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.

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , . Bookmark the permalink.

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