WARM UP
What is a function.
How are they used.
FUNCTIONS
OBJECTIVES
Work with functions that are defined algebraically,
numerically, or verbally
Understand and explain explain algebraic functions
Review functions learned in Algebra 1 and Algebra 2.
Graph functions
IMPORTANT TERMS &
CONCEPTS
Function
Expressing mathematical ideas graphically,
algebraically, numerically & verbally.
Mathematical model
Dependent variable
Domain
Range
asymptote
Extrapolation
Interpolation
Functions
If you pour a cup of coffee, it cools more
rapidly at first, than less rapidly, finally
approaching room temperature.
Since there is one and only one temperature at any given
time, the temperature is called a function of time.
You can show the relationship
between coffee temperature and
time graphically.
The graph shows the temperature
y, as function of time x. At x = 0,
the coffee has just been poured
Functions
The graph shows that as time goes
on, the temperature levels off, until
it is so close to room temperature,
20 degree centigrade, that you can’t
tell the difference.
This graph might have come from
an algebraic equation,
y  20  70(0.8)
x
From the equation, you can find numerical
information. If you enter the equation into
your grapher, then use the table feature,
you can find these temperatures rounded
to 0.1 degrees.
x (min)
0
54
10
15
20
y (°C)
90
2.9
27.5
22.5
20.8
MATHEMATICAL MODELS
Functions that are used to make predictions and
interpretations about something in the real world
are called mathematical models.
Temperature is the dependent variable because the
temperature of the coffee depends on the time is has
been cooling.
Time is the independent variable. You cannot change
time simply by changing coffee temperature.
Always plot the independent variable on the horizontal
axis and dependent variable on the vertical axis.
GRAPHING TERMS
The set of values the independent variable of a
function can have are called domain. In the cup
of coffee example, the domain is the set of
non-negative numbers or x > 0.
The set of values of the dependent
variable corresponding to the
domain is called the range of the
function.
Time is If you don’t drink the coffee
(which would end the domain) the
range is the set of temperatures
between 20 C and 90 C, including 90
degrees centigrade or 20 < x < 90.
The horizontal line at 20 is called the asymptote. The graph
gets arbitrarily close to the asymptote but never touches it.
EXAMPLE 1
The time it takes you to get home from a football game is
related to how fast you drive. Sketch a reasonable graph
showing how this time and speed are related. Tell the
domain and range of the functions.
Solution: It seems reasonable to assume that the time it takes
depends on the speed you drive. So you must plot time on
the vertical axis and speed on the horizontal axis.
To see what the graph should
look like, consider what happens
to the time as the speed varies.
Pick a speed value and plot a
point for the corresponding time.
EXAMPLE 1 CONTINUED
Then pick a faster speed. Because time will
be shorter, plot a point closer to the
horizontal axis.
For a slower speed, the
time will be longer. Plot
a point farther from the
horizontal axis.
Finally, connect the points with a smooth
curve, since it is possible to drive at any speed
within the speed limit.
The graph never touches either axis, as shown.
If the speed were zero, you would never get
home. The length of time would be infinite.
Also, no matter how fast you drive, it will
always take some time to get there. You
cannot arrive instantaneously.
Domain: 0 < speed < speed limit
Range: time > minimum time at
speed limit