# confidential 1 solving quadratic equations by graphing solving quadratic equations by graphing

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CONFIDENTIAL 2

Warm UpWarm Up

Write an equation in point-slope form for the line with the given slope that contains the given point.

1) slope = -3; (-2, 4)

2) slope = 0; (2, 1)

3) slope = 1; (2, 3) 2

CONFIDENTIAL 3

Every quadratic function has a related quadraticequation. A quadratic equation is an equation

that can be written in the standard form ax2 + bx + c = 0,

where a, b, and c are real numbers and a ≠ 0.

Solving Quadratic Equations by GraphingSolving Quadratic Equations by Graphing

Notice that when writing a quadratic function as its related quadratic equation,

you replace y with 0. So y = 0.

y = ax2 + bx + c

0 = ax2 + bx + c

ax2 + bx + c = 0

CONFIDENTIAL 4

One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function

may have two, one, or no zeros.

Solving Quadratic Equations by Graphing

Step 1 Write the related function.

Step 2 Graph the related function.

Step 3 Find the zeros of the related function.

CONFIDENTIAL 5

Solving Quadratic Equations by GraphingSolving Quadratic Equations by Graphing

Solve each equation by graphing the related function.

A) 2x2 - 2 = 0

Step 1 Write the related function.

2x2 - 2 = y, or y = 2x2 + 0x - 2

Step 2 Graph the related function.

• The axis of symmetry is x = 0.

• The vertex is (0, -2) .

• Two other points are (1, 0) and (2, 6).

• Graph the points and reflect them across the axis of symmetry.

CONFIDENTIAL 6

Step 3 Find the zeros of the related function.

The zeros appear to be -1 and 1.

Check

Substitute -1 and 1 for x in the quadratic equation.

2x2 - 2 = 0

2-(1)2 - 2 02(1) – 2 0

2(1) – 2 00 0

2x2 - 2 = 0

2(1)2 - 2 02(1) – 2 0

2(1) – 2 00 0

CONFIDENTIAL 7

Solve each equation by graphing the related function.

B) -x2 - 4x - 4 = 0

Step 1 Write the related function.

y = -x2 - 4x - 4 = 0

Step 2 Graph the related function.

• The axis of symmetry is x = -2.

• The vertex is (-2, 0).

• The y-intercept is -4.

• Another point is (-1, -1).

• Graph the points and reflect them across the axis of symmetry.

CONFIDENTIAL 8

Step 3 Find the zeros of the related function.

The only zero appears to be -2.

Check

y = - x2 - 4x - 4

0- -(2)2- 4-(2 )-40 - (4) + 8 -40 -4 + 40 0

CONFIDENTIAL 9

C) x2 + 5 = 4x

Step 1 Write the related function.

Step 2 Graph the related function Use a graphing calculator.

x2 - 4x + 5 = 0

y = x2 - 4x + 5

Step 3 Find the zeros of the related function.

The function appears to have no zeros.The equation has no real-number solutions.

CONFIDENTIAL 10

Now you try!

Solve each equation by graphing the related function.

1a. x2 - 8x - 16 = 2x2

1b. 6x + 10 = - x2

1c. -x2 + 4 = 0

CONFIDENTIAL 11

Aquatics ApplicationAquatics Application

A dolphin jumps out of the water. The quadratic function y = -16x2 + 20x models the dolphin’s height above the water after x seconds. About how long is

the dolphin out of the water?

When the dolphin leaves the water, its height is 0, and when the dolphin reenters the water, its height is

0. So solve 0 = -16x2 + 20x to find the times when the dolphin leaves and reenters the water.

Step 1 Write the related function.

o = -16x2 + 20x y = -16x2 + 20x

CONFIDENTIAL 12

Step 3 Find the zeros of the related function.

x = 1.25 y = 0

y = -16x2 + 20x

Step 2 Graph the related function. Use a graphing calculator.

The zeros appear to be 0 and 1.25. The dolphin leaves the water at 0 seconds and reenters the water at 1.25 seconds.

Check

0 - =16x2 + 20x

0- 16(1.25)2- 20(1.25)0 -16(1.5625) + 250 -25 + 250 0

Substitute 1.25 for x in the quadratic equation.

CONFIDENTIAL 13

Now you try!

2) Another dolphin jumps out of the water. Thequadratic function y = -16x2 + 32x models the

dolphin’s height above the water after x seconds. About how long is the dolphin out of the water?

CONFIDENTIAL 14

Assessment

1) x2 - 4 = 0

Solve each equation by graphing the related function.

2) x2 = 16

3) -2x2 - 6 = 0

CONFIDENTIAL 15

4) - x2 + 12x - 36 = 0

Solve each equation by graphing the related function.

6) 2x2 = 3x2 - 2x - 8

5) - x2 = -9

CONFIDENTIAL 16

Solve each equation by graphing the related function.

7) x2 - 6x + 9 = 0

8) 8x = -4x2 - 4

9) x2 + 5x + 4 = 0

CONFIDENTIAL 17

10) A baseball coach uses a pitching machine to simulate pop flies during practice. The baseball is shot out of the pitching machine with a velocity of 80 feet per second. The quadratic function y = -16x2 + 80x

models the height of the baseball after x seconds. How long is the baseball in the air?

CONFIDENTIAL 18

Let’s review

Every quadratic function has a related quadraticequation. A quadratic equation is an equation

that can be written in the standard form ax2 + bx + c = 0,

where a, b, and c are real numbers and a ≠ 0.

Solving Quadratic Equations by GraphingSolving Quadratic Equations by Graphing

Notice that when writing a quadratic function as its related quadratic equation,

you replace y with 0. So y = 0.

y = ax2 + bx + c

0 = ax2 + bx + c

ax2 + bx + c = 0

CONFIDENTIAL 19

One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function

may have two, one, or no zeros.

Solving Quadratic Equations by Graphing

Step 1 Write the related function.

Step 2 Graph the related function.

Step 3 Find the zeros of the related function.

CONFIDENTIAL 20

Solving Quadratic Equations by GraphingSolving Quadratic Equations by Graphing

Solve each equation by graphing the related function.

A) 2x2 - 2 = 0

Step 1 Write the related function.

2x2 - 2 = y, or y = 2x2 + 0x - 2

Step 2 Graph the related function.

• The axis of symmetry is x = 0.

• The vertex is (0, -2) .

• Two other points are (1, 0) and (2, 6).

• Graph the points and reflect them across the axis of symmetry.

CONFIDENTIAL 21

Step 3 Find the zeros of the related function.

The zeros appear to be -1 and 1.

Check

Substitute -1 and 1 for x in the quadratic equation.

2x2 - 2 = 0

2-(1)2 - 2 02(1) – 2 0

2(1) – 2 00 0

2x2 - 2 = 0

2(1)2 - 2 02(1) – 2 0

2(1) – 2 00 0

CONFIDENTIAL 22

Aquatics ApplicationAquatics Application

A dolphin jumps out of the water. The quadratic function y = -16x2 + 20x models the dolphin’s height above the water after x seconds. About how long is

the dolphin out of the water?

When the dolphin leaves the water, its height is 0, and when the dolphin reenters the water, its height is

0. So solve 0 = -16x2 + 20x to find the times when the dolphin leaves and reenters the water.

Step 1 Write the related function.

o = -16x2 + 20x y = -16x2 + 20x

CONFIDENTIAL 23

Step 3 Find the zeros of the related function.

x = 1.25 y = 0

y = -16x2 + 20x

Step 2 Graph the related function. Use a graphing calculator.

The zeros appear to be 0 and 1.25. The dolphin leaves the water at 0 seconds and reenters the water at 1.25 seconds.

Check

o = -16x2 + 20x

0- 16(1.25)2- 20(1.25)0 -16(1.5625) + 250 -25 + 250 0

Substitute 1.25 for x in the quadratic equation.