Find the function with x-intercepts (2, 0) and (-4, 0), which also goes through the point (-3, 10).
Answers
Given : A function whose x-intercepts (2, 0) and (-4, 0), which also goes through the point (-3, 10).
To Find : Function
Solution:
Assuming function to be Quadratic
as it has 2 x intercepts
(2, 0) and (-4, 0)
f(x) = ax² + bx + c
f(2) = 0
=> 4a + 2b + c = 0
f(-4) = 0
=> 16a - 4b + c = 0
=> 12a - 6b = 0
=> b = 2a
4a + 2b + c = 0
=> 4a + 4a + c = 0
=> 8a + c = 0
also goes through the point (-3, 10)
=> f(-3) = 10
=> 9a - 3b + c = 10
=> 9a - 6a + c = 10
=> 3a + c = 10
8a + c = 0
3a + c = 10
=> 5a = -10
=> a = - 2
Hence c = 16
b = -4
-2x² - 4x + 16
f(x) = -2x² - 4x + 16
= -2(x - 2)(x + 4)
Another way to solve
as function passes through (2, 0) and (-4, 0)
hence f(x) = a(x - 2)(x -(-4))
=> f(x) = a (x - 2)(x + 4)
f(-3) = 10
=> 10 = a(-3 - 2)(-3 + 4)
=> 10 = a(-5) (1)
=> a = - 2
f(x) = -2(x - 2)(x + 4)
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