Chapter 7, Problem 44RE

To determine

To calculate: the equation of the parabola $y=a{x}^2+bx+c$ that passes through the given points and then verify the result by graphing the parabola.

Answer to Problem 44RE

The equation of the parabola is $y=2x^2+x-5$

The graph is

  

Explanation of Solution

Given information:

The equation of the parabola is given as $y=a{x}^2+bx+c$ and it passes through the points $\left(2,5\right),\left(1,-2\right),\left(0,-5\right)$ from the figure.

  

Calculation:

First substitute $\left(2,5\right)$ in the given equation

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

Now substitute $\left(1,-2\right)$ the next point to get other equation.

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

Now substitute $\left(0,-5\right)$ the last point

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

Substitute c value in both the equations $\left(i\right)$ and $\left(ii\right)$ and solve them.

  $\begin{array}{l}4a+2b+\left(-5\right)=5\\ 4a+2b=10\\ 2a+b=5\cdots \left(iii\right)\\ a+b+\left(-5\right)=-2\\ a+b=3\cdots \left(iv\right)\end{array}$

Subtract equations $\left(iii\right)$ and $\left(iv\right)$ to eliminate a term.

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

Substitute a in $\left(iv\right)$ and get value of b

  $\begin{array}{l}a+b=3\\ 2+b=3\\ b=1\end{array}$

The equation of the parabola is $y=2x^2+x-5$

The graph is as follows

  

Conclusion:

The equation of the parabola is $y=2x^2+x-5$