Question

The graph of f(x) = 2x + 1 is shown below. Explain how to find the average rate of change between x = 0 and x = 2.

Graph of 2 to the power of x plus 1

Answer 1

I believe the correct given equation is in the form:

f (x) = 2^x + 1 where x is an exponent of the number 2

Now to solve for the average rate of change of the given function from points x = 0 and x = 2, we must first assume that the graph between those two points is linear. With that, we can solve for the average rate of change using the slope formula, that is:

average rate of change = slope = [f (x2) – f (x1)] / (x2 – x1)

Therefore,

average rate of change = [(2^2 + 1) – (2^0 + 1)] / (2 – 0)

average rate of change = [5 – 2] / 2

average rate of change = 3 / 2

average rate of change = 1.5

Answer 2

The last description actually clarifies the given equation. The equation should be written as: f(x) = 2ˣ +1. The x should be in the exponent's place.

The average rate of change, in other words, is the slope of the curve at certain points. In equation, the slope is equal to Δy/Δx. It means that the slope is the change in the y coordinates over the change in the x coordinate. So, we know the denominator to be: 2-0 = 2. To determine the numerator, we substitute x=0 and x=2 to the original equation to obtain their respective y-coordinate pairs.

f(0)= 2⁰+1 = 2
f(2) = 2² + 1 = 5

So, the slope is equal to:

Average rate of change = (5 - 2)/(2 - 0)
Average rate of change = 3/2 or 1.5