Download Mod6 Notes - Westwind Alternate School

Mod 6: Linear Functions
Slope
-The slope – denoted by m – refers to the steepness of the line. The steeper the line, the
greater the slope.
-In order to calculate the slope (m) of a line we must find the vertical distance (the rise) and the
horizontal distance (the run)
-In the following diagram, we are looking to calculate the slope of the line AC. In order to do
that, we must first find the vertical change (the change in Y) of the line between points A and
points C, which we call the rise. On the diagram, the rise is drawn between points B and C.
Counting the change from point B up to C, we calculate the rise to be 6.
-We must then calculate the horizontal distance of the line (the change in X), which in this case
is the distance from point A to B. This distance is called the run and in our diagram the run is 8.
We can then say that the slope of this line is
rise
slope 
run
OR Slope (m) = Change in Y
Change in x
6
3
therefore, m 
m
8
4
Typically we leave the slope as a reduced fraction and not in decimal form.
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Example:
F
D
C
B
A
When you are given a line, you can select any two points along the line. (Why is it possible to
select any two random points along the line?)
Count the following rise and runs on the graph.
Slope of AB
rise
mAB 
run
m
3
4
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Slope ofAC
rise
mAC 
run
m
6
8
m
3
4
Slope of CD
rise
mCD 
run
m
3
4
Slope of AE
rise
mAE 
run
m
Slope of BD
rise
mBD 
run
12
16
m
6
8
3
4
m
3
4
m
Example:
What do we notice to be different about the graph?
When we go to calculate the slope and we start at the first point to calculate the vertical rise,
we instead have to count down to the second point, instead of up. Therefore, when we give
the rise, we put is in as a negative.
rise
slope 
run
5
m
4
Any line that is decreasing to the right will have a negative slope.
Two Special Slopes
- 1) Horizontal lines are special because they always have a change of 0 in y (or f(x)). The
change in x is any real number. What happens when you divide 0 by any value that represents a
change in x? Zero divided by any number is equal to 0 so that is the slope of a horizontal line.
Using a skiing analogy, the slope of a horizontal line is much like cross country skiing. There is a
zero slope because you are moving ahead but you are not rising.
Note: The slope is usually represented y the letter m.
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-2) Vertical lines have a special slope as well. The rise is unlimited and the run is zero.
Therefore, the change in x or the denominator in the slope formula is always 0. What happens
when you divide a number by zero? Try it in your calculator and see what you get. You should
get an error reading. Dividing by zero is not defined, and therefore, we say that the slope of a
vertical line is undefined.
-Also note:
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Graphing Using Intercepts
-You have already graphed linear functions using a table of values. Now, look at two more
methods for graphing linear functions. The first method is graphing using the x and y
intercepts of the line. This is a great method to use when the equation of the line is in
general from.
Using this form allows easy algebraic identification of the x-intercept and the y-intercept.
When you know the intercepts, you can graph the line using these two points.
Find the x-intercept and the y-intercept of the following lines given in general form.
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Hortizontal and Vertical Lines
-The slope of a horizontal line is zero and the slope of a vertical line is undefined. When you
write the equation of these lines, these slopes have a strange effect on the slope-intercept
formula.
The horizontal line below has the following points:
-The equation of a horizontal line is always y = b, where b is the y-intercept of the line. In this
example, the equation of the graph is y = -1.
-The vertical line below has the following points:
-You probably already noticed that all x values are 1, and the y values, or the range, are all real
numbers. The slope of this line is undefined because the change in x is 0.
In this case, the value of y does not matter, as x is always equal to 1. Therefore, the equation of
this line is x = 1, and the x-intercept is (1, 0).
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Converting to general form
Example:
Write the following equations in general form and identify the x-intercept and y-intercept.
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Slope Intercept Formula
An example of a linear function in slope-intercept form is: y = 2x + 3.
For this line, the slope is 2 and the y-intercept is 3.
Example: Use what you know about slope-intercept form to identify the slope and the yintercept of each line.
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Example:
As you may have guessed, equations do not always come in slope-intercept form.
Study the questions below.
Write the following equations in slope-intercept form and identify the slope and yintercept.
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Graph Using Slope Intercept Form
After identifying the slope and y-intercept, you combine that process with the graphing of
lines using a point and a slope.
Example:
Writing Equations in Slope-Intercept Form
-Using the slope of a line and the y-intercept, you can write the equation of the line in slopeintercept form. Remember, the slope is m and the y-intercept is b.
Example 1
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Example 2
Using Point-Slope Form
-Another form of a linear equation is point-slope form. This form allows you to write the
equation of the line when you are given a point on the line and the slope of the line.
-In this equation, y and x are not to be substituted because they will be part of the equation
when you finish the manipulations. Use a given point (x1, y1) and the slope "m" to
determine the equation in slope-intercept form.
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Example 1:
Write the equation of the line passing through the point (-2, 4) with a slope of 6 in slopeintercept form.
Example 2:
Use the graph to determine the equation of the line.
Write in function notation.
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Determine the Equation of a Line Given Two Points
-When writing the equation of a line in slope-intercept form, you will realize that the steps
become more complex when you are given two points with no slope. Think about what is
missing: the slope. In the last example, you used the slope formula to find the slope given
two points. That is the only step to add to the examples above to write the equation of the
line given two points.
Example: Point A (10, -15) and point B (-2, -12) are two points on a line. What is the
equation of that line?
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Example:
The equation for the speed (not height) of a ball that is thrown straight up in the air is given
by v = 43 - 10t, where v is the velocity (in metres per second) and t is the number of
seconds after the ball is thrown. With what inital velocity was the ball thrown? What is the
meaning of the slope?
The slope is -10 (which is the number in front of v). This means the object is slowing down
at a rate of 10 meters per second.
Slope as Rate of Change
-Slope is the rate of change of f(x) (range) as compared to the rate of change of x (domain).
When real world values are put to this concept, how rates of changes are derived for
applications in mathematics become obvious. An example of looking at slope as a rate of
change is distance compared to time. Time is the domain and distance is the range.
When you look at a slope of a line on a distance-time graph, you will notice how slopes can
appear different. The slope of the line determines the speed. (Speed is the rate at which
distance changes with respect to time). The steeper the slope, the greater the speed. If the
slope is gradual, then the speed is low. Look at the examples of a car travelling a distance
over time.
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In the first graph, the slope is very steep. This means that the car is traveling at a fast rate of
speed. In the second graph, the slope is more gradual. This means that the car is driving at a
slow rate of speed.
There are many ways that we can think of slope as a rate of change. If you were to
brainstorm some common two-unit measures that you encounter, you may come up with a
list like this:
km/h
cm/day (Rain or snowfall)
km/litre
$/h
Understanding rates will help you interpret graphs of real-world values.
Interpreting Graphs
-The following graph has a domain of time in hours and a range of distance in kilometers.
One point on the blue line shows distance of 70 km was traveled in 7 hours. Another point
on the graph would be a distance of 10 km in 1 hour. Using these points, answer the
following:
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What is the slope of the line graphed?
What would the units be for the solution?
Since distance was measured in km and time was measured in hours, the slope will be in
km/hour. The answer will be slope is 10 km/hr.
In this example, the speed is very slow. The slope of the line appears to be a bit steep, but
the scale on the axes has much to do with how the data appears on the graph.
Discrete or Continuous Data
- Continuous Data: data that can take on all possible values. The graph of continuous data
is a line or curve.
Ex. The table shows the value of a piano over time.
Age (years)
0
1
2
3
Value ($)
5000
5325
5671.13
6039.75
The graph will look like:
Why is it logical to join the data points in
the graph?
Because time is continuous. We do not
jump from one year to the next
automatically.
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-Discrete Data: cannot take on all possible values.
Ex. Graph the relationship between the price per ticket and the number of tickets
purchases.
Number of Tickets Purchased
1
2
3
4
5 or more
Price per ticket($)
45
43
35
30
25
Graph:
50
45
40
40
35
30
30
Price
Per
Ticket
($)
25
25
25
25
25
25
20
10
0
0
2
4
6
8
10
Number of Tickets Purchased
In this example, it would not be logical to connect points, as we cannot purchase any
‘portion’ of a ticket. This is an example of discrete data.
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Slope practice question
Example:
Cameron puchases a car for $15 000. The value of the car, v, can be represented by the
formula v = 15 000 - 1 000t, where t is the time (age) of the car in years.
Complete the table of values:
What is an appropriate domain for this graph?
The domain should include only positive values (time) and should be limited to 15 years
because the value of the car becomes 0 in 15 years. What is an appropriate range for this graph?
The range represents the value of the car and should start at 15 000 (the maximum
possible value) and end at 0 (the minimum possible value). Graph the function: Be sure you label both axes and make an appropriate scale! What is the intercept for the time axis?
(15,0) This means that in 15 years the value of the car will be $0. What is the intercept for the value axis?
(0,15 000) This means that the initial value of the car is $15 000. You can use the graph to determine the value of the car at different intervals of time.
If Cameron has had the car for five years, what is the car worth?
The point on the graph that has a time value of five years shows a value of $10000
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Slope of Parallel and Perpendicular Lines
-Slopes may be used to determine if lines are parallel or perpendicular.
-Lines that are parallel do not intersect.
-Perpendicular lines intersect at a right angle.
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Example:
Find the slope of each line to determine if the lines are parallel, perpendicular, or neither.
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