Write a cubic function whose graph passes through the points (−4, 0), (4, 0), (0, 6) and (2, 0)

Question

Write a cubic function whose graph passes through the points (−4, 0), (4, 0), (0, 6) and (2, 0)

f(x) =

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Answer 1

f(x) = a(x+4)(x-4)(x-2) Develop general equation based on given zeroes

f(0) = a(0+4)(0-4)(0-2) Point (0,6) ==> When x=0, y=6

6 = a(4)(-4)(-2)

6 = a(32)

6/32 = a

3/16 = a

f(x) = (3/16)(x+4)(x-4)(x-2) Particular equation

Answer 2

The zeros (x-intercepts) are -4, 4, and 2 so that means the factors of the cubic are (x + 4)(x - 4)(x - 2) however, there can be a multiplier in front of those so we can use "a" as the multiplier and write:

y = a(x + 4)(x - 4)(x - 2)

To find the value of "a", we can plug in the other data point (0, 6) and solve for "a":

6 = a(0 + 4)(0 - 4)(0 - 2)

6 = a•4•(-4)•(-2)

6 = 32a

a = 6/32 = 3/16

So, the cubic function is: y = 3/16(x + 4)(x - 4)(x - 2)

This can be multiplied out if needed. (x + 4)(x - 4) = x² - 16 and then (x² - 16)(x - 2) = x³ - 2x² -16x + 32 and then 3/16(x³ - 2x² -16x + 32) = 3/16x³ - 3/8x² - 3x + 6 so:

y = 3/16x³ - 3/8x² - 3x + 6