Download Chapter Five: Linear Functions

Chapter Five: Linear Functions
Section One: Linear Functions and Graphs
A relation is any relationship between two sets of numbers. We can show this
relationship with a table, graph, mapping, set of ordered pairs, or equation. We have
already talked about some relations back in chapter one. A function is a special relation
where each x is paired with only one y. The set of all the x’s in a relation is the domain.
The set of all the y’s is the range.
EX1: Determine whether each set of ordered pairs is a function. Describe the domain and
range for each.
a. 1,3 2,5 3,7  4,9 
 4,55,43,41,5
c.  2,3 2, 3 4,2  2, 4 
b.
A function that defines a straight line as a graph is called a linear function.
EX2: Complete each ordered pair so that it is a solution to 3x  y  15
a.  3,? 
b.  ?,3
c.  0,?
d.  5,? 
EX3: The table below shows the amount that a company charges for a raft rental for up
to 10 hours. An initial deposit is included in the amounts shown. Write an equation for
the function. Write a set of ordered pairs for the function. Identify the domain and range
values.
Time
1
2
3
4
5
6
7
8
9
10
Cost 18.50 22.00 25.50 29.00 32.50 36.00 39.50 43.00 46.50 50.00
We can also represent a function with a graph.
EX4: The graph shows the distance that a car travels over
time. Identify the dependent and independent variables, and
describe the domain and range. Write an equation for the
linear function.
EX5: The graph shows the distance that a tour bus
travels over time. Identify the dependent and
independent variables, and describe the domain and
range. Write an equation for the linear function.
Section Two: Defining Slope
Slope is the measure of how steep or slanted a line is. It is the ratio of how much the line
rise 

rises to how much the line runs  slope  m 
.
run 

EX1: Yara and Nidal are hiking up a hill. The run of Yara’s step is 32 inches, and the rise
is 8 inches. The run for Nidal’s step is 36 inches, and the rise is 9 inches. Find the slope
of the lines representing each hiker’s pathway up the hill.
The hikers in the previous example were hiking uphill. This is a positive slope. However,
there are four different types of slope as illustrated in the examples below:
Positive Slope
Negative Slope
Zero Slope
Undefined Slope
EX2: The coordinate plane at the right shows the
position of a ladder resting against the side of a
building. Find the slope of the ladder.
When we are given two points in the coordinate plane we can find slope with the
previous formula or we can use a new shorter formula without even having to graph.
slope  m 
rise y2  y1

run x2  x1
EX3: Find the slope of the line connecting the following points.
a.  2,4  and  5, 6 
b.  2, 1 and 1,3
c.  3, 3 and 1, 3
d.  2, 5 and  2,7 
EX4: Graph the line with the following slopes that pass through the following points
3
a. m  and 1,2 
4
4
b. m   and  2, 1
3
c. m  0 and  3, 2 
d. m  undefined and  3, 4 
Section Three: Rate of Change and Direct Variation
A car is traveling down the road. A table of its movement is shown below. Create a graph
of the scenario.
Time
(hours)
Distance
(miles)
0
1
2
3
4
5
0
60
120
180
240
300
How fast is the automobile moving? What is the slope of the line? Notice that the values
are the same. The slope of a line is a rate of change or a speed.
EX1: The graph below shows the distance a certain train travels at a constant speed. What
is the speed of the train?
One type of rate of change is direct variation. A direct variation equation is any equation
that is of the form y  k  x where k is the average rate of change which we call the
constant of variation.
EX2: If y varies directly as x and y  8 when x  4 , find the constant of variation and
write an equation of direct variation.
EX3: If y varies directly as x and y  36 when x  9 , find x when y  48 .
EX4: If y varies directly as x and y  35 when x  7 , find y when x  84 .
Direct variations are proportional!
EX5: Work examples 3 and 4 using proportions
Section Four: The Slope-Intercept Form
John wants to rent a jet ski at the beach. The rental costs $10 plus $5 for each hour.
Complete the table for the cost of the jet ski.
Hours
0
1
2
3
4
Cost
Pick two points from your table and find the slope of the line. Write an equation that
represents the situation. Graph. Is there a relationship between the slope, the graph, and
the equation?
The intersection point of a graph and the y-axis is known as the y-intercept. The x part of
the point where the line crosses the y-axis is 0,  0,? . One of the most important
equations related to linear equations is the slope-intercept form: y=mx+b . We can very
easily find the equation of a line using this formula where m is the slope and b is the yintercept.
EX1: Identify the slope and y-intercept in the following equations.
a. y  3x - 4
2
b. y   x  2
3
EX2: Use the slope-intercept for to construct the graph of the equation
a. y  2 x  5
2
b. y  x  8
5
EX3: Find the equation of each line graphed
a.
b.
From the previous example we see that it is very easy to find the equation of a line if we
have the slope and the y-intercept. We can also find the equation given a point and a
slope. Plugging these three values into the slope intercept form we can find the yintercept.
EX4: Find the equation of the line with the following slope passing through the given
point.
3
a. m  and  4,3
4
2
b. m   and  5, 1
5
If we are given two points instead of a slope and a point, we will have to use the slope
formula: m 
y2  y1
x2  x1 .
EX5: Find the equation of the line passing through the two points.
a.  4,7  and 10,0
b.  3,3 and  5,7 
As mentioned before, the y-intercept is the point where a graph crosses the y-axis. One
way to find the y-intercept is to plug zero into the equation for x. The x-intercept is the
point where a graph crosses the x-axis. To find an x-intercept we can plug in zero for y.
EX6: Identify the x-intercept and y-intercept of the following equations. Use these
intercepts to graph the equation.
a. y  4 x  8
b. y  3x  15
The equations of horizontal and vertical lines are easy to find and work with. The
equation of a horizontal line is y  b where b is the y-intercept. The equation of a vertical
line is x  a where a is the x-intercept.
EX7: Plot the two points and connect them with a line. Find the equation of the line.
a.  2, 4 and  3, 4 
b.  5,1 and  3,1
c.  2, 3 and  2,5
d.  4,1 and  4,5
Section Five: The Standard and Point-Slope Forms
Any linear equation can be written in another form called standard form: Ax  By  C .
We will make A, B, and C all whole numbers. Remember that we can move a term across
the equal sign by simply changing its sign.
EX1: Write each equation in standard form.
a. y  3 x  1
b. 5 y  2  4 x
4
c. x  2 y  3
5
EX2: A donation of $300 was made to the Classical Music Club for the purpose of
building their collection of CDs. At the local music store, some classical CDs cost $10
and others cost $15. The club wants to know how many of each type they can buy with
$300. Write an equation in standard form to model the situation. Graph the equation.
Note: the equation in the previous example can be reduced. This makes working with the
equation much easier.
Point-slope form  y  y1  m  x  x1   is another way of finding the equation of a line
when we are given a point and a slope or two points. We simply plug in the slope for m
and the point for  x1, y1  .
EX3: Write the equation of the line with the following slope that passes through the
included point. Write your answer in slope-intercept and in standard form.
a. m  3 and  2,3
b. m 
4
and 10, 4
5
EX4: Write the equation of the line that passes through the two points. Write your answer
in slope-intercept and in standard form.
a.  5,65 and  7,71
b.  3, 6  and  5, 2 
Section Six: Parallel and Perpendicular Lines
Before discussing parallel and perpendicular lines, we will review how to find the slope
of a line.
EX1: Find the slope of the following lines
a. y  4 x  5
b. 20 x  5 y  13
2
c. x  6 y  1
3
Parallel lines are lines that stretch forever and never cross. Two lines are parallel if and
only if they have the same slope.
EX2: Are the following lines parallel. Verify with a graph.
a. y  3x  2 and y  3x  1
b. 4 x  3 y  12 and y  3x  1
We can find the equations of parallel lines by using previous methods.
EX3: Find the equation of a line that is parallel to y   x  7 with a y-intercept of -5.
Perpendicular lines are lines that cross at 90 degree angles. Two lines are perpendicular if
and only if their slopes are negative reciprocals (opposite flips) of each other.
EX4: Give the slope that would be perpendicular to the following slopes.
2
a.
3
1
b. 
5
c. 6
EX5: Write an equation in slope-intercept form for the line that has a y-intercept of -5
and that is perpendicular to the line y  2 x  3 .
EX6: Write an equation in slope-intercept form for a line containing the point  9, 4 
and…
a. parallel to the line x  3 y  30
1
b. perpendicular to the line  x  y  20
2