Section 2.4 More on Slope

Parallel and Perpendicular Lines

y=2x+7

Parallel Lines

Example Write the equation in slope intercept form for a line that is parallel to 3x-4y=12 and passing through (5,2).

Perpendicular Lines

Example Write the equation in slope intercept form for a line perpendicular to 3x-4y=12 and passing through (5,2).

Slope as Rate of Change

Definition- Slope is defined as the ratio of a change in y to a corresponding change in x.

Interpreting a real life situation The line graphs the percent of US adults who smoke cigarettes x years after 1997. a. Find the slope of the line segment from 1997 to 2007 Percent Adults (0,24.7) (10,19.5) b. What does this slope represent? The percent of US adult cigarette smokers is decreasing by .52 percent each year. The change is consistent each year. X years after 1997

The Average Rate of Change of a Function

The average rate of change of a function. If the graph of a function is not a straight line, the average rate of change between any two points is the slope of the line containing the two points. This line is called a secant line. Secant line

The slope of this line between the points (1,3.83) and (5,7.83) is

The slope of this line between the points (1,3.83) and (4,7.34) is Continuation of same problem

The slope of this line between the points (1,3.83) and (3,6.5) is Continuation of same problem

Let’s look at the different slopes from the point (1, 3.83). x y Slope of the secant line 3 6.5 1.34 4 7.34 1.17 5 7.83 1 Notice how the slope changes depending upon the point that you choose because this function is a curve, not a line. So the average rate of change varies depending upon which points you may choose. Continuation of same problem

Find the average rate of change of f(x)= x3

Example Find the average rate of change of f(x)=3x-1 from x1 =0 to x2=1 x1 =1 to x2=2 x1 =2 to x2=3

Average rate of change and the difference quotient Suppose x1=x and x2=x+h, then Do you recognize the difference quotient that we studied in section 2.2? You will study more about the difference quotient in future math classes.

Average Rate of Change Application Example When a person receives a drug injection, the concentration of the drug in the blood is a function of the hours elapsed after the injection. X represents the hours after the injection and f(x) represents the drug’s concentration in milligrams per 100 milliliters. a. Find the average rate of change of the drug’s concentration between the 1st and 4th hours. b. What does this value mean in terms of the drug’s concentration? concentration (1, 3.96) (4, .72) hours

Average Rate of Change Application Example Sometimes it takes a while for a drug to diffuse sufficiently to affect the desired organ. The curve below is a close approximation of the concentration in that organ compared to the time after the drug was taken. a. What is the average rate of change from the time the drug was taken until the first hour? b. What was the average rate of change from the second hour until the fourth.? c. What is the interpretation of each answer? concentration hours cncntr 0.00 0.00 0.50 6.14 1.00 8.10 1.50 8.54 2.00 8.43 2.50 8.13 3.00 7.77 3.50 7.41 4.00 7.05 4.50 6.71 5.00 6.38 hours

(a) (b) (c) (d)

(a) (b) (c) (d)

(a) (b) (c) (d)