Chapter 4.9, Problem 15E

To determine

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

Answer to Problem 15E

The required equations are $f\left(x\right)=\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$ , $g\left(x\right)=2\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$ and $h\left(x\right)=3\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$ .

Explanation of Solution

Given information:

The given points are:

  $\left(3,0\right)$ , $\left(4,0\right)$ and $\left(2,6\right)$ .

Formula used:

Equation of cubic root:

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

Calculation:

Consider the x values from the given graph

  $x_1=3$

  $x_2=4$

  $x_3=c$

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

  $y=\left(x-3\right)\left(x-4\right)\left(x-c\right)$

Consider the point $\left(2,6\right)$

Substitute $2$ for x and $6$ for y in the above equation.

  $\begin{array}{c}y=\left(2-3\right)\left(2-4\right)\left(2-c\right)\\ 6=\left(-1\right)\left(-2\right)\left(2-c\right)\\ 6=4-2c\\ 2c=-2\\ c=-1\end{array}$

Substitute $-1$ for c in the above equation.

  $f\left(x\right)=\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$

Substitute $2$ for a in the above equation, $f\left(x\right)=a\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$ .

  $g\left(x\right)=2\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$

Substitute $3$ for a in the above equation, $f\left(x\right)=a\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$ .

  $h\left(x\right)=3\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$

Hence, the required equations are $f\left(x\right)=\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$ , $g\left(x\right)=2\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$ and $h\left(x\right)=3\left(x-3\right)\left(x-4\right)\left(x-\left(-1\right)\right)$