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

Holt McDougal Algebra 1

8-5 Solving Quadratic Equations by Graphing8-5 Solving Quadratic Equations

by Graphing

Holt Algebra 1

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8-5 Solving Quadratic Equations by Graphing

Solve quadratic equations by graphing.

Objective



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

Solve the equation by graphing the related function.

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

Step 1 Write the related function.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 across the axis of symmetry.

(5, –8)

(4, –2)

●

●●

●

●

●

x = 3

(3, 0)



Example 1C: Solving Quadratic Equations by Graphing


2x2 + 4x = –3 Step 1 Write the related function.

y = 2x2 + 4x + 3 2x2 + 4x + 3 = 0

Step 2 Graph the function.Use a graphing calculator.


The function appears to have no zeros.



Check It Out! Example 1a Solve the equation by graphing the related function.

x2 – 8x – 16 = 2x2

Step 1 Write the related function.

y = x2 + 8x + 16

Step 2 Graph the function.• 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 across the axis of symmetry.

x = –4

(–4, 0) ●

(–3, 1) ●

(–2 , 4) ●●

●




–x2 + 4 = 0

Check It Out! Example 1c

Step 1 Write the related function.

y = –x2 + 4

Step 2 Graph the function.Use a graphing calculator.


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



Check It Out! Example 2

What if…? A dolphin jumps out of the water. The quadratic function y = –16x2 + 32 x models 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 function0 = –16x2 + 32xy = –16x2 + 32x




Use a graphing calculator.

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 + 32x0 –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 + 96t

models 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

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

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