Graphing Quadratic Equations and Finding Zeros

1. Warm Up 1. Graph y = x2 + 4x + 3. 2. Identify the vertex and zeros of the function above. vertex:(–2 , –1); zeros:–3, –1

2. 9-5 Solving Quadratic Equations by Graphing Holt Algebra 1

3. 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. 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

5. 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. x = 0 ● ● • The axis of symmetry is x = 0. • The vertex is (0, –18). • Two other points (2, –10) and • (3, 0) • Graph the points and reflectthem • across the axis of symmetry. (3, 0) ● ● (2, –10) ● (0, –18)

6. Check2x2 – 18 = 0 2x2 – 18 = 0 2(3)2 – 18 0 2(–3)2 – 18 0 2(9) – 18 0 2(9) – 18 0 18 – 18 0  18 – 18 0  0 0 Example 1A Continued Solve the equation by graphing the related function. 2x2 – 18 = 0 Step 3 Find the zeros. The zeros appear to be 3 and –3. Substitute 3 and –3 for x in the quadratic equation.

7. Step 1 Write the related function. y = –2x2 + 12x – 18 ● ● Example 1B: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. –12x + 18 = –2x2 x = 3 Step 2 Graph the function. (3, 0) ● ● (4, –2) • The axis of symmetry is x = 3. • The vertex is (3, 0). • Two other points (5, –8) and • (4, –2). • Graph the points and reflectthem • across the axis of symmetry. (5, –8) ● ●

8. 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. Example 1B Continued Solve the equation by graphing the related function. –12x + 18 = –2x2 Step 3 Find the zeros. The only zero appears to be 3.

9. Check y =x2 + 8x + 16 0 (–4)2 + 8(–4) + 16 0 16 – 32 + 16  0 0 Check It Out! Example 1a Continued Solve the equation by graphing the related function. x2 – 8x – 16 = 2x2 Step 3 Find the zeros. The only zero appears to be –4.

10. y = –x2 + 4 Step 2 Graph the function. Use a graphing calculator. Check It Out! Example 1c Solve the equation by graphing the related function. –x2 + 4 = 0 Step 1 Write the related function. Step 3 Find the zeros. The function appears to have zeros at (2, 0) and (–2, 0).

11. Check It Out! Example 1c Continued Solve the equation by graphing the related function. –x2 + 4 = 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.

12. Step 1 Write the related function 0 = –16t2 + 12t y = –16t2 + 12t 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.

13. 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. Example 2 Continued Step 2 Graph the function. Use a graphing calculator. The frog is off the ground for about 0.75 seconds.

14. 0 –16(0.75)2 + 12(0.75) 0 –16(0.5625) + 9 0 –9 + 9 0 0 Example 2 Continued Check 0 = –16t2 + 12t Substitute 0.75 for x in the quadratic equation. 

15. 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