Evaluate the quadratic functions given and then find the difference between consecutive pairs of the function's values, these are the first differences.
B
b
Take a look at the results found in Part A. Can you identify any pattern relating the x^2-term coefficient and the common second differences?
C
c
What can we tell by using the relationship found in Part B? How can it be used for a data set with common second differences?
A
a
i. f(x) = x^2-3
ii. f(x) = 3x^2
iii. f(x) = 4x^2-5x
B
b
See solution.
C
c
See solution.
Practice makes perfect
a
We are asked to choose five x-values and use them to find the second differences of each of the given functions. We will start with the quadratic function shown below.
i. f(x)=x^2-3
First we need to choose the x-values we will use. For example, x=-2, x= -1, x=0, x=1, and x=2. Now we can use a table to find the corresponding function's values.
x
f(x)=x^2-3
Simplify
- 2
f( - 2) = ( - 2)^2-3
f( - 2) = 1
- 1
f( - 1) = ( - 1)^2-3
f( - 1) = - 2
0
f( 0) = ( 0)^2-3
f( 0) = - 3
1
f( 1) = ( 1)^2-3
f( 1) = - 2
2
f( 2) = ( 2)^2-3
f( 2) = 1
We can create a second table with the x-values and the corresponding function's values we just found. To find the first differences, we need to take pairs of the function's values and find their difference.
If we repeat the process by taking pairs of first differences and subtract them, we can find the second differences.
As we can see, the second differences are a common constant.
We need to repeat the same process for the second function.
ii. f(x) = 3x^2
Finally, let's do the same for the third function.
ii. f(x) = 4x^2-5x
b
Let's take a look at the results from Part A.
Function
Common second difference
f(x) = 1x^2-3
2 = 2* 1
f(x) = 3x^2
6 = 2* 3
f(x) = 4x^2-5x
8 = 2* 4
As we can see, the common second differences are equal to the coefficient of the x^2-term times two.
c
According to the results found in Parts A and B, if a set of data pairs has common second differences we know that it fits a quadratic function. Furthermore, we could divide the common second difference by two to find the value of the x^2-term's coefficient.
