Chapter ISG, Problem 9.3.2P

(a)

To determine

To find: The solution of the equations.

Answer to Problem 9.3.2P

  $\begin{array}{l}m=36\\ n=13\end{array}$

Explanation of Solution

Given: The equations are $y=x^2+12x+m;y=2x^2-nx+72;y=-x^2+3x-10$

  

According to graph.

  $\begin{array}{l}\text{Vertex of }f\left(x\right)=\left(6.5,-12.5\right)\\ \text{Vertex of }g\left(x\right)=\left(2,-8\right)\\ \text{Vertex of }h\left(x\right)=\left(-6,0\right)\end{array}$

Now change each given function into vertex form.

  $\begin{array}{l}y=x^2+12x+m\\ y={\left(x+6\right)}^2+m-36\\ h\left(x\right)={\left(x+6\right)}^2+m-36\end{array}$

So,

  $\begin{array}{l}m-36=0\\ m=36\end{array}$

  $\begin{array}{l}y=2x^2-nx+72\\ y=2{\left(x-\frac{n}{2}\right)}^2+72-\frac{n^2}{4}=f\left(x\right)\\ \text{So, }\frac{n}{2}=6.5\\ n=13\end{array}$

  $\begin{array}{l}y=-x^2+3x-10\\ y=-{\left(x-\frac{3}{2}\right)}^2-\frac{31}{4}=h\left(x\right)\end{array}$

(b)

To determine

To choose: The appropriate value of m

Answer to Problem 9.3.2P

  $m=36$

Explanation of Solution

Given: The equations are $y=x^2+12x+m;y=2x^2-nx+72;y=-x^2+3x-10$

  

  $\begin{array}{l}y=x^2+12x+m\\ y={\left(x+6\right)}^2+m-36\\ h\left(x\right)={\left(x+6\right)}^2+m-36\end{array}$

According to graph $\text{Vertex of }h\left(x\right)=\left(-6,0\right)$

So, $m-36=0\Rightarrow m=36$

(c)

To determine

To choose: The appropriate value of m

Answer to Problem 9.3.2P

  $n=13$

Explanation of Solution

Given: The equations are $y=x^2+12x+m;y=2x^2-nx+72;y=-x^2+3x-10$

  

  $\begin{array}{l}y=2x^2-nx+72\\ y=2{\left(x-\frac{n}{2}\right)}^2+72-\frac{n^2}{4}=f\left(x\right)\end{array}$

According to graph $\text{Vertex of }f\left(x\right)=\left(6.5,-12.5\right)$

So, $\frac{n}{2}=6.5\Rightarrow n=13$