 Functions & Rate of Change MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts  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. 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: 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.  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 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 x1 = 1, and y1 = 4. From the second point, let x2 = 3 and y2 = 16.
Substitute into the formula: 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. 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:  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. 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 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:  