Chapter 7.3, Problem 53E

To determine

The equation of the parabola that passes through the points. The graphing utility to plot the points and graph the parabola.

Answer to Problem 53E

  $y=4x^2-2x+1$ .

Explanation of Solution

Given information:

The points are $(\frac{1}{2},1),(1,3),(2,13)$ .

Calculation:

The given points are $(\frac{1}{2},1),(1,3),(2,13)$ .

Write three equations of $y=a{x}^2+bx+c$ using the three points, and simplify them:

  $\begin{array}{}$ a{(\frac{1}{2})}^2+b(\frac{1}{2})+c=1$$ a{(1)}^2+b(1)+c=3$$ a{(2)}^2+b(2)+c=13$\end{array}$

  $\begin{array}{l}\frac{1}{4}a+\frac{1}{2}b+c=3\\ a+b+c=3\\ 4a+2b+c=13\end{array}$

Subtract equation second from first equation:

  $\begin{array}{l}\underset{\_}{\begin{array}{l}\frac{1}{4}a+\frac{1}{2}b+c=3\\ a+b+c=3\\ ---\end{array}}\\ -\frac{3}{4}a-\frac{1}{2}b=-2\end{array}$

Subtract equation third from second equation:

  $\begin{array}{l}\underset{\_}{\begin{array}{l}a+b+c=3\\ 4a+2b+c=13\\ ---\end{array}}\\ -3a-b=-10\end{array}$

Subtract the resultant equation one from another:

  $\begin{array}{l}\underset{\_}{\begin{array}{l}-3/4a-1/2b=-2\\ -3/2a-1/2b=5\\ +\text{ + }-\end{array}}\\ 3/4a=3\\ a=4\end{array}$

Find the value of $b$ and $c$ :

  $\begin{array}{l}-12-b=-10\\ b=-2\\ 4-2+c=3\\ c=1\end{array}$

The parabola is

  $y=4x^2-2x+1$ .

  

Thus, the parabola is $y=4x^2-2x+1$ .