5.5 The x-Intercepts of a Quadratic Relation In the James Bond film, The Man with the Golden Gun, a complicated spiral car jump was filmed in only one take. The exact speed of the car had been determined by a computer, so the stunt could be performed precisely as planned. For stunts like this and others, it is important for a stunt coordinator to model not only where a car will begin to spiral and its general path, but also where it will land. Investigate Tools • graphing calculator Compare the Equation of a Quadratic Relation to Its Graph 1. a) Graph the quadratic relation y = x2 + 10x + 16. What are the x-intercepts? b) Factor the expression on the right side of the relation. c) Compare the x-intercepts to the constant terms in the binomial factors of the factored form of the relation. What do you notice? d) Graph the factored relation in the same window. What do you notice? 2. Copy and complete the table. Find the x-intercepts without graphing. Relation y= x2 Factored Relation x-Intercepts + 10x + 21 y = x2 - 8x + 15 y = x2 + 2x - 24 y = x2 - 49 264 MHR • Chapter 5 FFCM11_CH05_FIN.indd 264 5/4/07 4:19:06 PM 3. Graph each factored relation from question 2. Use the graph to find intercept form • a quadratic relation of the form y = a(x - r)(x - s) • the constants, r and s, represent the x-intercepts of the relation zeros • the x-coordinates of the points where the graph of a relation crosses the x-axis • the x-intercepts of a relation • the values of x for which y = 0 Example 1 the x-intercepts. How do these x-intercepts compare to those you found in question 2? 4. Reflect Given a quadratic relation in the form y = x2 + bx + c, how can you find the x-intercepts without graphing? The relation y = x2 + 5x + 6 can be expressed as y = (x + 2)(x + 3). This is the intercept form of the quadratic relation. The x-intercepts are -2 and -3. In general, to find the x-intercepts of a quadratic relation y = ax2 + bx + c, first write the relation in intercept form, y = a(x - r)(x - s). The x-intercepts are at x = r and x = s. Since y = 0 at the points where the graph crosses the x-axis, the x-intercepts are also called the zeros of a quadratic relation. Factor to Find the Zeros of a Quadratic Relation Factor each quadratic relation. Use the factors to find the zeros. Then, sketch the graph using the zeros and the y-intercept. Refer to the a-value to decide if the parabola opens upward or downward. a) y = 4x2 + 4x - 168 b) y = -3x2 + 24x - 48 c) y = x2 - 8x d) y = x2 + 3x + 20 Solution a) y = 4x2 + 4x - 168 = 4[x2 + x - 42] Factor out the greatest common factor, 4. = 4(x + 7)(x - 6) Factor the resulting trinomial. The zeros are at x = -7 and x = 6. From the original relation, the y-intercept is -168. To sketch the graph of the relation, plot the zeros and the y-intercept. Since a > 0, the graph opens upward. 5.5 The x-Intercepts of a Quadratic Relation • MHR FFCM11_CH05_FIN.indd 265 265 5/4/07 4:19:07 PM y 20 (-7, 0) -10 -8 -6 -4 -2 0 2 (6, 0) 4 6 8 10 x -20 -40 -60 -80 y = 4x2 + 4x - 168 -100 -120 -140 (0, -168) -160 b) y = -3x2 + 24x - 48 = -3[x2 - 8x + 16] Factor out the greatest common factor, -3. = -3(x - 4)(x - 4) = -3(x - 4)2 Since both factors are the same, there is only one zero at x = 4. The intercept form of this relation is the same as the vertex form. The vertex is (4, 0); this is also the x-intercept of the graph. From the original relation, the y-intercept is -48. Since a < 0, the graph opens downward. y -4 -2 0 -20 2 (4, 0) 4 6 8 10 12 14 x y = -3x2 + 24x - 48 -40 (0, -48) -60 -80 -100 -120 -140 -160 266 MHR • Chapter 5 FFCM11_CH05_FIN.indd 266 5/4/07 4:19:07 PM c) y = x2 - 8x = x(x - 8) = (x + 0)(x - 8) The greatest common factor is x. You can also write the factors this way. The zeros of this relation are at x = 0 and x = 8. From the original relation, the y-intercept is 0. Since a > 0, the graph opens upward. y 20 16 y = x2 - 8x 12 8 4 (0, 0) -4 -2 0 2 4 6 8 (8, 0) 10 12 14 x -4 -8 -12 -16 d) y = x2 + 3x + 20 To factor this relation, try to find two values whose product is 20 and whose sum is 3. There are no such values. There are two possible reasons for this: • the relation has zeros that are not integers • the relation has no zeros Graph the relation to determine which reason applies to the relation. The graph of y = x2 + 3x + 20 does not cross the x-axis, so the relation has no zeros. 5.5 The x-Intercepts of a Quadratic Relation • MHR FFCM11_CH05_FIN.indd 267 267 5/4/07 4:19:07 PM Example 2 Different Forms of a Quadratic Relation Consider the quadratic relation y = 3(x + 4)2 - 108. a) What do you know about the graph of the given relation? Graph this relation. b) Write the relation in standard form. What do you know about the graph of the relation from the standard form? Graph the relation. c) Write the relation in intercept form. What do you know about the graph of the relation given the intercept form? Graph the relation. d) Compare the graphs of each form of the relation. Check your graphs using technology. Solution a) From the given relation, the vertex is at (-4, -108). Since a = 3, the graph opens upward. y -10 -8 -6 -4 -2 0 -20 2 4 6 8 10 x y = 3(x + 4)2 - 108 -40 -60 -80 -100 (-4, -108) b) y = 3(x + 4)2 - 108 = 3(x + 4)(x + 4) - 108 = 3[x2 + 4x + 4x + 16] - 108 = 3x2 + 12x + 12x + 48 - 108 = 3x2 + 24x - 60 From the standard form, the y-intercept is -60. y -10 -8 -6 -4 -2 0 -20 2 4 6 8 10 x y = 3x2 + 24x - 60 -40 -60 (0, -60) -80 -100 268 MHR • Chapter 5 FFCM11_CH05_FIN.indd 268 5/4/07 4:19:08 PM c) y = 3x2 + 24x - 60 = 3[x2 + 8x - 20] = 3(x - 2)(x + 10) From the intercept form, the zeros, or x-intercepts, are at x = 2 and x = -10. y (-10, 0) -10 -8 -6 -4 -2 0 2 (2, 0) 4 6 8 10 x -20 -40 y = 3(x - 2)(x + 10) -60 -80 -100 d) The graphs are identical for all forms of the relation. Use a graphing calculator to check. Example 3 Projectile Motion A football is kicked from ground level. Its path is given by the relation h = -4.9t2 + 22.54t, where h is the ball’s height above the ground, in metres, and t is the time, in seconds. a) Write the relation in intercept form. b) Use the intercept form of the relation. Make a table of values with times from 0.5 s to 3.5 s, in increments of 0.5 s. c) Use the intercept form of the relation to find the zeros. d) Plot the zeros and the points from the table of values. Draw a smooth curve through the points. e) When did the ball hit the ground? Explain how you found your answer. 5.5 The x-Intercepts of a Quadratic Relation • MHR FFCM11_CH05_FIN.indd 269 269 5/4/07 4:19:08 PM Solution a) There is no constant term. Factor out -4.9t. h = -4.9t2 + 22.54t = -4.9t(t - 4.6) b) Time (s) Height (m) 0.5 10.045 1.0 17.640 1.5 22.785 2.0 25.480 2.5 25.725 3.0 23.520 3.5 18.865 c) The intercept form is h = -4.9t(t - 4.6). The zeros are at t = 0 and t = 4.6. d) Height of a Football y 30 Height (m) 25 20 15 10 5 0 1 2 3 4 5 x Time (s) e) The zeros represent the times when the ball was on the ground. One of the zeros is at t = 0, which is when the ball was kicked. The other zero is at t = 4.6, which is when the ball landed. The ball hit the ground at 4.6 s. 270 MHR • Chapter 5 FFCM11_CH05_FIN.indd 270 5/4/07 4:19:08 PM Key Concepts • Given a quadratic relation in intercept form, y = a(x - r)(x - s), the zeros, or x-intercepts, are r and s. • The vertex, standard, and intercept forms of a quadratic relation give the same parabola when graphed. Discuss the Concepts D1. What do you know about the graph given each form of a quadratic relation? a) vertex form, y = a(x - h)2 + k b) standard form, y = ax2 + bx + c c) intercept form, y = a(x - r)(x - s) D2. The x-intercepts of a quadratic relation are at x = -3 and x = 5, and a = 5. Explain how you would find the standard form of the quadratic relation. Practise A For help with questions 1 to 5, refer to Example 1. 1. Find the x-intercepts of each quadratic relation. a) y 6 0 2 4 6 8 x -6 -12 -18 -24 b) y -8 -6 -4 -2 0 x -2 -4 -6 -8 -10 5.5 The x-Intercepts of a Quadratic Relation • MHR FFCM11_CH05_FIN.indd 271 271 5/4/07 4:19:08 PM 2. Find the zeros of each quadratic relation. a) y b) y 24 12 20 10 16 8 12 6 8 4 4 2 -8 -4 0 4 8 -4 -2 0 x 2 4 x 3. Find the zeros of each quadratic relation. a) y = (x - 5)(x + 3) b) y = (x - 4)(x - 1) c) y = 5(x - 9)(x - 9) d) y = 3(x - 7)(x + 6) e) y = -2(x + 8)(x + 2) f) y = -3x(x + 5) 4. Find the zeros by factoring. Check by graphing the intercept form and the standard form of each relation. a) y = x2 + 10x + 16 b) y = x2 - 2x - 35 c) y = x2 - 6x - 7 d) y = 5x2 - 125 e) y = 3x2 + 39x + 108 f) y = 2x2 - 28x + 98 5. Find the zeros by factoring. Check by graphing the intercept form and the standard form of each relation. a) y = 4x2 - 16x b) y = 5x2 - 125x c) y = -5x2 + 5x + 360 d) y = -x2 - 18x - 81 e) y = -3.9x2 + 19.5x f) y = 7.5x2 + 90x + 270 For help with questions 6 and 7, refer to Example 2. 6. Which relations have more than one zero? Explain how you know. a) y = 3(x - 15)2 + 2 b) y = -5(x + 2)2 + 9 c) y = -(x - 8)2 - 6 d) y = 9(x + 3)2 - 10 7. Given each quadratic relation in vertex form, express the relation in standard form and in intercept form. Then, check your answers by graphing all three forms. a) y = (x + 5)2 - 4 b) y = (x - 3)2 - 36 c) y = -2(x + 4)2 + 8 d) y = 6(x + 2)2 - 6 e) y = 3(x - 4)2 - 48 f) y = -4(x - 5)2 + 100 272 MHR • Chapter 5 FFCM11_CH05_FIN.indd 272 5/4/07 4:19:08 PM Apply B 8. A skateboarder jumps a gap that is 1.3 m wide. Her path can be modelled by the relation h = -1.25d2 + 1.875d, where h is her height above the ground and d is her horizontal distance from the edge of the gap, both in metres. a) Write the relation in intercept form. b) Determine the zeros of the relation. Will the skateboarder make it across the gap? Explain. c) Copy and complete the table. Horizontal Distance (m) Height (m) 0 0.25 0.50 0.75 1.00 1.25 1.50 d) Estimate the maximum height the skateboarder reached during her jump. e) Graph the relation. 9. A second skateboarder jumps off a ledge. His path is modelled by the relation h = -0.8d2 + 0.8d + 1.6, where h is his height above the ground and d is his horizontal distance from the ledge, both in metres. a) What is the height of the ledge? b) Factor to find the zeros of the relation. c) At what point will the skateboarder land on the ground? d) Graph the relation. 5.5 The x-Intercepts of a Quadratic Relation • MHR FFCM11_CH05_FIN.indd 273 273 5/4/07 4:19:09 PM 10. In a target game at an amusement park, players launch a beanbag toward a bucket using a mallet and a small seesaw. The path of a beanbag that lands directly in the bucket can be modelled by the relation h = -0.65d2 + 1.625d, where h is the beanbag’s height above the table and d is the beanbag’s distance from the seesaw, both in metres. a) Find the zeros of the relation. b) How far is the bucket from the seesaw? c) Find the beanbag’s maximum height above the table to the nearest tenth of a metre. 11. The path of a stunt car can be modelled by the relation h = -0.1d2 + 0.5d + 3.6 where h is the car’s height above the ground and d is the car’s horizontal distance from the edge of the ramp, both in metres. a) Find the zeros of the relation. b) How far from the ramp will the car land? c) Suppose the stunt is done inside a sound studio with a ceiling height of 10 m. Will the car hit the ceiling? Explain your reasoning. 274 MHR • Chapter 5 FFCM11_CH05_FIN.indd 274 5/4/07 4:19:10 PM Extend C 12. A quadratic relation of the form y = ax2 + bx + c that cannot be factored might still have zeros. Another method for finding the zeros of a quadratic relation is to use the quadratic formula: _______ -b ±√b2 - 4ac x = __ 2a Use the quadratic formula to find the zeros of each relation. a) y = 3x2 + 21x + 30 b) y = 16x2 - 40x - 75 c) y = 2x2 + 5x - 6 13. A quadratic relation of the form y = ax2 + bx + c has zeros if the _______ -b ±√b2 - 4ac __ 2a is greater than or equal to zero. Determine if each relation has zeros. expression b2 - 4ac in the quadratic formula x = a) y = 5x2 + 3x + 15 b) y = 25x2 + 60x + 36 c) y = 7x2 - 10x + 5 14. A cannonball is shot from ground level with an initial velocity of 20 m/s. Ignoring air resistance, its path can be modelled by the 0.0125 d2 + (tan θ)d, where h is the cannonball’s relation h = -_ (cos θ)2 height above the ground, in metres, d is the horizontal distance from the cannon, in metres, and θ is the cannon’s angle of elevation, in degrees. What angle of elevation will allow the cannonball to travel the greatest distance? θ 5.5 The x-Intercepts of a Quadratic Relation • MHR FFCM11_CH05_FIN.indd 275 275 5/4/07 4:19:12 PM

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