Chapter 4.9, Problem 13E

To determine

To find: The error in the formation of a cubic function.

Answer to Problem 13E

The required equation is $y=3\left(x+6\right)\left(x-1\right)\left(x-3\right)$ .

Explanation of Solution

Given information:

The given points are:

  $\left(-6,0\right)$ , $\left(1,0\right)$ , $\left(3,0\right)$ and $\left(0,54\right)$ .

Formula used:

Equation of cubic root:

  $y=a\left(x-x_1\right)\left(x-x_2\right)\left(x-x_3\right)$

Consider the x values from the given graph

  $x_1=-6$

  $x_2=1$

  $x_3=3$

Substitute the above values in the equation of cube root $y=a\left(x-x_1\right)\left(x-x_2\right)\left(x-x_3\right)$

  $\begin{array}{c}y=a\left(x-\left(-6\right)\right)\left(x-1\right)\left(x-3\right)\\ =a\left(x+6\right)\left(x-1\right)\left(x-3\right)\end{array}$

Consider the point $\left(0,54\right)$

Substitute $0$ for x and $54$ for y in the above equation.

  $\begin{array}{c}y=a\left(x+6\right)\left(x-1\right)\left(x-3\right)\\ 54=a\left(0+6\right)\left(0-1\right)\left(0-3\right)\\ 54=18a\\ a=3\end{array}$

Substitute $3$ for a in the above equation.

  $y=3\left(x+6\right)\left(x-1\right)\left(x-3\right)$

Hence, the required equation is $y=3\left(x+6\right)\left(x-1\right)\left(x-3\right)$