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Holt McDougal Algebra 1

8-5Solving Quadratic Equations by Graphing8-5

Solving Quadratic Equations by Graphing

Holt Algebra 1

Warm Up

Lesson Presentation

Lesson Quiz



8-5Solving Quadratic Equations by Graphing

Warm Up

1. Graph y = x2 + 4x + 3.

2. Identify the vertex and zeros of the function above.

vertex:(–2 , –1); zeros:–3, –1



Solve quadratic equations by graphing.

Objective



quadratic equation

Vocabulary



Every quadratic function has a related quadratic equation. 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.

y = ax2 + bx + c0 = ax2 + bx + c

ax2 + bx + c = 0

When writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0.



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.



Example 1A: Solving Quadratic Equations by Graphing

Solve the equation by graphing the related function.

2x2 – 18 = 0

Step 1 Write the related function.

2x2 – 18 = y, or y = 2x2 + 0x – 18

Step 2 Graph the function.

• The axis of symmetry is x = 0.

• The vertex is (0, –18).

• Two other points (2, –10) and

(3, 0)

• Graph the points and reflect them

across the axis of symmetry.

(3, 0)

●x = 0

(2, –10) ●

(0, –18)●

●

●



Example 1A Continued


Step 3 Find the zeros.

2x2 – 18 = 0

The zeros appear to be 3 and –3.

Substitute 3 and –3

for x in the quadratic

equation. 0 0

Check 2x2 – 18 = 0

2(3)2 – 18 0

2(9) – 18 0

18 – 18 0 ✓

2x2 – 18 = 0

2(–3)2 – 18 0

2(9) – 18 0

18 – 18 0 ✓



Example 1B: Solving Quadratic Equations by Graphing


–12x + 18 = –2x2


y = –2x2 + 12x – 18


• The axis of symmetry is x = 3.• The vertex is (3, 0). • Two other points (5, –8) and

(4, –2).• Graph the points and reflect them


(5, –8)

(4, –2)

●

●●

●

●

●

x = 3

(3, 0)



Example 1B Continued



The only zero appears to be 3.

Checky = –2x2 + 12x – 18

0 –2(3)2 + 12(3) – 18

0 –18 + 36 – 18

0 0 ✓

You can also confirm the solution by using the Table

function. Enter the function and press

When y = 0, x = 3. The x-intercept is 3.

–12x + 18 = –2x2



Example 1C: Solving Quadratic Equations by Graphing


2x2 + 4x = –3


y = 2x2 + 4x + 3

2x2 + 4x + 3 = 0


Use a graphing calculator.


The function appears to have no zeros.



Example 1C: Solving Quadratic Equations by Graphing


2x2 + 4x = –3

The equation has no real-number solutions.

Check reasonableness Use the table function.

There are no zeros in the Y1 column.

Also, the signs of the values in this

column do not change. The function

appears to have no zeros.



Check It Out! Example 1a


x2 – 8x – 16 = 2x2


y = x2 + 8x + 16


• The axis of symmetry is x = –4.• The vertex is (–4, 0). • The y-intercept is 16.• Two other points are (–3, 1) and

(–2, 4).• Graph the points and reflect them


x = –4

(–4, 0)

●(–3, 1) ●

(–2 , 4) ●●

●




Check It Out! Example 1a Continued


The only zero appears to be –4.

Check y = x2 + 8x + 16

0 (–4)2 + 8(–4) + 16

0 16 – 32 + 16

0 0 ✓

x2 – 8x – 16 = 2x2




6x + 10 = –x2


y = x2 + 6x + 10

Check It Out! Example 1b


• The axis of symmetry is x = –3 .• The vertex is (–3 , 1). • The y-intercept is 10.• Two other points (–1, 5) and

(–2, 2)• Graph the points and reflect them


x = –3

(–3, 1) ●

(–2, 2) ●

(–1, 5) ●

●

●




6x + 10 = –x2

Check It Out! Example 1b Continued


There appear to be no zeros.

You can confirm the solution

by using the Table function.

Enter the function and press

There are no negative

terms in the Y1 column.




–x2 + 4 = 0

Check It Out! Example 1c


y = –x2 + 4




The function appears to have zeros at (2, 0) and (–2, 0).




The equation has two real-number solutions.

Check reasonableness Use the table function.

There are two zeros in the Y1

column. The function appears to

have zeros at –2 and 2.

Check It Out! Example 1c Continued

–x2 + 4 = 0



Example 2: Application

A frog jumps straight up from the ground. The quadratic function f(t) = –16t2 + 12tmodels the frog’s height above the ground after t seconds. About how long is the frog in the air?

When the frog leaves the ground, its height is 0, and when the frog lands, its height is 0. So solve 0 = –16t2 + 12t to find the times when the frog leaves the ground and lands.

Step 1 Write the related function

0 = –16t2 + 12t

y = –16t2 + 12t



Example 2 Continued



Step 3 Use to estimate the zeros.

The zeros appear to be 0 and 0.75.

The frog leaves the ground at 0 seconds and lands at 0.75 seconds.

The frog is off the ground for about 0.75 seconds.



Check 0 = –16t2 + 12t

0 –16(0.75)2 + 12(0.75)

0 –16(0.5625) + 9

0 –9 + 9

0 0✓

Substitute 0.75 for t

in the quadratic

equation.

Example 2 Continued



Check It Out! Example 2

What if…? A dolphin jumps out of the water. The quadratic function y = –16x2 + 32 xmodels the dolphin’s height above the water after x seconds. 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 + 32x to find the times when the dolphin leaves and reenters the water.

Step 1 Write the related function

0 = –16x2 + 32x

y = –16x2 + 32x





Step 3 Use to estimate the zeros.

The zeros appear to be 0 and 2.

The dolphin leaves the water at 0 seconds and reenters at 2 seconds.

The dolphin is out of the water for 2 seconds.

Check It Out! Example 2 Continued



Check It Out! Example 2 Continued

Check 0 = –16x2 + 32x

0 –16(2)2 + 32(2)

0 –16(4) + 64

0 –64 + 64

0 0✓

Substitute 2 for x in

the quadratic

equation.



Lesson Quiz

Solve each equation by graphing the related function.

1. 3x2 – 12 = 0

2. x2 + 2x = 8

3. 3x – 5 = x2

4. 3x2 + 3 = 6x

5. A rocket is shot straight up from the ground. The quadratic function f(t) = –16t2 + 96tmodels the rocket’s height above the ground after t seconds. How long does it take for the rocket to return to the ground?

2, –2

–4, 2

no solution

1

6 s

8-5 solving quadratic equations by graphing · holt mcdougal algebra 1 8-5 solving quadratic...

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