We want to find a cubic function that satisfies the given conditions. Let's think about what the given information tells us.
A cubic function is a 3-degree function — this degree tells us that the function can have one or three real zeros.
The function passes through (0,6) — this tells us the location of the y-intercept
Let's notice that we do not have any information about the zeros, so we can choose any values we want, except for 0, which is already a y-intercept, and then find such a that the function will be passing through the given point. Let's recall the general form of a factored cubic equation.
f(x)=a(x-p)(x-q)(x-r)In the above equation, p, q, and r are the x-intercepts of the function. For simplicity we will let p= 1, q= 2 and r=3.
f(x)=a(x- 1)(x- 2)(x-3)
Now, since the function passes through the point (0,6), we can substitute 0 for x and 6 for f(x) in our partial equation, and solve for a.
Knowing that a=- 1 we can write the equation of our cubic function.
f(x)=- 1(x-1)(x-2)(x-3)
⇕
f(x)=- (x-1)(x-2)(x-3)
Finally, let's rewrite the function in order to obtain its standard form.
