Mysterious Function

Observe the table of domains and ranges for a mysterious function. Given the table of values, what is the range for a domain of $-10.5$?

 Domain Range -13 -352 -5 -56 -1 -4 0 -1 7 -92 20 -781

Source: mathcontest.olemiss.edu 6/30/2008

SOLUTION

We plot the given 6 data points to help us visualize the graph of the mysterious function. The plot seems to suggest that the function might be a quadratic function of the form $y=ax^2+bx+c$.

Calculate $\mathbf{c}$
When $x=0,y=-1$:
$-1=a\cdot 0^2+b\cdot 0+c$
$-1=c$

Calculate $\mathbf{b}$ and $\mathbf{c}$
When $x=-1,y=-4$:
$-4=a\left (-1\right )^2+b\left (-1\right )-1$
$-3=a-b$                                   (1)

When $x=-5,y=-56$:
$-56=a\left (-5\right )^2+b\left (-5\right )-1$
$-55=25a-5b$
$-11=5a-b$                              (2)

Multiply Eq. (1) by $-1$ and add it to Eq. (2) yields:
$3=-a+b$
$-11=5a-b$
——————–
$-8=4a$
$-2=a$

Substitute $-2\textup{ for }a$ in Eq. (1) yields:
$-3=a-b$
$-3=-2-b$
$1=b$

Equation of the function
So the equation of the function is
$y=-2x^2+x-1$

Check
When $x=-13, y=-352$?
$y=-2\left (-13\right )^2+\left (-13\right )-1$
$=-2\left (169\right )-14$
$=-338-14$
$=-352$. YES

When $x=-5, y=-56$?
$y=-2\left (-5\right )^2+\left (-5\right )-1$
$=-2\left (25\right )-6$
$=-50-6$
$=-56$. YES

We should also check the remaining 4 data points: $\left (-1,-4\right ),\left (0,-1\right ),\left (7,-92\right ),\left (20,-781\right )$.

Graph of the function

What is the value of the function when $\mathbf{x=-10.5}$?
$y=-2x^2+x-1$
$=-2\left (-10.5\right )^2+\left (-10.5\right )-1$
$=-2\left (110.25\right )-11.5$
$=-220.5-11.5$
$=-232$.

Answer: $-232$.