Solving Quadratic Equations by Graphing CONFIDENTIAL

Warm Up Write an equation in point-slope form for the line with the given slope that contains the given point. 1) slope = -3; (-2, 4) 2) slope = 0; (2, 1) 3) slope = 1; (2, 3) 2 CONFIDENTIAL

Solving Quadratic Equations by Graphing 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 + c 0 = ax2 + bx + c ax2 + bx + c = 0 Notice that when writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. CONFIDENTIAL

Solving Quadratic Equations by Graphing 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. Solving Quadratic Equations by Graphing Step 1 Write the related function. Step 2 Graph the related function. Step 3 Find the zeros of the related function. CONFIDENTIAL

Solving Quadratic Equations by Graphing Solve each equation by graphing the related function. A) 2x2 - 2 = 0 Step 1 Write the related function. 2x2 - 2 = y, or y = 2x2 + 0x - 2 Step 2 Graph the related function. • The axis of symmetry is x = 0. • The vertex is (0, -2) . • Two other points are (1, 0) and (2, 6). • Graph the points and reflect them across the axis of symmetry. CONFIDENTIAL

Step 3 Find the zeros of the related function. The zeros appear to be -1 and 1. Check 2x2 - 2 = 0 2(-1)2 - 2 0 2(1) – 2 0 0 0 2x2 - 2 = 0 2(1)2 - 2 0 2(1) – 2 0 0 0 Substitute -1 and 1 for x in the quadratic equation. CONFIDENTIAL

Solve each equation by graphing the related function. B) -x2 - 4x - 4 = 0 Step 1 Write the related function. y = -x2 - 4x - 4 = 0 Step 2 Graph the related function. • The axis of symmetry is x = -2. • The vertex is (-2, 0). • The y-intercept is -4. • Another point is (-1, -1). • Graph the points and reflect them across the axis of symmetry. CONFIDENTIAL

Step 3 Find the zeros of the related function. The only zero appears to be -2. Check y = - x2 - 4x - 4 0 -(-2)2 -4(-2) -4 0 - (4) + 8 -4 0 -4 + 4 0 0 CONFIDENTIAL

Step 1 Write the related function. C) x2 + 5 = 4x Step 1 Write the related function. x2 - 4x + 5 = 0 y = x2 - 4x + 5 Step 2 Graph the related function Use a graphing calculator. Step 3 Find the zeros of the related function. The function appears to have no zeros. The equation has no real-number solutions. CONFIDENTIAL

Solve each equation by graphing the related function. Now you try! Solve each equation by graphing the related function. 1a. x2 - 8x - 16 = 2x2 1b. 6x + 10 = - x2 1c. -x2 + 4 = 0 CONFIDENTIAL

Step 1 Write the related function. Aquatics Application A dolphin jumps out of the water. The quadratic function y = -16x2 + 20x models the dolphin’s height above the water after x seconds. About 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 + 20x to find the times when the dolphin leaves and reenters the water. Step 1 Write the related function. o = -16x2 + 20x y = -16x2 + 20x CONFIDENTIAL

Step 2 Graph the related function. Use a graphing calculator. x = 1.25 y = 0 y = -16x2 + 20x Step 2 Graph the related function. Use a graphing calculator. Step 3 Find the zeros of the related function. The zeros appear to be 0 and 1.25. The dolphin leaves the water at 0 seconds and reenters the water at 1.25 seconds. Check 0 = -16x2 + 20x 0 -16(1.25)2 -20(1.25) 0 -16(1.5625) + 25 0 -25 + 25 0 0 Substitute 1.25 for x in the quadratic equation. CONFIDENTIAL

2) Another dolphin jumps out of the water. The Now you try! 2) Another dolphin jumps out of the water. The quadratic function y = -16x2 + 32x models the dolphin’s height above the water after x seconds. About how long is the dolphin out of the water? CONFIDENTIAL

Solve each equation by graphing the related function. Assessment Solve each equation by graphing the related function. 1) x2 - 4 = 0 2) x2 = 16 3) -2x2 - 6 = 0 CONFIDENTIAL

Solve each equation by graphing the related function. 4) - x2 + 12x - 36 = 0 5) - x2 = -9 6) 2x2 = 3x2 - 2x - 8 CONFIDENTIAL

Solve each equation by graphing the related function. 7) x2 - 6x + 9 = 0 8) 8x = -4x2 - 4 9) x2 + 5x + 4 = 0 CONFIDENTIAL

10) A baseball coach uses a pitching machine to simulate pop flies during practice. The baseball is shot out of the pitching machine with a velocity of 80 feet per second. The quadratic function y = -16x2 + 80x models the height of the baseball after x seconds. How long is the baseball in the air? CONFIDENTIAL

Solving Quadratic Equations by Graphing Let’s review Solving Quadratic Equations by Graphing 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 + c 0 = ax2 + bx + c ax2 + bx + c = 0 Notice that when writing a quadratic function as its related quadratic equation, you replace y with 0. So y = 0. CONFIDENTIAL

Solving Quadratic Equations by Graphing 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. Solving Quadratic Equations by Graphing Step 1 Write the related function. Step 2 Graph the related function. Step 3 Find the zeros of the related function. CONFIDENTIAL

Solving Quadratic Equations by Graphing Solve each equation by graphing the related function. A) 2x2 - 2 = 0 Step 1 Write the related function. 2x2 - 2 = y, or y = 2x2 + 0x - 2 Step 2 Graph the related function. • The axis of symmetry is x = 0. • The vertex is (0, -2) . • Two other points are (1, 0) and (2, 6). • Graph the points and reflect them across the axis of symmetry. CONFIDENTIAL

Step 3 Find the zeros of the related function. The zeros appear to be -1 and 1. Check 2x2 - 2 = 0 2(-1)2 - 2 0 2(1) – 2 0 0 0 2x2 - 2 = 0 2(1)2 - 2 0 2(1) – 2 0 0 0 Substitute -1 and 1 for x in the quadratic equation. CONFIDENTIAL

Step 1 Write the related function. Aquatics Application A dolphin jumps out of the water. The quadratic function y = -16x2 + 20x models the dolphin’s height above the water after x seconds. About 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 + 20x to find the times when the dolphin leaves and reenters the water. Step 1 Write the related function. o = -16x2 + 20x y = -16x2 + 20x CONFIDENTIAL

Step 2 Graph the related function. Use a graphing calculator. x = 1.25 y = 0 y = -16x2 + 20x Step 2 Graph the related function. Use a graphing calculator. Step 3 Find the zeros of the related function. The zeros appear to be 0 and 1.25. The dolphin leaves the water at 0 seconds and reenters the water at 1.25 seconds. Check o = -16x2 + 20x 0 -16(1.25)2 -20(1.25) 0 -16(1.5625) + 25 0 -25 + 25 0 0 Substitute 1.25 for x in the quadratic equation. CONFIDENTIAL

You did a great job today! CONFIDENTIAL