bullet Linear Functions:
When working with straight lines (linear functions) you saw the "slope and rate of change".
This concept will be the same when applied to linear functions.

rate33.gif
Δyx is read "delta y over delta x". The word "delta" means "the change in".

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 "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 "rate of change" is not constant.

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

 

rate7

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ex1 Finding rate of change from a table.

A function is shown in the table at the right.
Find the 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 x1 = 1, and y1 = 4. From the second point, let x2 = 3 and y2 = 16.
Substitute into the formula:    rate32
x
y
0
1
1
4
2
9
3
16
The 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 rate of change from a graph.

A function called g (x) is shown in the graph at the right.
Find the 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 x1 = 1, and y1 = 1. From the second point, let x2 = 4 and y2 = 2.
Substitute into the formula:    rate322

rate15
The 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 rate of change from a word problem.

A ball thrown in the air has a height of y = - 16x² + 50x + 3 feet after x seconds.
a) What are the units of measurement for the rate of change of y?
b) Find the rate of change of y between x = 0 and x = 2?
balltoss2
Solution:
a) The rate of change will be measured in feet per second. (This will be the velocity of the ball).
b) Start by plugging in x = 0 and x = 2 to the equation to find the accompanying y-values.
x = 2:   y = -16(2)² + 50(2) + 3 = 39,  which gives (2,39)
x = 0:   y = -16(0)² + 50(0) + 3 = 3,  which gives (0,3)
Now, use the rate of change formula:
rate334

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