Chapter 11.2, Problem 73AYU

To determine

To find: The function $y=a{x}^2+bx+c$, whose graph contains the points $\left(1,2\right), \left(-2,-7\right)$ and $\left(2,-3\right)$.

Answer to Problem 73AYU

Solution:

$y=-2x^2+x+3$

Explanation of Solution

Given:

The points $\left(1,2\right), \left(-2,-7\right)$ and $\left(2,-3\right)$.

Calculation:

The function is: $y=a{x}^2+bx+c$.

Since the above function passes through the points $\left(1,2\right), \left(-2,-7\right)$ and $\left(2,-3\right)$; we get the following equations in $a,b,c$:

$2=a+b+c$

$-7=a{\left(-2\right)}^2+b\left(-2\right)+c$

$4a-2b+c=-7$

$-3=a{\left(2\right)}^2+b\left(2\right)+c$

$4a+2b+c=-3$

Thus we get,

$a+b+c=2$

$4a-2b+c=-7$

$4a+2b+c=-3$

Subtracting the second equation from the third equation we get,

$4b=4$

$b=1$

Substituting $b=1$ in the first equation we get,

$a+1+c=2$

$a+c=1$

Adding the second and third equations we get,

$8a+2c=-10$

$4a+c=-5$

Now we have,

$4a+c=-5$

$a+c=1$

Solving the above equations we get,

$a=-2,c=3$

Thus we get, $a=-2,b=1,c=3$.

Therefore the required function will be;

$y=-2x^2+x+3$