MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

Warm Ups

Factor each quadratic.

6 1 7x x

Thursday, April 20, 2023

6 3 1 2x x

21. 6 41 7x x 22. 18 30 12x x

23. 4 15 9x x

(4 3)( 3)x x

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

2

Thursday, April 20, 2023

Essential Question:How do we find the roots or zeros of quadratic functions of the form y = ax2 + bx +c ?

Lesson 3.4B

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.3

Solutions of a quadratic equation or function.

If y = f (x) is a quadratic function and a is a real number then the following statements are equivalent.

1. x = a is a zero of f.

2. x = a is a root of f.

3. x = a is a solution of the quadratic equation f (x) = 0.

4. (x – a) is a factor of the quadratic f (x).

5. (a, 0) is an x-intercept of the graph of y = f (x).

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 4

22 3 2x x We must set the equation equal to zero before we factor.

1.2 2

22 3 2 0x x

+3

- 4

+4 -1 2 1x 2 0x 2 1 0x

1 1 2 1x

2 0x

2x 2 2

1

2x

-2 -2

The solutions are and .1

2x 2x

22 4 1 2 0x x x 2 2x x 1 2 0x

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 5

We must set the equation equal to zero before we factor.

2. 10 8x

29 21 10 0x x

+21

+90

+15 +6 3 5x 3 2 0x 3 5 0x

-5 -53 5x

3 2 0x

3 2x 3 3

5

3x

-2 -2

The solutions are and .5

3x

2

3x

29 11 18 10 8x x x 10 8 10 8x x

3 32

3x

29 15 6 10 0x x x 3 3 5x x 2 3 5 0x

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 6

We must substitute zero for “y” and solve the quadratic equation.

3.20 3 28x x

+3

- 28

+7 -4 7x 4 0x

7 0x -7 -7

4 0x

4x 7x +4 +4

The zeros are and .7x 4x

2 3 28y x x

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 7

4.

-15

+36

-12 -3 4 3x 3 0x 4 3 0x

3 3 4 3x

3 0x

3x 4 4

3

4x

+3 +3

The roots are and .3

4x 3x

24 15 9y x x We must substitute zero for “y” and solve the quadratic equation.

20 4 15 9x x 20 4 12 3 9x x x

4 3x x 3 3 0x

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 8

5.

+4

+4

+2 +2 2 1x 2 1 0x 2 1 0x

-1 -12 1x

2 21

2x The x-intercept is .

1

2x

We must substitute zero for “y” and solve the quadratic equation.

20 4 4 1x x

24 4 1y x x

If you solve the other equation you will get the same solution.We have only one unique solution!

24 2 2 1 0x x x 2 2 1x x 1 2 1 0x

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 9

6.

-7

+6

-6 -1

The x-intercepts are and .6x

We must substitute zero for “y” and solve the quadratic equation.

20 2 14 12x x 22 14 12y x x

20 2 7 6x x 6x 1 0x

6 0x +6 +6

1 0x

1x 6x +1 +1

1x

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

x

y

x

y

A.

B.

Which of the following is the graph of the function f(x) = (x + 3)(x – 3) ?

x

y

C.

7.

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

Find the x-intercepts( )2

3 16x + =8.

( )( ) 3 3 16x x+ + =2 3 3 9 16x x x+ + + =

2 6 9 16x x+ + =2 6 7 0x x+ - =

( )( )7 1 0x x+ - =7, 1x x=- =

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

Find the zeroes of the function2 2 32x =

9.

2 16x =2 16x =

4x =±

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

2

If the area of a rectangle is represented

by the function  4 15 9,which

of the following factors could represent

the length of the rectangle?

y x x

. (4 3)

. ( 3)

. (4 3)

. ( 4)

a x

b x

c x

d x

10.

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

Find the maximum value of a quadratic function

The height y (in feet) of a baseball t seconds after it is hit is given by this function:

Baseball

y = –16t2 + 96t + 3

Find the maximum height of the baseball.

SOLUTION

The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.

7.

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

Find the maximum value of a quadratic function

y = – 16t2 + 96t +3 Write original function.

The vertex is (3, 147), so the maximum height of the baseball is 147 feet.

ANSWER

2

bx

a

96

2 16x

3x

2(3) 16(3) 96(3) 3y

(3) 147y

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula.

SOLUTION

The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.

y = – 16t2 + 80t +2 Write original function.

Suppose the height of the baseball is given by

y = – 16t2 + 80t + 2. Find the maximum height of the baseball.

Find the maximum value of a quadratic function

The vertex is (2.5, 102), so the maximum height of the baseball is 102 feet.

8.

MM2A3 Students will analyze quadratic functions in the forms f(x) = ax2 +bx + c and f(x) = a(x – h)2 = k.

MM2A4b Find real and complex solutions of equations by factoring, taking square roots, and applying the quadratic formula. 17

THE END

Homework page 81 # 1 – 3 all, 16 – 21 all.