Solving Quadratic Equations by Graphing 8-5 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Objective Solve quadratic equations by graphing.

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 reflect them across the axis of symmetry. (3, 0) ● ● (2, –10) ● (0, –18)

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. 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  Substitute 3 and –3 for x in the quadratic equation.

Example 1B: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. –12x + 18 = –2x2 Step 1 Write the related function. y = –2x2 + 12x – 18 x = 3 Step 2 Graph the function. (3, 0) ● ● ● 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. (4, –2) ● (5, –8) ●

Example 1C: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. 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. Step 3 Find the zeros. The function appears to have no zeros.

Solve the equation by graphing the related function. Check It Out! Example 1a Solve the equation by graphing the related function. x2 – 8x – 16 = 2x2 Step 1 Write the related function. x = –4 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. ● ● (–2 , 4) ● ● (–3, 1) ● (–4, 0)

Check It Out! Example 1c Solve the equation by graphing the related function. –x2 + 4 = 0 Step 1 Write the related function. y = –x2 + 4 Step 2 Graph the function. Use a graphing calculator. Step 3 Find the zeros. 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 function 0 = –16x2 + 32x y = –16x2 + 32x

Check It Out! Example 2 Continued Step 2 Graph the function. 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 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 + 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