THE LINEAR FUNCTION
The table gives the information about the height of the winning vault pole in the Olympic
Games during the early 1900’s.
Year
Height (inches)
1900
130
1904
138
1908
146
1912
154
o What is the increase every four years? Every eight years?
o If the event happened every year, with the information you have, what would the
increase per year be?
Because the increase per year is constant (rate of change of height per year is 2), this
function has a linear behavior. The change in h is the same when the time t changes by 1.
These are two ways to present the information considering the constant rate of change of
the height with respect to time.
t (time)
h (height)
1900
130+2*0
1904
130+2*4
1908
130+2*8
1912
130+2*12
t (time)
h (height)
0
130+2*0
4
130+2*4
8
130+2*8
12
130+2*12
In the second table we should keep in mind that t=0 (initial time) corresponds to 1900.
Let’s define a function by a formula to describe this situation. To do this let’s call d the
distance and t the time after 1900.
h (t) =130+2t
EXERCISE 1
Use the formula for the vault pole to determine each of the following values. Interpret
them and indicate whether or not it is a reasonable value for the situation.
h (5) =
h(-4) =
h(50) =
h(t) =150
The main characteristic of a function y=f(x) with linear behavior is that the rate of change
of y with respect to x is always constant. This constant is called the slope of the line.
Denoting the change in x as ∆x, and the corresponding change in y as ∆y, and, we have
y Change in y Rise


 Slope of the line
x Change in x Run
Linear Functions. Revised 2/6/2016
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The graphs below display a line with slope 2. The one on the left shows the rate of change
of y when x changes by 1, and the one on the right shows the rate of change when x
changes by 4. The rate of change is exactly the same.
From these observations we can think of the slope of a line as the change in y, when the
value of x is increased by 1. This interpretation will be very useful for geometric
interpretations later on.
The equation of a linear function can be written in two forms:
Slope-Y-intercept
Standard form
y = m x+ b
c x +d y = a
m is the slope, b the y-intercept
a, b, c constants
For the line displayed above we have m=2 and b=2 then the equation is y=2x+2, which
can also be written as y-2x=2 (standard form).
EXERCISE 2
A straight line has four possible positions as shown in the graph, which can be described
as:
1. The linear function is increasing, or the line goes up from left to right, or the line
has positive slope
2. The linear function is decreasing, or the line goes down from left to right, or the
line has negative slope
3. The linear function is constant, or the line is horizontal, or the line has slope zero.
4. The line is vertical (it is not a function anymore), or the slope is undefined.
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Positive slope
Negative slope
Zero slope
Undefined slope
slope
Each statement above describes a line using equivalent statements. Justify those
equivalences
How do we know a data set has a linear behavior?
It can be determined that a function given by a table is linear in either of the two ways:
1. Observe that for equal change in the independent variable the dependent variable
change by the same quantity.
2. Determine the rate of change of the function using consecutive values of the
independent variable. If it is the same, the function is linear.
EXERCISE 3
1. Consider the data in the following tables:
x
0
3
6
9
12
y
8
13
18
23
28
x
-1
3
10
20
50
y
7
-1
-15
-35
-95
x
0
1
2
3
4
y
-3
1
5
9
12
x
3
0
-3
-7
-11
y
1
2
3
4
5
1.1. Which of them have linear behavior? (No need to find formulas yet!). Explain
in terms of rates of change.
1.2. Consider the data sets with linear behavior. Pick a point and use the slope you
found in part (2) to find three more point.
1.3. Find the algebraic expression to describe the data sets which are linear.
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1.4. Use technology to plot the data.
1.5. Graph the mathematical model you found together with the data and see that the
data points satisfy the mathematical model.
2. A car is 30 kilometers away from a city and start moving at a constant velocity of 50
km/hour.
2.1. Generate a table to determine the distance from the city during the first 4 hours,
recording the information every half and hour (∆t = 0.5)
2.2. Find a mathematical expression that represents the distance of the car from the
city as a function of time, in hours.
2.3. Use the mathematical expression to determine:
2.3.1. How far the car is after 3.5 hours.
2.3.2. How long it takes the car to be 135 km away from the city.
3. We know that water freezing point is zero in the Celsius scale ( 00C ) which
corresponds to 32 in the Fahrenheit scale (320F), and that the water boiling point is
100 in the Celsius scale (0C) or 212 in the Fahrenheit scale (0F).
3.1. Write F (Fahrenheit) degrees as a linear function of C (Celsius) degrees
3.2. Write C degrees as a linear function of F degrees.
3.3. In each case interpret the meaning of the slope. Make sure you refer to the units.
4. Two lines are perpendicular if the product of their slopes is -1.
4.1. Find the equation of the line passing through the points (-1, 2) and (3, 4.5).
4.2. Find the equation of the line perpendicular to the line you found in 4.1 that goes
through the point (0,3)
4.3. Find the points which are 5 units from (-1, 2), and lie on a line perpendicular to
the line in 4.1.
5. The point (3, 4) is on a line with slope -0.5.
5.1. Use the interpretation of the slope (do not find the equation of the line) to find
three more points on the line.
5.2. Find the equation of the line and use it to find three more points on the line.
5.3. Find the expression that will characterize all the lines parallel to this line.
5.4. Use the expression from 5.2 to determine the equation of parallel passing through
the point (3,-10).
6. Find the equation of the line perpendicular to the line y  10 and passing through
the point (2,50)
7. Consider the line passing through the points (2,5) and (2,0).
7.1. Sketch the line and find its equation.
7.2. Find the equation of the line parallel to it and passing through the point (0,2)
7.3. Find the equation of the line perpendicular to it and passing through the point
(0,2)
Linear Functions. Revised 2/6/2016
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