9-4 Solving Quadratic Equations by Graphing Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

9-4 Solving Quadratic Equations by Graphing Warm Up 1. Graph y = x 2 + 4x Identify the vertex and zeros of the function above. vertex:(–2, –1); zeros:–3, –1

9-4 Solving Quadratic Equations by Graphing 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Also covered: 23.0 California Standards

9-4 Solving Quadratic Equations by Graphing Every quadratic function has a related quadratic equation. The standard form of a quadratic equation is ax 2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. y = ax 2 + bx + c 0 = ax 2 + bx + c When writing a quadratic function as its related quadratic equation, you replace y with 0.

9-4 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.

9-4 Solving Quadratic Equations by Graphing Additional Example 1A: Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. 2x 2 – 18 = 0 Step 1 Write the related function. 2x 2 – 18 = y, or y = 2x 2 + 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) ● ● ●

9-4 Solving Quadratic Equations by Graphing Additional Example 1A Continued Step 3 Find the zeros. 2x 2 – 18 = 0 The zeros appear to be 3 and –3. Substitute 3 and –3 for x in the original equation. 0 2(3) 2 – (9) – – 18 0 Check 2x 2 – 18 = 0 2x 2 – 18 = 0 The solutions of 2x 2 – 18 = 0 are 3 and –3. 2(–3) 2 – (9) – – Solve the equation by graphing the related function.

9-4 Solving Quadratic Equations by Graphing Additional Example 1B: Solving Quadratic Equations by Graphing –12x + 18 = –2x 2 Step 1 Write the related function. Step 2 Graph the function. y = 2x 2 – 12x x 2 – 12x + 18 = 0 Use a graphing calculator. Step 3 Find the zeros. The only zero appears to be 3. This means 3 is the only root of 2x 2 – 12x Solve the equation by graphing the related function.

9-4 Solving Quadratic Equations by Graphing Additional Example 1C: Solving Quadratic Equations by Graphing 2x 2 + 4x = –3 Step 1 Write the related function. y = 2x 2 + 4x + 3 Step 2 Graph the function. The axis of symmetry is x = –1. The vertex is (–1, 1). Two other points (0, 3) and (1, 9). Graph the points and reflect them across the axis of symmetry.  (–1, 1)     (0, 3) (1, 9) (–2, 3) (–3, 9) Solve the equation by graphing the related function.

9-4 Solving Quadratic Equations by Graphing Additional Example 1C Continued Step 3 Find the zeros. The function appears to have no zeros. 2x 2 + 4x = –3 The equation has no real-number solutions. Solve the equation by graphing the related function.

9-4 Solving Quadratic Equations by Graphing Check It Out! Example 1a Solve the equation by graphing the related function. x 2 – 8x – 16 = 2x 2 Step 1 Write the related function. y = x 2 + 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) ● ● ●

9-4 Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. Check It Out! Example 1a Continued Step 3 Find the zeros. The only zero appears to be –4. Check y = x 2 + 8x (–4) 2 + 8(–4) – x 2 – 8x – 16 = 2x 2 Substitute –4 for x in the quadratic equation.

9-4 Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. 6x + 10 = –x 2 Step 1 Write the related function. y = x 2 + 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) ● ● ●

9-4 Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. x 2 + 6x + 10 = 0 Check It Out! Example 1b Continued The equation has no real-number solutions. Step 3 Find the zeros. The function appears to have no zeros

9-4 Solving Quadratic Equations by Graphing Solve the equation by graphing the related function. –x = 0 Check It Out! Example 1c Step 1 Write the related function. y = –x 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).

9-4 Solving Quadratic Equations by Graphing Recall from Chapter 7 that a root of a polynomial is a value of the variable that makes the polynomial equal to 0. So, finding the roots of a quadratic polynomial is the same as solving the related quadratic equation.

9-4 Solving Quadratic Equations by Graphing Additional Example 2A: Finding Roots of Quadratic Polynomials Find the roots of x 2 + 4x + 3 Step 1 Write the related equation. 0 = x 2 + 4x + 3 Step 2 Write the related function. Step 3 Graph the related function. y = x 2 + 4x + 3 The axis of symmetry is x = –2. The vertex is (–2, –1). Two other points are (–3, 0) and (–4, 3) Graph the points and reflect them across the axis of symmetry. y = x 2 + 4x + 3      (–2, –1) (–3, 0) (–4, 3)

9-4 Solving Quadratic Equations by Graphing Additional Example 2A Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. The zeros appear to be –3 and –1. This means –3 and –1 are the roots of x 2 + 4x + 3. Check 0 = x 2 + 4x (–3) 2 + 4(–3) – = x 2 + 4x (–1) 2 + 4(–1) – 4 + 3

9-4 Solving Quadratic Equations by Graphing Additional Example 2B: Finding Roots of Quadratic Polynomials Find the roots of x 2 + x – 20 Step 1 Write the related equation. 0 = x 2 + x – 20 Step 2 Write the related function. Step 3 Graph the related function. y = x 2 + 4x – 20 The axis of symmetry is x = –. The vertex is (–0.5, –20.25). Two other points are (1, –18) and (2, –15) Graph the points and reflect them across the axis of symmetry. y = x 2 + 4x – 20 (–0.5, –20.25).  (1, –18)  (2, –15)   

9-4 Solving Quadratic Equations by Graphing Additional Example 2B Continued Find the roots of x 2 + x – 20 Step 4 Find the zeros. The zeros appear to be –5 and 4. This means –5 and 4 are the roots of x 2 + x – 20. Check 0 = x 2 + x – (–5) 2 – 5 – – 5 – 20 0 = x 2 + x – – – 20

9-4 Solving Quadratic Equations by Graphing Additional Example 2C: Finding Roots of Quadratic Polynomials Find the roots of x 2 – 12x + 35 Step 1 Write the related equation. 0 = x 2 – 12x + 35 y = x 2 – 12x + 35 Step 2 Write the related function. Step 3 Graph the related function. The axis of symmetry is x = 6. The vertex is (6, –1). Two other points (4, 3) and (5, 0) Graph the points and reflect them across the axis of symmetry. (6, –1).  (4, 3)  (5, 0)   

9-4 Solving Quadratic Equations by Graphing Additional Example 2C Continued Find the roots of x 2 – 12x + 35 Step 4 Find the zeros. The zeros appear to be 5 and 7. This means 5 and 7 are the roots of x 2 – 12x Check 0 = x 2 – 12x – 12(5) – = x 2 – 12x – 12(7) –

9-4 Solving Quadratic Equations by Graphing Check It Out! Example 2a Find the roots of each quadratic polynomial. x 2 + x – 2 Step 1 Write the related equation. 0 = x 2 + x – 2 Step 2 Write the related function. Step 3 Graph the related function. y = x 2 + x – 2 The axis of symmetry is x = –0.5. The vertex is (–0.5, –2.25). Two other points (–1, –2) and (–2, 0) Graph the points and reflect them across the axis of symmetry.  (–0.5, –2.25). (–1, –2)  (–2, 0)    y = x 2 + x – 2

9-4 Solving Quadratic Equations by Graphing Find the roots of each quadratic polynomial. Step 4 Find the zeros. The zeros appear to be –2 and 1. This means –2 and 1 are the roots of x 2 + x – 2. Check 0 = x 2 + x – (–2) 2 + (–2) – – 2 – 2 0 = x 2 + x – (1) – – 2 Check It Out! Example 2a Continued

9-4 Solving Quadratic Equations by Graphing Check It Out! Example 2b Find the roots of each quadratic polynomial. 9x 2 – 6x + 1 Step 1 Write the related equation. 0 = 9x 2 – 6x + 1 Step 2 Write the related function. Step 3 Graph the related function. y = 9x 2 – 6x + 1 y = 9x 2 – 6x + 1  (, 0). The axis of symmetry is x =. The vertex is (, 0). Two other points (0, 1) and (, 4) Graph the points and reflect them across the axis of symmetry. (, 4)  (0, 1)   

9-4 Solving Quadratic Equations by Graphing Find the roots of each quadratic polynomial. Step 4 Find the zeros. Check It Out! Example 2b Continued There appears to be one zero at. This means that is the root of 9x 2 – 6x + 1. Check 0 = 9x 2 – 6x ( ) 2 – 6( ) – 2 + 1

9-4 Solving Quadratic Equations by Graphing Check It Out! Example 2c Find the roots of each quadratic polynomial. 3x 2 – 2x + 5 Step 1 Write the related equation. 0 = 3x 2 – 2x + 5 y = 3x 2 – 2x + 5 Step 2 Write the related function. Step 3 Graph the related function. y = 3x 2 – 2x + 5 The axis of symmetry is x =. The vertex is (, ). Two other points (1, 6) and (, ) Graph the points and reflect them across the axis of symmetry.  (1, 6)    

9-4 Solving Quadratic Equations by Graphing Find the roots of each quadratic polynomial. Step 4 Find the zeros. Check It Out! Example 2c Continued There appears to be no zeros. This means that there are no real roots of 3x 2 – 2x + 5.

9-4 Solving Quadratic Equations by Graphing Additional Example 3: Application A frog jumps straight up from the ground. The quadratic function f(t) = –16t t 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 = –16t t to find the times when the frog leaves the ground and lands. Step 1 Write the related function. 0 = –16t t y = –16t t

9-4 Solving Quadratic Equations by Graphing Additional Example 3 Continued Step 2 Graph the function. Use a graphing calculator. Step 3 Use to estimate the zeros. The zeros appear to be 0 and 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.

9-4 Solving Quadratic Equations by Graphing Check 0 = –16t t 0 –16(0.75) (0.75) 0 –16(0.5625) – Substitute 0.75 for t in the quadratic equation. Additional Example 3 Continued

9-4 Solving Quadratic Equations by Graphing Check It Out! Example 3 What if…? A dolphin jumps out of the water. The quadratic function y = –16x x models the dolphin’s height above the water after x seconds. About how long is the dolphin out of the water? Check your answer. When the dolphin leaves the water, its height is 0, and when the dolphin reenters the water, its height is 0. So solve 0 = –16x x to find the times when the dolphin leaves and reenters the water. Step 1 Write the related function 0 = –16x x y = –16x x

9-4 Solving Quadratic Equations by Graphing 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 about 2 seconds. Check It Out! Example 3 Continued

9-4 Solving Quadratic Equations by Graphing Check It Out! Example 3 Continued Check 0 = –16x x 0 –16(2) (2) 0 –16(4) – Substitute 2 for x in the quadratic equation.

9-4 Solving Quadratic Equations by Graphing Lesson Quiz Solve each equation by graphing the related function. 1.3x 2 – 12 = 0 2.x 2 + 2x = 8 3.3x – 5 = x 2 4.3x = 6x 5.A rocket is shot straight up from the ground. The quadratic function f(t) = –16t t 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 ø 1 6 s