and simplify, if possible:

Standard Form Slide 16 / 222

Write 3x - 2 = 5x in standard form:

Standard Form Slide 17 / 222

Similar to Quadratic Equations, the standard form of a Quadratic Function is y = ax2 + bx + c, where a ≠ 0. Notice, a can be positive or negative.

Standard Form Slide 18 / 222

When graphed, a quadratic function will make the shape of a parabola. The parabola will open upward if a > 0 or downward if a < 0.

Graph

SLIDE 4

Slide 19 / 222

The domain of a quadratic function is all real numbers.

Domain Slide 20 / 222 Slide 21 / 222

If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value of the vertex. The range of this quadratic is

Range Slide 22 / 222

An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form

Axis of Symmetry Slide 23 / 222

The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeros, roots or solutions and solution sets. Each quadratic function will have 0, 1, or 2 or real solutions. 2 real solutions no real solutions 1 real solution

X-Intercepts Slide 24 / 222

1 If a parabola opens downward, the vertex is the highest value on the parabola. True False

SLIDE 5

Slide 25 / 222

2 If a parabola opens upward then...

A

a > 0

B

a < 0

C

a = 0

Slide 26 / 222

3 The vertical line that divides a parabola into two symmetrical halves is called...

A

discriminant

B

quadratic equation

C

axis of symmetry

D

vertex

E

maximum

Slide 27 / 222 Slide 28 / 222 Slide 29 / 222 Slide 30 / 222

What is the range of the quadratic function below?

SLIDE 6

Slide 31 / 222

Combining Transformations (REVIEW)

Return to Table of Contents

Slide 32 / 222 Slide 33 / 222

Let the graph of f(x) be Graph y = 2f(.5x+1) - 2

Combining Transformations Slide 34 / 222

Let the graph of f(x) be Graph y =(-

1/2 )f(2x + 1) + 2

Combining Transformations Slide 35 / 222

Let the graph of f(x) be Graph y = 3f(-.5x - 2) + 1

Combining Transformations

Answer

Slide 36 / 222

Let the graph of f(x) be Graph y = (-1/2)f(-x + 2) +1

Combining Transformations

SLIDE 7

Slide 37 / 222

Consider the graph y = x 2 and the rules for stretches and shrinks, Graph

Graph the Transformation Slide 38 / 222

7 Given the graph of h(x), which of the following graphs is y = 2h(-x+1) - 3?

A B C D

Slide 39 / 222

8 Given the graph of h(x), which of the following graphs is y = -0.5h(2x - 1) + 2?

A B C D

Slide 40 / 222

Graph Quadratic Functions

Return to Table

f Contents

Slide 41 / 222 Graph by Following Five Steps:

Step 1 - Find Axis of Symmetry Step 2 - Find Vertex Step 3 - Find y-intercept Step 4 - Locate another point Step 5 - Reflect and Connect

Slide 42 / 222

SLIDE 8

Slide 43 / 222 Slide 44 / 222 Slide 45 / 222 Slide 46 / 222 Slide 47 / 222

9 What is the axis of symmetry for y = x 2 + 2x - 3 (Step 1)?

A

x = 1

B

x = -1

C

x = 2

D

x = -3

Slide 48 / 222

10 What is the vertex for y = x 2 + 2x - 3 (Step 2)?

A

(-1, -4)

B

(1, -4)

C

(-1, -6)

D

(1, -6)

SLIDE 9

Slide 49 / 222

11 What is the y-intercept for y = x 2 + 2x - 3 (Step 3)?

A

(0 , -3)

B

(0 , 3)

Slide 50 / 222

Practice: Graph

Graph Slide 51 / 222

Practice: Graph

Graph Slide 52 / 222

Practice: Graph

Graph Slide 53 / 222

Solve Quadratic Equations by Graphing

Return to Table of Contents

Slide 54 / 222

When asked to solve a quadratic equation, there are several ways to do so. One way to solve a quadratic equation in standard form is to find the zeros of the related function by graphing. A zero is the point at which the parabola intersects the x-axis. A quadratic function may have one, two or no zeros.

Solve by Graphing

SLIDE 10

Slide 55 / 222

How many zeros do the parabolas have? What are the values of the zeros? No zeroes 2 zeroes; x = -1 and x=3 1 zero; x=1

click click click

Solve by Graphing Slide 56 / 222

Every quadratic function has a related quadratic equation. A quadratic equation is used to find the zeroes of a quadratic

function. When a function intersects the x-axis its y-value is

zero. y = ax2 + bx + c Quadratic Function 0 = ax2 + bx + c ax2 + bx + c = 0 Quadratic Equation When writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0.

Vocabulary Slide 57 / 222

One way to solve a quadratic equation in standard form is to find the zeros or x-intercepts of the related function. Solve a quadratic equation by graphing: Step 1 - Write the related function. Step 2 - Graph the related function. Step 3 - Find the zeros (or x-intercepts) of the related function.

Solve by Graphing Slide 58 / 222

2x2 - 18 = 0 2x2 - 18 = y y = 2x2 + 0x - 18 Step 1 - Write the Related Function

Solve by Graphing Slide 59 / 222

Use the same five-step process for graphing The axis of symmetry is x = 0. The vertex is (0, -18). The y-intercept is (0, -18). Since the vertex is the y-intercept, locate two

ther points by substituting values for x. We'll

use (2,-10) and (3,0) Graph these points and use reflection across the axis of symmetry. Connect all points with a smooth curve. Step 2 - Graph the Function y = 2x2 + 0x – 18

Solve by Graphing Slide 60 / 222

(3,0) # x = 0 (2,-10) # (0,-18) # # # Step 2 - Graph the Function y = 2x2 + 0x – 18

Solve by Graphing

SLIDE 11

Slide 61 / 222

The zeros appear to be 3 and -3.

(3,0)

#

x = 0 (2,-10)

#

(0,-18)

# # #

Step 3 - Find the zeros y = 2x2 + 0x – 18

Solve by Graphing Slide 62 / 222

The zeros are 3 and -3. Substitute 3 and -3 for x in the quadratic equation. Check 2x2 – 18 = 0 2(3)2 – 18 = 0 2(9) - 18 = 0 18 - 18 = 0 0 = 0

#

Step 3 - Find the zeros

y = 2x2 + 0x – 18 2(-3)2 – 18 = 0 2(9) - 18 = 0 18 - 18 = 0 0 = 0

#

Solve by Graphing Slide 63 / 222

12

A B C

y = -2x2 + 12x - 18 Solve the equation by graphing the related function and identifying the zeros.

12x + 18 = -2x2

Step 1: Which of these is the related function? y = 2x2 - 12x - 18 y = -2x2 - 12x + 18

Slide 64 / 222

13 What is the axis of symmetry? Formula: -b 2a y = -2x2 + 12x - 18

A

x = -3

B

x = 3

C

x = 4

D

x = -5

Slide 65 / 222

14 What is the vertex? y = -2x2 + 12x - 18

A

(3,0)

B

(-3,0)

C

(4,0)

D

(-5,0)

Slide 66 / 222

15 What is the y-intercept? y = -2x2 + 12x - 18

A

(0,0)

B

(0, 18)

C

(0, -18)

D

(0, 12)

SLIDE 12

Slide 67 / 222

16

A B C D

If two other points are (5, -8) and (4, -2), what does the graph of y = -2x2 + 12x - 18 look like?

Slide 68 / 222

17 Find the zero(s) y = -2x2 + 12x - 18

A

18

B

4

C

3

D

8

Slide 69 / 222

Solve Quadratic Equations by Factoring

Return to Table of Contents

Slide 70 / 222

In addition to graphing, there are additional ways to find the zeros or x-intercepts of a quadratic. This section will explore solving quadratics using the method of factoring. A complete review of factoring can be found in the Fundamental Skills of Algebra (Supplemental Review) Unit.

Fundamental Skills of Algebra (Supplemental Review) Click for Link

Solve by Factoring Slide 71 / 222

Review of factoring - Factoring is simply rewriting an expression in an equivalent form which uses multiplication. To factor a quadratic, ensure that you have the quadratic in standard form: ax2+bx+c=0 Tips for factoring quadratics: · Check for a GCF (Greatest Common Factor). · Check to see if the quadratic is a Difference of Squares or other special binomial product.

Solve by Factoring Slide 72 / 222

Examples: Quadratics with a GCF: 3x2 + 6x in factored form is 3x(x + 2) Quadratics using Difference of Squares: x2 - 64 in factored form is (x + 8)(x - 8) Additional Quadratic Trinomials: x2 - 12x +27 in factored form is (x - 9)(x - 3) 2x2 - x - 6 in factored form is (2x + 3)(x - 2)

Solve by Factoring

SLIDE 13

Slide 73 / 222

Practice: To factor a quadratic trinomial of the form x

2 + bx + c, find two factors

f c whose sum is b.

Example - To factor x

2 + 9x + 18, look for factors of 18 whose sum is 9.

(In other words, find 2 numbers that multiply to 18 but also add to 9.) Factors of 18 Sum

Solve by Factoring Slide 74 / 222

Practice: Factor x2 + 4x - 12, look for factors of -12 whose sum is 4. (in other words, find 2 numbers that multiply to -12 but also add to 4.) Factors of -12 Sum

Solve by Factoring Slide 75 / 222 Zero Product Property

= 0

x

? ?

Imagine this: If 2 numbers must be placed in the boxes and you know that when you multiply these you get ZERO, what must be true?

Slide 76 / 222 Zero Product Property

For all real numbers a and b, if the product of two quantities equals zero, at least one of the quantities equals zero.

If a b = 0 then a = 0 or b = 0 Slide 77 / 222

Example: Solve x2 + 4x - 12 = 0

1. Factor the trinomial.
2. Using the Zero Product Property, set each factor equal to zero.
3. Solve each simple equation.

Now... combining the 2 ideas of factoring with the Zero Product Property, we are able to solve for the x-intercepts (zeros) of the quadratic.

Solve by Factoring Slide 78 / 222

Example: Solve x2 + 36 = 12x Remember: The equation has to be written in standard form (ax2 + bx + c).

Solve by Factoring

SLIDE 14

Slide 79 / 222

18 Solve

A

x = -7

B

x = -5

C

x = -3

D

x = -2

E

x = 2

F

x = 3

G

x = 5

H

x = 6

I

x = 7

J

x = 15

Slide 80 / 222

19 Solve

A

m = -7

B

m = -5

C

m = -3

D

m = -2

E

m = 2

F

m = 3

G

m = 5

H

m = 6

I

m = 7

J

m = 15

Slide 81 / 222

20 Solve

A

h = -12

B

h = -4

C

h = -3

D

h = -2

E

h = 2

F

h = 3

G

h = 4

H

h = 6

I

h = 8

J

h = 12

Slide 82 / 222

21 Solve

A

d = -7

B

d = -5

C

d = -3

D

d = -2

E

d = 0

F

d = 3

G

d = 5

H

d = 6

I

d = 7

J

d = 37

Slide 83 / 222

Berry Method to factor Step 1: Calculate ac. Step 2: Find a pair of numbers m and n, whose product is ac, and whose sum is b. Step 3: Create the product . Step 4: From each binomial in step 3, factor out and discard any common factor. The result is the factored form. Example: Solve When a does not equal 1, check first for a GCF, then use the Berry Method.

Berry Method to Factor Slide 84 / 222

Use the Berry Method. a = 8, b = 2, c = -3

4 and 6 are factors of -24 that add to +2

Solve Step 1 Step 2 Step 3 Step 4 Discard common factors

Berry Method to Factor

SLIDE 15

Slide 85 / 222

Use the Zero Product Rule to solve. Solve

Berry Method to Factor Slide 86 / 222

Solve Use the Berry Method. a = 4, b = -15, c = -25

Berry Method to Factor Slide 87 / 222

Use the Zero Product Rule to solve. Solve

Berry Method to Factor Slide 88 / 222

Solve

Berry Method to Factor Slide 89 / 222

Application of the Zero Product Property

In addition to finding the x-intercepts of quadratic equations, the Zero Product Property can also be used to solve real world application problems. Return to Table of Contents

Slide 90 / 222

Example: A garden has a length of (x+7) feet and a width of (x +3) feet. The total area of the garden is 396 sq. ft. Find the width of the garden.

Application

SLIDE 16

Slide 91 / 222

22 The product of two consecutive even integers is

48. Find the smaller of the two integers.

Hint: Two consecutive integers can be expressed as x and x + 1. Two consecutive even integers can be expressed as x and x + 2.

Slide 92 / 222

23 The width of a rectangular swimming pool is 10 feet less than its length. The surface area of the pool is 600 square feet. What is the pool's width?

Slide 93 / 222

24 A science class designed a ball launcher and tested it by shooting a tennis ball straight up from the top of a 15-story building. They determined that the motion

f the ball could be described by the

function: , where t represents the time the ball is in the air in seconds and h(t) represents the height, in feet, of the ball above the ground at time t. What is the maximum height of the ball? At what time will the ball hit the ground? Find all key features and graph the function.

Students type their answers here

Problem is from: Click link for exact lesson.

Slide 94 / 222

25 A ball is thrown upward from the surface of Mars with an initial velocity of 60 ft/sec. What is the ball's maximum height above the surface before it starts falling back to the surface? Graph the function. The equation for "projectile motion" on Mars is:

Students type their answers here

Slide 95 / 222

Solve Quadratic Equations Using Square Roots

Return to Table of Contents

Slide 96 / 222

SLIDE 17

Slide 97 / 222 Slide 98 / 222

What if x2 has a coefficient other than 1? Example: Solve 4x2 = 20 using the square roots method.

Solve Using Square Roots Slide 99 / 222

26 When you take the square root of a real number, your answer will always be positive. True False

Slide 100 / 222

27 If x2 = 16, then x =

A

4

B

2

C

2

D

26

E

4

Slide 101 / 222

28 Solve using the square root method.

A 5 B

20

C

4

D

2

E

5

F

2

G

4

H

20

Slide 102 / 222

29 If y2 = 4, then y =

A

4

B

2

C

2

D

26

E

4

SLIDE 18

Slide 103 / 222 Slide 104 / 222 Slide 105 / 222

32 If (3g - 9)2 + 7= 43, then g =

A B C D E

Slide 106 / 222

Challenge: Solve (2x - 1)² = 20 using the square root method.

Solve Using Square Roots Slide 107 / 222

33 A physics teacher put a ball at the top of a ramp and let it roll toward the floor. The class determined that the height of the ball could be represented by the equation, ,where the height, h, is measured in feet from the ground and time, t, in

seconds. Determine the time it takes the ball to

reach the floor.

Students type their answers here

Problem is from: Click link for exact lesson.

Slide 108 / 222

34 A rock is dropped from a 1000 foot tower. The height

f the rock as a function of time can be modeled by

the equation: . How long does it take for the rock to reach the ground?

Students type their answers here Answer

SLIDE 19

Slide 109 / 222

Solving Quadratic Equations by Completing the Square

Return to Table of Contents

Slide 110 / 222

In algebra, "Completing the Square" is a technique for changing a quadratic expression from standard form: ax2 + bx + c to the vertex/graphing form: a(x + h)2 + k. It can also be used as a method for solving quadratic equations.

Completing the Square Slide 111 / 222 Slide 112 / 222 Slide 113 / 222

35 Find (b/2)2 if b = 14.

Slide 114 / 222

36 Find (b/2)2 if b = 10.

SLIDE 20

Slide 115 / 222

37 Find (b/2)2 if b = -12.

Slide 116 / 222

38 Complete the square to form a perfect square trinomial. x2 + 18x + ?

Slide 117 / 222

39 Complete the square to form a perfect square trinomial. x2 - 6x + ?

Slide 118 / 222

Step 1 - Write the equation in the form x2 + bx = c. Step 2 - Find (b ÷ 2)2. Step 3 - Complete the square by adding (b ÷ 2)2 to both sides of the equation. Step 4 - Factor the perfect square trinomial. Step 5 - Take the square root of both sides. Step 6 - Write two equations, using both the positive and negative square root and solve each equation.

Completing the Square Slide 119 / 222

Let's look at an example to solve: x2 + 14x -15 = 0 Step 1 - Rewrite Equation Step 2 - Find (b/2)2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve How can you check your solutions?

Completing the Square Slide 120 / 222

Let's look at an example to solve: Step 1 - Rewrite Equation Step 2 - Find (b/2)2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve How can you check your solutions?

Completing the Square

x2 - 2x - 2 = 0

SLIDE 21

Slide 121 / 222

40 Solve the following by completing the square : x2 + 6x = -5

A

5

B

2

C

1

D

5

E

2

Slide 122 / 222

41 Solve the following by completing the square : x2 - 8x = 20

A

10

B

2

C

1

D

10

E

2

Slide 123 / 222 Slide 124 / 222

Challenge: Step 1 - Rewrite Equation Step 2 - Find (b/2)2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve

Completing the Square

3x2 - 10x = -3

*Note: There is no GCF to factor out like the previous example.

Slide 125 / 222

Challenge: Step 1 - Rewrite Equation Step 2 - Find (b/2)2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve

Completing the Square

4x2 - 17x + 4 = 0

*Note: There is no GCF to factor out.

Slide 126 / 222

Challenge:

Completing the Square

6x2 - 25x - 25 = 0

*Note: There is no GCF to factor out.

SLIDE 22

Slide 127 / 222

43 Solve the following by completing the square :

A B C D E

Slide 128 / 222

Return to Table of Contents

Solve Quadratic Equations by Using the Quadratic Formula

Slide 129 / 222

At this point you have learned how to solve quadratic equations by: Today we will be given a tool to solve ANY quadratic equation, and it ALWAYS works! Many quadratic equations may be solved using these methods. Though completing the square works for any quadratic equation, it can be cumbersome to repeatedly use the algorithm. · graphing · factoring · using square roots and · completing the square

Solving Quadratics Slide 130 / 222

Now try Completing the Square on the standard form of a quadratic equation.

Completing the Square

Step 1 - Rewrite Equation Step 2 - Find (b/2)2 Step 3 - Add the result to both sides Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Write 2 Equations & Solve

Slide 131 / 222

Steps 2 and 3 - Find (b/2)2 , Add the result to both sides, simplify Step 1 - Rewrite Equation and factor out a Step 4 - Factor & Simplify Step 5 - Take Square Root of both sides Step 6 - Solve for x

Completing the Square Slide 132 / 222

SLIDE 23

Slide 133 / 222

Example 1: Solve 2x2 + 3x - 5 = 0 a = 2 b = 3 c = -5 Once you identify the values of a, b, and c, simply substitute into the quadratic formula and simplify as much as possible. How can you check your answers?

Quadratic Formula Slide 134 / 222

Example 2: Solve: 2x = x2 - 3

To use the Quadratic Formula, the equation must be in standard form (ax2 + bx +c = 0). Identify a, b, and c, then substitue into the formula and simplify. Don't forget to check your results!

Quadratic Formula Slide 135 / 222

44 Solve the following equation using the quadratic formula:

A

5

B

4

C

3

D

2

E

1

F

1

G

2

H

3

I

4

J

5

Slide 136 / 222

45 Solve the following equation using the quadratic formula:

A

5

B

4

C

3

D

2

E

1

F

1

G

2

H

3

I

4

J

5

Slide 137 / 222

46 Solve the following equation using the quadratic formula:

A

5

B

4

C

3

D

2

E

1.5

F

1.5

G

2

H

3

I

4

J

5

Slide 138 / 222

Example 3:

Solve using the quadratic formula, and simplify the result. x

2 - 2x - 4 = 0

Quadratic Formula

SLIDE 24

Slide 139 / 222

47 Find the larger solution to the equation.

Slide 140 / 222

48 Find the smaller solution to the equation.

Slide 141 / 222

Work in small groups to solve the quadratic equation using the following different methods. Factoring Quadratic Formula Completing the Square Graphing

Which Method Slide 142 / 222

Factoring Quadratic Formula Completing the Square Graphing Work in small groups to solve the quadratic equation using the following different methods.

Which Method Slide 143 / 222

The Discriminant

Return to Table of Contents

Slide 144 / 222

2 real solutions no real solutions 1 real solution Recall what it means to have 0, 1, or 2 solutions/zeros/roots

Solutions

SLIDE 25

Slide 145 / 222

At times, it is not necessary to solve for the zeros or roots of a quadratic function, but simply to know how many roots exist (zero, one, or two). The quickest way to determine how many solutions a quadratic has, algebraically, is to calculate what's called the discriminant. It may look familiar, as the discriminant is a part of the quadratic formula.

The Discriminant Slide 146 / 222 Slide 147 / 222

Other important tips before practice: · The square root of a positive number has two solutions. · The square root of zero is 0. · The square root of a negative number has no real solution.

The Discriminant Slide 148 / 222 Slide 149 / 222

If b2 - 4ac > 0 (POSITIVE) the quadratic has two real solutions If b2 - 4ac = 0 (ZERO) the quadratic has one real solution If b2 - 4ac < 0 (NEGATIVE) the quadratic has no real solutions CONCLUSION:

The Discriminant

click to reveal

Slide 150 / 222

49 What is the value of the discriminant of 2x2 - 2x + 3 = 0 ?

SLIDE 26

Slide 151 / 222

50 Use the discriminant to find the number of solutions for 2x2 - 2x + 3 = 0 A B 1 C 2

Slide 152 / 222

51 What is the value of the discriminant of x2 - 8x + 4 = 0 ?

Slide 153 / 222

52 Use the discriminant to find the number

f solutions for x2 - 8x + 4 = 0

A B 1 C 2

Slide 154 / 222

Vertex Form

Return to Table of Contents

Slide 155 / 222 Vertex Form

A quadratic equation in vertex form: So far, we have been using quadratics in standard form. However, sometimes when graphing, it is more useful to write them in Vertex Form.

Slide 156 / 222

Vertex Form shows the location of the vertex (h , k). The a still tells the direction of opening. And the axis of symmetry is x = h. Example: Find the vertex, direction of opening and the axis of symmetry for the graph of: A quadratic function written in vertex form:

Vertex Form

SLIDE 27

Slide 157 / 222

Find the vertex, direction of openness and the axis of symmetry for each. A. B. C.

Vertex Form Slide 158 / 222

D. E.

Vertex Form

Find the vertex, direction of openness and the axis of symmetry for each.

Slide 159 / 222

53 Find the vertex for the graph of

A B C D

Slide 160 / 222

54 Find the direction of opening for the graph of

A

up

B

down

C

left

D

right

Slide 161 / 222 Slide 162 / 222

SLIDE 28

Slide 163 / 222

57 Give the direction of opening for the graph of

A

up

B

down

C

left

D

right

Slide 164 / 222

58 Give the axis of symmetry for the graph of

Slide 165 / 222 Slide 166 / 222

60 Give the direction of openness of

A

up

B

down

C

left

D

right

Slide 167 / 222

61 The axis of symmetry for the graph of is ______ .

Slide 168 / 222

SLIDE 29

Slide 169 / 222 Slide 170 / 222 Slide 171 / 222 Converting from Standard Form to Vertex Form

To convert from standard form to vertex form, we need to recall the method for completing the square. Step 1 - Write the equation in the form y = x2 + bx + ___ + c - ___ Step 2 - Find (b ÷ 2)2 Step 3 - Write the result from Step 2 in the first blank and in the second blank. Step 4 - Rewrite the first three terms as a perfect square.

Slide 172 / 222 Slide 173 / 222 Slide 174 / 222

65 What is the vertex form of:

A B C D

SLIDE 30

Slide 175 / 222 Slide 176 / 222 Slide 177 / 222

Compare the following functions based on information from the

equations. What do the graphs have in common? How are

they different? Sketch both graphs to confirm your conclusions.

Comparing Functions Slide 178 / 222

Write two different quadratic equations whose graphs have vertices at (3.5, -7).

Two Functions Slide 179 / 222

What if "a" does not equal 1? Step 1 - Write the equation in the form y = ax2 + bx +__+ c - __ Step 2 - Factor: y = a(x2 + (b/a)x +__)+ c - __ Step 3 - Find (b/a ÷ 2)2 Step 4 - Put your answer from Step 3 in the first blank and multiply Step 3 by a to fill in the second blank. Step 5 - Write trinomial as perfect square.

Standard Form to Vertex Form Slide 180 / 222

SLIDE 31

Slide 181 / 222 Slide 182 / 222 Slide 183 / 222 Slide 184 / 222 Slide 185 / 222 Geometric Definition of a Parabola

A parabola is a locus* of points equidistant from a fixed point, the focus, and a fixed line, the directrix.

*locus is just a fancy word for set.

Every parabola is symmetric with respect to a line through the focus and perpendicular to the

directrix. The vertex
f the parabola is the

"turning point" and is

n the axis of

symmetry.

Slide 186 / 222

Axis of Symmetry Directrix Focus L1 L2 L1=L2

Focus and Directrix of a Parabola

Every point on the parabola is the same distance from the directrix and the focus.

SLIDE 32

Slide 187 / 222 Parabolas

The parts are the same for all parabolas,regardless of the direction in which they open. Directrix Axis of Symmetry Vertex Focus y=ax2+bx+c Vertex Focus Directrix Axis of Symmetry x=ay2+by+c

Slide 188 / 222 Slide 189 / 222

General Form y= ax

2 + bx + c

x= ay

2 +by + c

Vertex Form y= a(x - h)

2 +k

x= a(y - k)

2 + h

Opens a>0 opens up a<0 opens down a>0 opens to the right a<0 opens to the left Axis of Symmetry x = h y = k Vertex (h , k) (h , k) Directrix Focus

Parabola Summary Slide 190 / 222

71 What is the vertex of ?

A

(-3, 2)

B

(-3, -2)

C

(2, 3)

D

(-2, -3)

Slide 191 / 222

72 What is the vertex of ?

A

(3, 2)

B

(-3, -2)

C

(2, 3)

D

(-2, -3)

Slide 192 / 222

SLIDE 33

Slide 193 / 222 Slide 194 / 222 Slide 195 / 222 Slide 196 / 222 Slide 197 / 222 Slide 198 / 222

78 What is the vertex of y= x

2 - 8x +21?

A

(4, 5)

B

(-4, 5)

C

(-5, 4)

D

(5, 4)

SLIDE 34

Slide 199 / 222

79 What is the equation of the directrix for the following equation?

A

y = 2

B

y = -4

C

x = 3

D

x = -5

Slide 200 / 222

80 Where is the focus for the following equation?

A

(-3 , 5)

B

(3 , 5)

C

(5 , 3)

D

(5 , -3)

Slide 201 / 222 Slide 202 / 222

82 What is the equation of the parabola with vertex (2,3) and directrix y = 4?

A y = 4(x - 2)2 + 3

y = -1/4(x - 2)2 + 3 x = 4(y - 2)2 + 3 x = 1/4(y - 2)2 + 3

B C D

Slide 203 / 222

Key Terms Solve Quadratic Equations by Factoring Application of - - PDF document

Algebra II

Key Terms

Explain Characteristics

Functions

Combining Transformations (REVIEW)

Graph Quadratic Functions

Solve Quadratic Equations by Graphing

#

#

Solve Quadratic Equations by Factoring

= 0

? ?

Application of the Zero Product Property

Solve Quadratic Equations Using Square Roots

Solving Quadratic Equations by Completing the Square

Solve Quadratic Equations by Using the Quadratic Formula

The Discriminant

Vertex Form

More Application Problems Using Quadratics