holt mcdougal algebra 1 9-5 solving quadratic equations by graphing 9-5 solving quadratic equations...

Holt McDougal Algebra 1

9-5 Solving Quadratic Equations by Graphing9-5 Solving Quadratic Equations

by Graphing

Holt Algebra 1

Warm UpWarm Up

Lesson PresentationLesson Presentation

Lesson QuizLesson Quiz



9-5 Solving 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

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 1B Continued



The only zero appears to be 3.Check y = –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 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.



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 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) ●●

●




Check It Out! Example 1a Continued


The only zero appears to be –4.

Check y = x2 + 8x + 160 (–4)2 + 8(–4) + 16 0 16 – 32 + 16 0 0

x2 – 8x – 16 = 2x2




6x + 10 = –x2

Step 1 Write the related function.y = x2 + 6x + 10

Check It Out! Example 1b

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

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

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).




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 + 12t models 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 function0 = –16t2 + 12ty = –16t2 + 12t



Example 2 Continued


Use a graphing calculator.

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 + 12t0 –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 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 9-5 solving quadratic equations by graphing 9-5 solving quadratic equations...

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