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 (1,0) and (7,0) — this tells us the location of two of the real zeros.
Using the general form of a factored cubic equation, we can write the factored form of a cubic function that passes through the given points.
f(x)=a(x-p)(x-q)(x-r)In the above equation, p, q, and r are the x-intercepts of the function. We are told that the points ( 1,0) and ( 7,0) lie on the graph. Therefore, 1 and 7 are roots of the equation.
f(x)=a(x- 1)(x- 7)(x-r)
Since we do not know the third root, we can choose whatever r we want. For simplicity, let r=
f(x)=a(x- 1)(x- 7)(x- )
⇕
f(x)=ax(x-1)(x-7)
Since a does not have any effect on the roots, we can choose any value. For simplicity, we will let a= 1.
f(x)= 1x(x-1)(x-7)
⇕
f(x)=x(x-1)(x-7)
Finally, let's rewrite the function in order to obtain its standard form.
Note that this is only one of the infinitely many functions with the given characteristics. Any function passing through (1,0) and (7,0) will be correct.
