How do I find #f(x)# if its graph includes #{(1, 5), (2, 8), (3, 11)}#?

To start off, you want to find the first differences. This means that you find the differences in "y" values: (First differences constant = linear, second differences constant = quadratic)

Of 5, 8, and 11, the differences are 3 every time. This means that the function will be LINEAR . The form of a linear equation is y=mx+b , or f(x)=mx + b where "m" is the slope and "b" is the y intercept.

Since you know the first differences are 3, this means that the slope (m) is 3. Now you can sub this into the linear equation as follows:

y = 3x + b

Now, take any of the points you are given and sub them into their corresponding variables. I will use (2,8). 8 should be the "y", and 2 should be the "x"

8 = 3(2) + b

Simplifying this:

8 = 6 + b (do a little algebra)

b = 2 --> Now you have m and b, so putting it together the function would be:

f(x) = 3x +2 or y = 3x + 2

To test this, you can do a left side right side check by subbing in one of the points into the equation. I will use the third point (11 is y and 3 is x):

11 = 3(3) + 2

11 = 9 + 2

11 = 11 --> Therefore, you know the equation is correct!

Hope this helps! :)