
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.
Δy/Δx 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:
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.


NonLinear Functions:
When working with nonlinear 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 is computed.


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 x_{1} = 1, and y_{1} = 4. From the second point, let x_{2} = 3 and y_{2} = 16.
Substitute into the formula:


The rate of change is 6 over 1, or just 6.
The yvalues change 6 units every time the xvalues change 1 unit, on this interval. 
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 x_{1} = 1, and y_{1} = 1. From the second point, let x_{2} = 4 and y_{2} = 2.
Substitute into the formula:


The rate of change is 1 over 3, or just 1/3.
The yvalues change 1 unit every time the xvalues change 3 units, on this interval. 
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?


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 yvalues.
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:

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