Chapter 7.3, Problem 77E

To determine

To find the equation of the parabola $y=a{x}^2+bx+c$ which passes through the points $\left(-1,1\right),\left(0,-4\right),\left(1,-13\right)$ .

Answer to Problem 77E

The equation of the parabola is $y=-2x^2-7x-4$ and also the points $\left(-1,1\right),\left(0,-4\right),\left(1,-13\right)$ passes through the parabola,

  

Explanation of Solution

Given information :

General form of parabola, $y=a{x}^2+bx+c$ which passes through the points $\left(-1,1\right),\left(0,-4\right),\left(1,-13\right)$ .

Calculation:

Consider, the general form $y=a{x}^2+bx+c$ and the points $\left(-1,1\right),\left(0,-4\right),\left(1,-13\right)$ .

Substitute the point $\left(-1,1\right)$ for $\left(x,y\right)$ in $y=a{x}^2+bx+c$

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

Substitute the point $\left(0,-4\right)$ for $\left(x,y\right)$ in $y=a{x}^2+bx+c$

  $\begin{array}{l}\Rightarrow -4=a{\left(0\right)}^2+b\left(0\right)+c\\ \Rightarrow -4=c\end{array}$

Substitute the point $\left(1,-13\right)$ for $\left(x,y\right)$ in $y=a{x}^2+bx+c$

  $\begin{array}{l}\Rightarrow -13=a{\left(1\right)}^2+b\left(1\right)+c\\ \Rightarrow -13=a+b+c\end{array}$

Hence,

  $\begin{array}{l}a-b+c=1---(1)\\ a+b+c=-13---(2)\\ c=-4---(3)\end{array}$ are the set of equation

Solving for $a,b$ substitute $c=-4$ in (1) and (2),

  $\begin{array}{c}a-b=1+4\\ a+b=-13+4\end{array}\Rightarrow \begin{array}{c}a-b=5\\ a+b=-9\end{array}$

By subtracting the above equations,

  $a=-2$

Now substitute $a=-2$ and $c=-4$ in $a+b+c=-13$ ,

  $\begin{array}{l}\Rightarrow \left(-2\right)+b+\left(-4\right)=-13\\ \Rightarrow b=-13+6\\ \Rightarrow b=-7\end{array}$

Therefore, $a=-2,b=-7,c=-4$ . Substitute the values in the general form.

  $\begin{array}{l}y=(-2)x^2+\left(-7\right)x+\left(-4\right)\\ y=-2x^2-7x-4\end{array}$

The equation of the parabola is, $y=-2x^2-7x-4$

By graphing the parabola and plotting the points in the graph,

  

It is clear that the points $\left(-1,1\right),\left(0,-4\right),\left(1,-13\right)$ passes through the parabola $y=-2x^2-7x-4$ .