LINEAR FUNCTIONS

Example #1:  What is the location of Point M if each line in the grid represents one unit? Point M is located at (–3, 4).

Example #2:  State the ordered pairs shown in the table (h, w). The set of ordered pairs (h, w) represented in this table would be: {(68, 125), (64, 118), (65, 112), (72, 145), (64, 126), (67,130), (66, 128)} *Notice in this example that there are repeated numbers in the height column.  Because of this, the chart and set of ordered pairs only represents a relation. {(68, 125), (64, 118), (65, 112), (72, 145), (64, 126), (67,130), (66, 128)}

Example #3:  State the ordered pairs shown in the table (x, y). The set of ordered pairs (x, y) represented in the table would be: {(11, 63), (12, 64), (13, 65), (14, 70), (15, 72), (16, 72)} *Notice in this example that there are no repeated values in the first column.  Because of this, the chart and the set of ordered pairs represent a function.

Example #4:  An employee for Elton’s Electric Company charges $125 for an estimate and $30 per hour for repairs.  This situation can be represented by y = 125 + 30x.  The employee is permitted to work no more than 38 hours per job and his time is rounded to the nearest half hour.  Fill in the chart with appropriate domain (x) and range (y) values based on the given terms of Elton's Electronic Company for the employee.

State the domain values.

Example #6: Complete each ordered pair so that it is a solution to 3x + 2y = 1. a)  (1, ?) b)   c)  (?, 8)

Example #1:  From the graph below, determine the slope of the line. - Put your pencil on the red point. - Move straight up (vertical rise) until your pencil is in the same line as the black point. (2 units) - Move right (horizontal run) until you reach the black point. (3 units) You have now determined the slope of the line to be .

Example # 2:  From the graph below, determine the slope of the line. - Put your pencil on the red point. - Move straight down (vertical rise) until your pencil is in the same line as the black point. (–3 units) - Move right (horizontal run) until you reach the black point.  (6 units) You have now determined the slope of the line to be .  This slope can be reduced to .  *Notice that if you count down one unit (rise) from the red point and right two units (run), you will be on a point of the line.

Lines with positive slopes rise to the right.	Lines with negative slopes rise to the left.

At this point, we are going to learn how to find the slope of a line by using two points that lie on the line.

The definition of slope states that given two points, (x₁, y₁) and (x₂, y₂), the formula for finding the slope of a line containing these points is:

m =

Example #3:  Find the slope of the line containing the point A(–2, –6) and B(3, 5). Let point B = (x2, y2) and point A = (x1, y1) . (x2, y2) = (3, 5)               x2= 3, y2 = 5 (x1, y1) = (–2, –6)           x1= –2, y1 = –6 m =   m =   m = The slope (m) of the line is .

Example # 4:  Find the slope of the line containing the point C(5, –2) and D(8, –2). Let point D = (x2, y2) and point C = (x1, y1) . (x2, y2) = (8, –2)          x2= 8, y2 = –2 (x1, y1) = ( 5, –2)          x1= 5, y1 = –2 m =   m =   m =   m = = 0 The slope (m) of the line is 0.  This line is a horizontal line.

Example #5:  Find the slope of the line containing the point E(9, 4) and F(9, 1). Let point F = (x2, y2) and point E = (x1, y1) . (x2, y2) = (9, 1)           x2= 9, y2 = 1 (x1, y1) = ( 9, 4)          x1= 9, y1 = 4 m =       m =       m = *Division by zero is undefined. The slope (m) of the line is undefined.  This line is a vertical line.

Stop!  Go to Questions #18-29 about this section, then return to continue on to the next section.


Graphing a Line on a Coordinate Plane Using a Point and the Slope

Example #1:  Graph the line containing the point (–1, –3) and having a slope of m = . 1.  Plot the point (–1, –3). 2.  Use the rise (3 units) over run (4 units) ratio for slope to plot a second point. 3.  Draw a line through the points with a straightedge.

The y-coordinate for every point on a horizontal line is the same.   Choose two points on the line and use the slope formula m = , to determine the slope. m = The slope of every horizontal line is 0.

The x-coordinate for every point on a vertical line is the same. Choose two points on the line and use the slope formula m = , to determine the slope. m = undefined or no slope. The slope of every vertical line is undefined!

Example 1:  The graph shows the distance a bicyclist travels at a constant speed.  Find the speed of the bicyclist. Select two points and use the slope formula. Let (x1, y1) = (0.5, 5)   Let (x2, y2) = (1.5, 15)     The bicyclist travels 10 miles for every hour he travels.

Example 2:  Brenton works at a local fast food restaurant and is examining his recent pay checks (amounts do not reflect any type of deductions).  Brenton's time worked and salary is shown in chart below.

Do these ordered pairs represent a function?

"Click here" to check the answer.

Let’s find the rate of change using Brenton's information:

Let’s try another set of data points:

This procedure can be done with any of the data points in the chart.  So what does the 8.25 represent in terms of the data?  In this problem, the top number is the change in dollars earned and the bottom number is the change in hours worked.  In mathematics, the symbol Δ (delta)  means change. means Thus, the 8.25 represents the dollars per hour that Brenton is being paid to work at the local fast food restaurant. Brenton makes $8.25 per hour for his wages.  (rate of change is dollars / hour)

Example 3:  To make the commute to work, Brenton must ride a bus part of the way  and walk part of the way.  Look at the graph below and describe what the graph represents in terms of Brenton's commute to work and rate of change.  Think about what transpires realisticallyon his commute. Solution:       Notice that the first line in the graph shows a rate of change that is not as great as shown elsewhere in the graph.  This line represents Brenton's walk to the bus stop. Brenton must then wait for the bus at the bus stop.  At this point, the line in the graph is flat; time is increasing but the miles are not.  The last line in the graph shows the greatest rate of change.  This line represents Brenton's ride on the bus (the bus travels faster – more miles per minute than walking or waiting). The rate of change would be miles per minutes in this problem.

Example 1:  If y varies directly as x and y = 6 when x = 0.3, find the constant of variation and write an equation of direct variation. 1.  Substitute x and y in to = k and solve for k. = k   20 = k   2.  Use the constant of variation, k, to write an equation of direct variation. y = kx   y = 20x

Example 3:  A cell phone company charges $10 for the first 250 minutes of usage.  Given this is an example of direct variation, find the constant of variation and then determine the cost of 400 minutes of use. Solution:       1.  Substitute x and y in to = k and solve for k.             y = number of minutes            x = charges in dollars = k   25 = k The constant of variation (k) is 25.   2.  Use the constant of variation, k, to write an equation of direct variation. y = kx   y = 25x   3.  Use the direct variation equation to determine the cost of 400 minutes. y = 400 minutes x = cost in dollars     y = 25x Direct variation equation     400 = 25x Substitute: y = 400     16 = x Divide both sides by 25.     x = 16 Apply the reflexive property. The cost for 400 minutes is $16.00.