Question

What is the average rate of change between

X=1 and X=2?

X=2 and X=3?

X=3 and X=4?

Answer 1

Answer:  The average rate of change between x = 1 to x = 2 is 2, from x = 2 to x = 3 is 4 and from x = 3 to x = 4 is 8.

Step-by-step explanation:  We are given to find the rate of change from

x = 1 to x = 2,  x = 2 to x = 3  and  x = 3 to x = 4.

We know that

The average rate of change between x = a to x = b for a function f(x) is given by

$A_v=\dfrac{f(b)-f(a)}{b-a}.$

For the given function f(x), we have from the graph that

f(1) = 2,  f(2) = 4,  f(3) = 8  and  f(4) = 16.

So, the average rate of change from x = 1 to x = 2 is given by

$A_{v1}=\dfrac{f(2)-f(1)}{2-1}=\dfrac{4-2}{1}=2.$

The average rate of change from x = 2 to x = 3 is given by

$A_{v2}=\dfrac{f(3)-f(2)}{3-2}=\dfrac{8-4}{1}=4.$

The average rate of change from x = 3 to x = 4 is given by

$A_{v3}=\dfrac{f(4)-f(3)}{4-3}=\dfrac{16-8}{1}=8.$

Thus, the average rate of change between x = 1 to x = 2 is 2, from x = 2 to x = 3 is 4 and from x = 3 to x = 4 is 8.

Answer 2

The average rate of change between:

$x = 1 and x = 2 is 2\\x = 2 and x = 3 is 4\\x = 3 and x = 4 is 8$

Recall:

Formula for finding average rate of change is given as:

$\frac{f(b) - f(a)}{b - a}$

Apply this formula to find the average rate of change of each given problem.

Average rate of change between $x = 1 and x = 2$

Let,

$a = 1\\b = 2$

From the graph find the of f(x) that corresponds to x = 1 and x = 2 respectively.

$f(a) = f(1) = 2\\f(b) = f(2) = 4$

Plug in the values into the formula for average rate of change:

Average rate of change $= \frac{4 - 2}{2 - 1} = \frac{2}{1} = 2$

Average rate of change between $x = 2 and x = 3$

Let,

$a = 2\\b = 3$

From the graph find the of f(x) that corresponds to x = 2 and x = 3 respectively.

$f(a) = f(2) = 4\\f(b) = f(3) = 8$

Plug in the values into the formula for average rate of change:

Average rate of change $= \frac{8-4}{3-2} = \frac{4}{1} = 4$

Average rate of change between $x = 3 and x = 4$

Let,

$a = 3\\b = 4$

From the graph find the of f(x) that corresponds to x = 3 and x = 4 respectively.

$f(a) = f(3) = 8\\f(b) = f(4) = 16$

Plug in the values into the formula for average rate of change:

Average rate of change $= \frac{16-8}{4-3} = \frac{8}{1} = 8$

Therefore, the average rate of change between:

$x = 1 and x = 2 is 2\\x = 2 and x = 3 is 4\\x = 3 and x = 4 is 8$

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