Chapter 9.9, Problem 9P

To determine

To determine the equation of a parabola that passes through the points $(-1,4)$ , $(2,7)$ and $(1,0)$ .

Answer to Problem 9P

  $y=3x^2-2x-1$ is the solution.

Explanation of Solution

Given:

A parabola that passes through the points $(-1,4)$ , $(2,7)$ and $(1,0)$ .

Formula Used:

The equation of a parabola is:

  $y=a{x}^2+bx+c$ where a, b and c are the co-efficient.

Calculation:

Given the parabola that passes through the points $(-1,4)$ , $(2,7)$ and $(1,0)$ .

We need to calculate the co-efficient for each of the three points by substituting the values of x and y for each point.

Hence, for the point $(-1,4)$ :

  $4=a\times {(-1)}^2+b\times (-1)+c$

  $4=a-b+c$

For the point $(2,7)$ :

  $7=a\times {(2)}^2+b\times (2)+c$

  $7=4a+2b+c$

For the point $(1,0)$ :

  $0=a\times {(1)}^2+b\times (1)+c$

  $0=a+b+c$

Solving the equation $4=a-b+c$ and $0=a+b+c$ we have:

  $4=2a+2c$

  $2=a+c$

  $c=2-a$

Putting the value of $c=2-a$ in the equation $0=a+b+c$ for getting the value of b, we have:

  $0=a+b+(2-a)$

  $b=-2$

Putting the value of $c=2-a$ and $b=-2$ in the equation $7=4a+2b+c$ for getting the value of a, we have:

  $7=4a+2\times (-2)+(2-a)$

  $7=4a-4+2-a$

  $3a=9$

  $a=3$

Putting the value of $a=3$ in the equation $c=2-a$ for getting the value of c, we have:

  $c=2-3$

  $c=-1$

Now, putting the values of a, b and c in the general equation of the parabola $y=a{x}^2+bx+c$ , we get:

  $y=3\times x^2+(-2)\times x+(-1)$

  $y=3x^2-2x-1$

Hence, the answer is $y=3x^2-2x-1$

Conclusion:

  $y=3x^2-2x-1$ is the solution.