Equations of LinesChapter 1Seciton4

Forms for the Equation of a Line Slope-Intercept form: m is the slope(rate of

change) and b is the y-intercept (initial value when x = 0).

Point-Slope form: m is the slope and is a known point.

Example 1 Write an equation for

the line passing thru (-1,5) and has a slope of ¾ .

Write the equation of the line passing thru (-1,5) and (2,4).

Appliance RepairAn appliance repairman charges $60 for a service call plus $25 per hr for each hour spent on the repair. Assuming his service call charges can be modeled by a linear function of hours spent on the repair, write the equation of the function.

Example 2

Example 3The number of people (in millions) in US prisons or jails grew at a constant rate from 1990 to 2000, with 1.15 million people incarcerated in 1990 and 1.91 million incarcerated in 2000. What is the rate of growth of people incarcerated from

1990 to 2000? Write a linear function that models the number of N,

prisoners as a function of the year x. The Bureau of Justice Statistics projected that 2.29

million people would be incarcerated in 2005. Does the model agree with this projection?

Vertical and Horizontal Lines

Example 4:Write the equations of the lines that pass thru the point (-1,5) and have a. Slope of 0

b. Undefined slope

• A vertical line has the form x = a, where a is a constant and the coordinate of any point on the line.

• A horizontal line has the form y = b where b is a constant and the coordinate of any point on the line.

Parallel and Perpendicular Lines• Two distinct non-vertical

lines that have the same slope are called parallel. If Line, L1 has slope m1 and Line, L2 has slope m2, then

• The slopes of perpendicular lines are negative reciprocals of each other. If Line, L1 has slope m1 and Line, L2 has slope m2, then

Write the equation of the line through (4,5) and

• Parallel to the line with equation 3x + 2y = -1.

• Perpendicular to the line with equation 3x + 2y = -1.

Example 5

General FormThe average rate of change of f(x) with respect to x over the interval from x = a to x = b (moving left to right) is calculated asAverage rate of change

= Change in f(x) values

change in x values

• General Form: Ax + By = C and A,B,C are whole numbers and A is positive. The slope, m = -A/B and the intercepts are (0, C/B ) and ( C/A, 0) .

Average Rate of Change for any Function

Example 6: Hybrid Vehicle Sales The total hybrid electric passenger vehicle sales for the years from 1997 to 2005 are shown in the figure. The number of sales can be modeled by the function

where S(x) is in thousands and x is in years after 1990.a. Find the average

rate of change between 2000 and 2006.

b. Interpret your answer.

c. Find the slope of the secant line joining the points (10, 12.692) and (16, 264.710). How does that compare to part a. ?

Difference Quotient

The average rate of change of the function f(x) from x to x + h is

Example 7