Chapter 7, Problem 45RE

Solution

To calculate: The coefficient of the function so that its graph goes through the given points.

The function is $f\left(x\right)=\frac{17}{840}{x}^3-\frac{33}{280}{x}^2-\frac{571}{420}x+\frac{386}{35}$ .

Given Information:

The points for the function are $\left(2,8\right),\left(4,5\right),\left(6,3\right),\left(9,4\right)$ .

Calculation:

Consider the given general function,

  $f\left(x\right)=a{x}^3+b{x}^2+cx+d$

Substitute the given point $\left(2,8\right)$ in the function $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ .

  $\begin{array}{c}8=a{\left(2\right)}^3+b{\left(2\right)}^2+2c+d\\ 8a+4b+2c+d=8\end{array}$

Substitute the given point $\left(4,5\right)$ in the function $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ .

  $\begin{array}{c}5=a{\left(4\right)}^3+b{\left(4\right)}^2+4c+d\\ 64a+16b+4c+d=5\end{array}$

Substitute the given point $\left(6,3\right)$ in the function $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ .

  $\begin{array}{c}3=a{\left(6\right)}^3+b{\left(6\right)}^2+6c+d\\ 216a+36b+6c+d=3\end{array}$

Substitute the given point $\left(9,4\right)$ in the function $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ .

  $\begin{array}{c}4=a{\left(9\right)}^3+b{\left(9\right)}^2+9c+d\\ 729a+81b+9c+d=4\end{array}$

Write the all obtained system as $AX=B$ matrix form.

  $A=\left[\begin{array}{cccc}8& 4& 2& 1\\ 64& 16& 4& 1\\ 216& 16& 4& 1\\ 729& 81& 9& 1\end{array}\right]$ , $X=\left[\begin{array}{c}a\\ b\\ c\\ d\end{array}\right]$ , and $B=\left[\begin{array}{c}8\\ 5\\ 3\\ 4\end{array}\right]$

Solve the equation $AX=B$ .

  $\begin{array}{c}{A}^{-1}AX=A^{-1}B\\ X=A^{-1}B\\ ={\left[\begin{array}{cccc}8& 4& 2& 1\\ 64& 16& 4& 1\\ 216& 36& 6& 1\\ 729& 81& 9& 1\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}8\\ 5\\ 3\\ 4\end{array}\right]\left[\text{using calculator}\right]\\ =\left[\begin{array}{c}17/840\\ -99/840\\ -1142/840\\ 9264/840\end{array}\right]\\ =\left[\begin{array}{c}17/840\\ -33/280\\ -571/420\\ 386/35\end{array}\right]\end{array}$

Substitute the obtained values in the function.

  $f\left(x\right)=\frac{17}{840}{x}^3-\frac{33}{280}{x}^2-\frac{571}{420}x+\frac{386}{35}$

Hence, the obtained function is $f\left(x\right)=\frac{17}{840}{x}^3-\frac{33}{280}{x}^2-\frac{571}{420}x+\frac{386}{35}$ .