Chapter 7, Problem 45RE

To determine

Find the equation of the parabola.

Answer to Problem 45RE

  $\boxed{y=2x^2+x-5}$

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 $\left(2,5\right),\left(1,-2\right)\&\left(0,-5\right)$

Let parabola passes through the points $\left(2,5\right)$

  $y=a{x}^2+bx+c$

  $\begin{array}{l}5=a{\left(2\right)}^2+b\left(2\right)+c\\ 4a+2b+c=\mathrm{5......}\left(1\right)\end{array}$

Parabola passes through the points $\left(1,-2\right)$

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

Parabola passes through the points $\left(0,-5\right)$

  $\begin{array}{l}-5=a{\left(0\right)}^2+b\left(0\right)+c\\ c=-\mathrm{5......}\left(3\right)\end{array}$

Now multiply equation $\left(2\right)$ by $-4$ and add with equation $\left(2\right)$

  $\left(1\right)$

  $\frac{\begin{array}{l}4a+2b+c=5\\ -4a-4b-4c=8\end{array}}{-2b-3c=13}$

The new system of equations is,

  $\begin{array}{l}4a+2b+c=5\\ -2b-3c=13\\ c=-5\end{array}$

Now substitute $c=-5$ in $-2b-3c=13$

  $\begin{array}{l}-2b-3\times -5=13\\ -2b=13-15\\ -2b=-2\\ b=1\end{array}$

Now substitute $c=-5\&b=1$ in equation $\left(1\right)$ .

  $\begin{array}{l}4a+2\times 1+\left(-5\right)=5\\ 4a=5+3\\ 4a=8\\ a=2\end{array}$

  $\boxed{y=2x^2+x-5}$

The graph shown that the points lie in the same parabola so verified.

  

Hence, the equation of the parabola $\boxed{y=2x^2+x-5}$