Algebra of Functions

Question

Consider the function f(x)=x2+4 and find the following:

a) The average rate of change between the points  (−1,f(−1)) and (1,f(1)) . 

b) The average rate of change between the points  (a,f(a)) and (b,f(b)) . 

c) The average rate of change between the points  (x,f(x)) and (x+h,f(x+h)) . 

Answer 1

To find the average rate of change, you just use the slope formula:

So, you just plug in whatever two points you have and simplify. Is there a particular thing you are struggling with? It may be hard to do the points that do not have numbers in them, like (x, f(x)) and (x+h, f(x+h)) but you do them the same way.

x₂ = "x + h"

x₁ = "x"

y₂ = f(x + h) = (x + h)² + 4 = x² + 2xh + h² + 4

y₁ = f(x) = x² + 4

So y₂ - y₁ = f(x + h) - f(x) = (x² + 2xh + h² + 4) - (x² + 4) = x² + 2xh + h² + 4 - x² - 4 = "2xh + h²"

And x₂ - x₁ = (x + h) - (x) = "h"

So, dividing, we get (2xh + h²)/h = 2x + h (factor out an "h" on top and cancel). That means the average rate of change for that problem is "2x + h"

Answer 2

The average rate of change between two points is change in y over change in x, so the general form (and thus answer to part (b), before you plug the variables into the given function) would be [f(b) - f(a)]/(b - a). By plugging in the values of a and b, and using the given definition of the function f, you should be able to get answers to the questions.

Part (a) is thus [f(1) - f(-1)]/(1-(-1)) = [5 - 5]/2 = 0/2 = 0.

Part (b) becomes [f(b) - f(a)]/(b - a) = [(b^2 + 4) - (a^2 + 4)]/(b - a) = [b^2 - a^2]/(b - a) = (b - a)(b + a)/(b - a) = b + a.

Part (c) becomes [f(x + h) - f(x)]/((x + h) - x) = [((x + h)^2 + 4) - (x^2 + 4)]/h = [(x^2 + 2hx + h^2) - x^2]/h = [2hx + h^2]/h = h(2x + h)/h = 2x + h.

I hope this helps!

Answer 3

a) [f(1) - f(-1)] / (1 - (-1))

b) [f(b) - f(a)] / (b - a)

c) [f(x+h)) - f(x)] / h