WARM UP
INVESTIGATION
Paper Cup Analysis
Find an equation for calculating the height of a stack of paper cups.
Instructions:
1. In your groups measure the height of stacks containing 5, 4, 3, 2
and 1.
2. Record the heights the nearest 0.1 cm. State what kind of cup
you used.
3. Use the sheets to record the data
4. Complete #1 - 11
FUNCTIONS
OBJECTIVES
Work with functions that are defined algebraically,
numerically, or verbally
IMPORTANT TERMS &
CONCEPTS
Function
Expressing mathematical ideas graphically, algebraically,
numerically & verbally.
Mathematical model
MATHEMATICAL OVERVIEW
In Algebra 1 and 2 you studied linear, quadratic,
exponential, power and other important functions. In
this course, you will learn general properties that apply
to all types of functions.
You will learn this in four ways:
Graphically
Algebraically
Numerically
Verbally
WHAT IS A FUNCTION?
Function
THE FOUR WAYS
GRAPHICALLY: This is the graph of a quadratic
function. The y-variable could represent the height
of an arrow at various times, x, after its release into
the air. For larger time values, the quadratic
function shows y is negative. These values may or
may not be reasonable in the real world.
ALGEBRAICALLY: The equation of the function is
y  4.9x2  20x  5
NUMERICALLY: The table shows corresponding x- and
y- values that satisfy the equation of the function:
x(s)
y(m)
0
5.0
1
20.1
VERBALLY: When the variables in a function stand for
things in the real world, the function is being used as
2
25.4
a mathematical model. The coefficients in the
3
20.9
2
equation of the
function
have
a
realy  4.9x  20x  5
world
For meaning.
example, the coefficient -4.9 is a constant that is a result of
the gravitational acceleration, 20 is the initial velocity and 5
reflects the initial height of the arrow.
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.
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
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. Give
the domain and range of the function.
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.
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. The pick a faster speed. Because the time
will be shorter, plot a point closer to the horizontal axis.
SOLUTION
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, because it is possible to drive at any
speed within the speed limit. The graph
never touches either axis, as the graph below
shows. If 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 you some time to get home.
You cannot arrive home instantaneously.
Domain: 0 ≤ speed ≤ speed limit
Range: time ≥ minimum time at speed limit
CHAPTER 1.1
ASSIGNMENTS
Paper Cup Activity
Concept Practice 1.1
Archery Problem: a – e
Mortgage Problem: a – e