a We are given a diagram showing the differences for the cubic function f(x)= x^3 -2x +5 for the x-values 0, 1, 2, 3, 4, and 5.
We can complete the diagram by finding the difference between the pairs of y-values indicated by the diagram.
b As we can see from the diagram found in Part A all the third differences are 6, so we can expect the same for the third difference corresponding to the new value f(6). Let's check this by adding the value f(6) = 6^3 -2* 6 +5 = 209 and finding the corresponding differences.
We can see that the difference is 6, as expected.
c Remember that for a linear function the first differences are constant. On the other hand, for a quadratic function the second differences are constant. This pattern suggest that the third differences are constant for cubic functions. We can check if this is the case by using a general cubic function.
f(x) = ax^3+bx^2+cx+d
Now let's evaluate this function at different x-values, just as we did with the exercise's function. We can use a table to organize this information.
x
a^3+bx^2+cx+d
Simplify
0
a( 0)^3+b( 0)^2+c( 0)+d
d
1
a( 1)^3+b( 1)^2+c( 1)+d
a+b+c+d
2
a( 2)^3+b( 2)^2+c( 2)+d
8a+4b+2c+d
3
a( 3)^3+b( 3)^2+c( 3)+d
27a+9b+3c+d
4
a( 4)^3+b( 4)^2+c( 4)+d
64a+16b+4c+d
5
a( 5)^3+b( 5)^2+c( 5)+d
125a+25b+5c+d
With this information we can calculate the first differences.
Let's now find the second differences.
Finally, we can find the third differences.
As we can see, the third differences are constant. Since we used a general cubic function with arbitrary parameter values, this is true for any cubic function!
