A student draws two parabolas on graph paper. Both parabolas cross the x-axis at (–4, 0) and (6, 0). The y-intercept of the first parabola is (0, –12). The y-intercept of the second parabola is (0, –24). What is the positive difference between the a values for the two functions that describe the parabolas? Write your answer as a decimal rounded to the nearest tenth.
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Answer:
0.5
Step-by-step explanation:
(x - h)² = 4p(y - k)
(-4 - h)² = -4pk
(6 - h)² = - 4pk
=> (-4 - h)² = (6 - h)²
=> 16 + h² + 8h = 36 + h² - 12h
=> 20h = 20
=> h = 1
(6 - h)² = - 4pk
=> 5² = - 4pk
=> 25 = - 4pk
Parabola 1
(x -1)² = 4p(y - k)
(0-1)² = 4p(-12 - k)
1 = -48p - 4pk
1 = -48p + 25
=> 48p = 24
=> p = 1/2
=> 25 = -4(1/2)k
=> k = -25/2
(x - 1)² = 2(y + 25/2)
=> y = ((x - 1)² - 5²)/2
=> y = (x + 4)(x-6)/2
Parabola 2
(x -1)² = 4p(y - k)
(0-1)² = 4p(-24 - k)
1 = -96p - 4pk
1 = -96p + 25
=> 96p = 24
=> p = 1/4
=> 25 = -4(1/4)k
=> k = -25
(x - 1)² = (y + 25)
=> y = (x - 1)² - 5²
=> y = (x + 4)(x-6)
Difference = 1 - 1/2 = 1/2 = 0.5
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