Using the general form of a factored cubic equation, we can write the factored form of the 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

(- 6, 0),

(- 3, 0),

and

(3, 0)

lie on the graph. Therefore,

- 6,

- 3,

and

3

are roots of the equation.

f (x) = a (x - (- 6)) (x - (- 3)) (x - 3) ⇕ f (x) = a (x + 6) (x + 3) (x - 3)

Finally, we will find the value of

a

using the fact that the curve also passes through the point

(0, - 9) .

We can substitute

0

for

x

and

- 9

for

f (x)

in our equation with the roots.

f (x) = a (x + 6) (x + 3) (x - 3)

$x = 0$ , $f (x) = - 9$

- 9 = a (0 + 6) (0 + 3) (0 - 3)

▼

Solve for

a

Add and subtract terms

- 9 = a (6) (3) (- 3)

- 9 = - 54 a

$LHS / - 54 = RHS / - 54$

\frac{- 9}{- 5 4} = a

$\frac{a}{b} = \frac{a / - 9}{b / - 9}$

\frac{1}{6} = a

Rearrange equation

a = \frac{1}{6}

Now that we know that

a = \frac{1}{6},

we can write the full equation of the cubic function.

f (x) = \frac{1}{6} (x + 6) (x + 3) (x - 3)

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