Chapter 7, Problem 46RE

To determine

Find the equation of the parabola.

Answer to Problem 46RE

  $\boxed{y=3x^2+11x-14}$

The graph is verified.

Explanation of Solution

Given information:

Find the equation of the parabola $y=a{x}^2+bx+c$ that passes through the points. Verify your result using a graphing utility.

  

Calculation:

Consider the given graph,

  

To find the value constants $a,b,c$ refer the graph that parabola passes through the points $\left(-5,6\right),\left(1,0\right)\&\left(2,20\right)$

Let parabola passes through the points $\left(-5,6\right)$

  $y=a{x}^2+bx+c$

  $\begin{array}{l}6=a{\left(-5\right)}^2+b\left(-5\right)+c\\ 25a-5b+c=\mathrm{6......}\left(1\right)\end{array}$

Parabola passes through the points $\left(1,0\right)$

  $\begin{array}{l}0=a{\left(1\right)}^2+b\left(1\right)+c\\ a+b+c=\mathrm{0......}\left(2\right)\end{array}$

Parabola passes through the points $\left(2,20\right)$

  $\begin{array}{l}20=a{\left(2\right)}^2+b\left(2\right)+c\\ 4a+2b+c=\mathrm{20......}\left(3\right)\end{array}$

Now subtract equation $\left(2\right)$ from equation $\left(1\right)$

  $\frac{\begin{array}{l}25a-5b+c=6\\ -a-b-c=0\end{array}}{\begin{array}{l}24a-6b=6\\ or\\ 4a-b=1\end{array}}$

Now subtract equation $\left(2\right)$ from equation $\left(3\right)$

  $\frac{\begin{array}{l}4a+2b+c=20\\ -a-b-c=0\end{array}}{3a+b=20}$

Now add $3a+b=20$ and $4a-b=1$

  $\frac{\begin{array}{l}3a+b=20\\ 4a-b=1\end{array}}{\begin{array}{l}7a=21\\ a=3\end{array}}$

Now put $a=3$ in equation $4a-b=1$

  $\begin{array}{l}4\times 3-b=1\\ -b=-11\\ b=11\end{array}$

Now put $a=3$ and $b=11$ in equation $\left(2\right)$

  $\begin{array}{l}a+b+c=0\\ 3+11+c=0\\ c=-14\end{array}$

The equation of the parabola $\boxed{y=3x^2+11x-14}$

use $TI-83$ calculator to verify the equation of parabola,

press $Y=$ button and enter the equation of parabola. The display will be,

  

Press the window button set the range of axis,

  

Press graph button to view the graph,

  

Press 2^nd and then TRACE button to choose 1and press enter.

  

Type $X=-5$ and press enter.

  

Type $X=1$ and press enter

  

Type $X=2$ and press enter

  

Hence, the graph is verified.

Hence, the equation of the parabola $\boxed{y=3x^2+11x-14}$