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Homework 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 6 seconds to return to the ground? Warm Up 1. Graph y = x2 + 4x + 3. 2. Identify the vertex and zeros of the function above. vertex:(–2 , –1); zeros:–3, –1 Every quadratic function has a related quadratic equation. The standard form of a quadratic equation is 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. Function: Equation: y = ax2 + bx + c 0 = ax2 + bx + c 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, or roots, of the related function. Recall that a quadratic function may have two, one, or no zeros. Additional 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. x=0 ● ● (3, 0) ● ● (2, –10) ● (0, –18) Additional 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. The solutions of 2x2 – 18 = 0 are 3 and –3. Check 2x2 – 18 = 0 2(3)2 – 18 0 2(9) – 18 0 18 – 18 0 0 0 Substitute 3 and –3 for x in the original equation. 2x2 – 18 = 0 2(–3)2 – 18 2(9) – 18 18 – 18 0 0 0 0 0 Additional 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 (–3, 9) Step 2 Graph the function. (1, 9) • The axis of symmetry is x = –1. • The vertex is (–1, 1). (–2, 3) (0, 3) • Two other points (0, 3) and (–1, 1) (1, 9). • Graph the points and reflect them across the axis of symmetry. Additional Example 1C Continued Solve the equation by graphing the related function. 2x2 + 4x = –3 Step 3 Find the zeros. The function appears to have no zeros. The equation has no real-number solutions. Partner Share! 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 ●(–2 , 4) ● ● ● ● (–3, 1) (–4, 0) Partner Share! 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. Check y = x2 + 8x + 16 0 0 0 (–4)2 + 8(–4) + 16 16 – 32 + 16 0 Substitute –4 for x in the quadratic equation. Partner Share! Example 1b Solve the equation by graphing the related function. 6x + 10 = –x2 Step 1 Write the related function. x = –3 y = x2 + 6x + 10 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. ● (–1, 5) ● ● ● ● (–2, 2) (–3, 1) Partner Share! Example 1b Continued Solve the equation by graphing the related function. x2 + 6x + 10 = 0 Step 3 Find the zeros. The function appears to have no zeros The equation has no real-number solutions. 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. Additional Example 2A: Finding Roots of Quadratic Polynomials Find the roots of x2 + 4x + 3 Step 1 Write the related equation. 0 = x2 + 4x + 3 y = x2 + 4x + 3 Step 2 Write the related function. y = x2 + 4x + 3 Step 3 Graph the related function. (–4, 3) • The axis of symmetry is x = –2. (–3, 0) • The vertex is (–2, –1). (–2, –1) • Two other points are (–3, 0) and (–4, 3) • Graph the points and reflect them across the axis of symmetry. 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 x2 + 4x + 3. Check 0 = x2 + 4x + 3 0 0 0 (–3)2 + 4(–3) + 3 9 – 12 + 3 0 0 = x2 + 4x + 3 0 0 0 (–1)2 + 4(–1) + 3 1–4+3 0 Additional Example 2B: Finding Roots of Quadratic Polynomials Find the roots of x2 + x – 20 Step 1 Write the related equation. 2 + 4x – 20 y = x 2 0 = x + x – 20 Step 2 Write the related function. y = x2 + 4x – 20 Step 3 Graph the related function. • 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. (2, –15) (1, –18) (–0.5, –20.25). Additional Example 2B Continued Find the roots of x2 + x – 20 Step 4 Find the zeros. The zeros appear to be –5 and 4. This means –5 and 4 are the roots of x2 + x – 20. Check 0 = x2 + x – 20 0 (–5)2 – 5 – 20 0 0 25 – 5 – 20 0 0 = x2 + x – 20 0 42 + 4 – 20 0 16 + 4 – 20 0 0 Additional Example 2C: Finding Roots of Quadratic Polynomials Find the roots of x2 – 12x + 35 Step 1 Write the related equation. 2 – 12x + 35 y = x 2 0 = x – 12x + 35 Step 2 Write the related function. y = x2 – 12x + 35 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. (4, 3) (5, 0) (6, –1). Additional Example 2C Continued Find the roots of x2 – 12x + 35 Step 4 Find the zeros. The zeros appear to be 5 and 7. This means 5 and 7 are the roots of x2 – 12x + 35. Check 0 = x2 – 12x + 35 0 = x2 – 12x + 35 0 52 – 12(5) + 35 0 0 25 – 60 + 35 0 0 0 0 72 – 12(7) + 35 49 – 84 + 35 0 Partner Share! Example 2a Find the roots of each quadratic polynomial. x2 + x – 2 y = x2 + x – 2 Step 1 Write the related equation. 0 = x2 + x – 2 Step 2 Write the related function. y = x2 + x – 2 (–2, 0) Step 3 Graph the related function. (–1, –2) (–0.5, –2.25). • 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. Partner Share! Example 2a Continued 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 x2 + x – 2. Check 0 = x2 + x – 2 0 (–2)2 + (–2) – 2 0 = x2 + x – 2 0 0 4–2–2 0 0 0 0 12 + (1) – 2 1+1–2 0 Partner Share! Example 2b Find the roots of each quadratic polynomial. 9x2 – 6x + 1 y = 9x2 – 6x + 1 Step 1 Write the related equation. 0 = 9x2 – 6x + 1 ( , 4) Step 2 Write the related function. y = 9x2 – 6x + 1 Step 3 Graph the related function. (0, 1) • The axis of symmetry is x = . ( , 0). • The vertex is ( , 0). • Two other points (0, 1) and ( , 4) • Graph the points and reflect them across the axis of symmetry. Partner Share! Example 2b Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. There appears to be one zero at is the root of 9x2 – 6x + 1. Check 0 = 9x2 – 6x + 1 0 9( )2 – 6( 0 1–2+1 0 0 )+1 . This means that Partner Share! Example 2c Find the roots of each quadratic polynomial. 3x2 – 2x + 5 y = 3x2 – 2x + 5 Step 1 Write the related equation. 0 = 3x2 – 2x + 5 Step 2 Write the related function. (1, 6) y = 3x2 – 2x + 5 Step 3 Graph the related function. • 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. Partner Share! Example 2c Continued Find the roots of each quadratic polynomial. Step 4 Find the zeros. There appears to be no zeros. This means that there are no real roots of 3x2 – 2x + 5. Lesson Review! Solve each equation by graphing the related function. 1. 3x2 – 12 = 0 2, –2 2. x2 + 2x = 8 –4, 2 3. 3x – 5 = x2 ø 4. 3x2 + 3 = 6x 1 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 6 seconds to return to the ground?