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

(- 5, 0),

(1, 0),

and

(4, 0)

lie on the graph. Therefore,

- 5,

1,

and

4

are roots of the equation.

f (x) = a (x - (- 5)) (x - 1) (x - 4) ⇕ f (x) = a (x + 5) (x - 1) (x - 4)

Finally, we will find the value of

a

using the fact that the curve also passes through the point

(2, - 2) .

We can substitute

2

for

x

and

- 2

for

f (x)

in our equation with the roots.

f (x) = a (x + 5) (x - 1) (x - 4)

$x = 2$ , $f (x) = - 2$

- 2 = a (2 + 5) (2 - 1) (2 - 4)

▼

Solve for

a

Add and subtract terms

- 2 = a (7) (1) (- 2)

- 2 = - 14 a

$LHS / - 14 = RHS / - 14$

\frac{- 2}{- 1 4} = a

$\frac{a}{b} = \frac{a / - 2}{b / - 2}$

\frac{1}{7} = a

Rearrange equation

a = \frac{1}{7}

Now that we know that

a = \frac{1}{7},

we can write the full equation of the cubic function.

f (x) = \frac{1}{7} (x + 5) (x - 1) (x - 4)

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