bullet Linear Functions:
You are already familiar with the concept of "average rate of change".
When working with straight lines (linear functions) you saw the "average rate of change" to be:

rate1

The word "slope" may also be referred to as "gradient", "incline" or "pitch", and be expressed as:
      rate4

A special circumstance exists when working with straight lines (linear functions), in that the "average rate of change" (the slope) is
constant. No matter where you check the slope on a straight line, you will get the same answer.

 

rate3
bullet Non-Linear Functions:

When working with non-linear functions, the "average rate of change" is not constant.

The process of computing the "average rate of change", however, remains the same as was used with straight lines: two points are chosen, and rate5 is computed.

FYI: You will learn in later courses that the "average rate of change" in non-linear functions is actually the slope of the secant line passing through the two chosen points. A secant line cuts a graph in two points.

rate7

When you find the "average rate of change" you are finding the rate at which (how fast) the function's y-values (output) are changing as compared to the function's x-values (input).

When working with functions (of all types), the "average rate of change" is expressed using function notation.

Average Rate of Change
For the function y = f (x) between x = a and x = b, therate8

While this new formula may look strange, it is really just a re-write of rate9.

Remember that y = f (x).
So, when working with points (x1, y1) and (x2, y2), we can also write them as

the points rate11.

Then our slope formula can be expressed as rate10.

If we rename x1 to be a, and x2 to be b, we will have the new formula.

The points are rate12, and the

rate13.

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ex1 Finding average rate of change from a table.

Function f (x) is shown in the table at the right.
Find the average rate of change over the interval 1 < x < 3.

Solution:

If the interval is 1 < x < 3, then you are examining the points (1,4) and (3,16). From the first point, let a = 1, and f (a) = 4. From the second point, let b = 3 and f (b) = 16.
Substitute into the formula: rate14
x
f (x)
0
1
1
4
2
9
3
16
The average rate of change is 6 over 1, or just 6.
The y-values change 6 units every time the x-values change 1 unit, on this interval.

 

ex2 Finding average rate of change from a graph.

Function g (x) is shown in the graph at the right.
Find the average rate of change over the interval
1 < x < 4.

Solution:

If the interval is 1 < x < 4, then you are examining the points (1,1) and (4,2), as seen on the graph. From the first point, let a = 1, and g (a) = 1. From the second point, let b = 4 and g (b) = 2.
Substitute into the formula: ARCnew
rate15
The average rate of change is 1 over 3, or just 1/3.
The y-values change 1 unit every time the x-values change 3 units, on this interval.

 

ex2 Finding average rate of change from a word problem.

A ball thrown in the air has a height of h(t) = - 16t² + 50t + 3 feet
after t seconds.
a) What are the units of measurement for the average rate of change
of h?
b) Find the average rate of change of h between t = 0 and t = 2?
balltoss2
Solution:
a) In the formula, rate18, the numerator (top) is measured in feet and the denominator (bottom) is measured in seconds. This ratio is measured in feet per second, which will be the velocity of the ball.
b) Start by finding h(t) when t = 0 and t = 2, by plugging the t values into h(t).
h(2) = -16(2)² + 50(2) + 3 = 39
h(0) = -16(0)² + 50(0) + 3 = 3
Now, use the average rate of change formula:
rate19

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