Download Chapter 2 Notes

The Language of Functions (2-1)
The table is an example of a relation, a pairing
of 2 variables.
We can represent a relation in many ways:
A set of ordered pairs: (
,
), (
,
)
A mapping
A table of values (or T-table):
x
y
A rule expressed in words or symbols:
y = 2x + 3
The independent variable is called the domain. (It is the allowable values of the
independent variable.) Most often it is the set of x-coordinates.
The dependent variable is called the range. (It is the possible results from the allowable
domain values.) Most often it is the set of y-coordinates.)
A function is a special type of relation where each first element is __________________
___________________________________.
Is our relation of (age, height) a function?
Quick way to test: Vertical Line Test ______
______________________________________
Examples:
1. State the domain and range, and determine whether the relation is a function.
A. (-1,2), (-2,2), (3,2), (2,-1)
B. (1,1), (1,2), (1,-2), (2,0), (3,-1)
C. y = x2
D. y =
2
x 1
E. y = x
2. The cost of renting a car is $35 per day plus $40/mile for mileage over 100 miles.
a. Which is true? “Cost is a function of miles driven.” Or “Miles driven is a function of
cost.”
b. Identify the variables:
Independent variable: _____________
Dependent variable : _______________
c. State the domain and range.
Domain: _______________
Range: ________________
d. Write an equation to represent the relationship between cost, C, and mileage, m.
e. Graph the function.
Function Notation:
f(x) , which is read as “f of x”, indicates the value of the dependent variable when the
independent variable is x.
For example: y = f(x) = 2x + 3
Function: _______
Independent Variable : __________
Dependent Variable: ____________
f(4) means finding the value of the dependent variable when the independent variable is
4. (That is, plugging 4 in for x.)
f(4) = _______________________
Examples:
3. Suppose f(t) = (2 – 3t)(t + 6)
a. Evaluate f(1), f(2), f(3)
b. Does f(1) + f(2) = f(3)?
4. Suppose h(x) = x2 a. h(-2)
ATM
1
x
, find:
b. h(4t)
c. h(a+5)
Linear Models (2-2)
THE HUMAN WAVE
In this activity, we will gather information about how long it takes people to do the
human wave.
1. We need to gather information by timing how long it takes different numbers of
people to perform the wave. We will fill out the following table as we gather the
data.
# of people
Time (seconds)
2. What is the independent variable? ________________ Label that column x.
3.
What is the dependent variable? _______________ Label that column y.
4. Make a scatter plot by plotting the above points on the following graph. Be sure
to give a scale and label each axis with appropriate values.
5. Sketch what you think would be the line that best fits the above graph. Find the
equation of the line in the form y = mx + b. (You should be able to estimate the
y-intercept and use two points to find the slope.) Show work.
EQUATION __________________________________________
6. Use your equation to predict how long it would take for the entire school to
perform the wave. Show work.
7. Use your equation to predict how many students would be able to complete the
wave in 5 seconds. ___________ How many would complete the wave in one
minute? Show work.
Definitions:
Interpolation:
Extrapolation:
8. Interpret the meaning of the slope of your equation.
9. Interpret the meaning of the y-intercept of your equation.
* * * * * * * * We will finish this after the next lesson * * * * * * * * * *
10. Use your calculator to find the equation of the line of best fit. Graph that line on
the scatter plot. How close was your line compared to the calculator’s line?
Calculator EQUATION: ____________________________
11. Using the line of best fit equation, determine how long it would take for the entire
school to line up and complete the human wave. How close was your prediction in
#6?
12. Using the line of best fit equation, determine how many students would be able to
complete the wave in 5 seconds. How close was your prediction in #7?
How close was your prediction for the number of students who could complete the
wave in one minute?
2.3
The Line of Best Fit
The three data points (2,5), (5,3) and (8,4) are graphed on each of the graphs below,
along with a line that could be considered to fit the data. We are going to look at a
procedure for constructing a linear model which fits a set of points better than any other
line.
The Line of Best Fit is also called the ___________________________________. It has
the ____________________________ for the sum of the squares of the errors.
The method for finding the line of best fit is called _____________________________ .
The center of gravity for the data set is one point that is always ____________________
___________________________________. To find the center of gravity, find the
________ of the x values and the ________ of the y values.
Examples:
(a) Find the center of gravity for the points above.
(b) Is the center of gravity on line A? _____ on line B? _________
(c) Based on your answer to part (b), what can you conclude about which line is a better
fit?
________________________________________________________________________
(d) Can you say for sure that either line is the line of best fit based on the answer to part
b)?_____________________________ ____________________________________
Using a calculator, we get the following equation for the line of best fit. (You will learn
to do this tomorrow.)
y = -.1667x + 4.833 or y =

1
5
x 4
6
6
Example:
(a) Graph the data points given at the beginning of the lesson and graph the line of best
fit.
obs
pred
x
y
y
(error)2
(b) Is this line a better fit than the first two? __________ How can you tell? _________
____________________________________
(c) Verify that the center of gravity is on the line of best fit.
ATM
2.4 Practice
Graphing Greatest Integer/Least Integer Functions
Name __________________
Graph each of the following functions. Identify: a. the domain, b. the range, and c. the
values at which the function is not continuous.
1) f ( x)  x  1
2) g ( x)  x  1
a. domain:____________________
a. domain:____________________
b. range: ____________________
b. range: ____________________
c. discontinuous points: _______________
_______________
c. discontinuous points:
3) h( x)  2[ x]
4) F ( x)  x  [ x]
a. domain:____________________
a. domain:____________________
b. range: ____________________
b. range: ____________________
c. discontinuous points: _______________
_______________
Advanced Topics
c. discontinuous points:
Using TI-83/TI-84 to Calculate Correlation
Coefficient “r”
To enter data into a list:
Press STAT > EDIT, then type all the values of x (domain in L1) and the range in L2
To clear a list:
Go to the spreadsheet by pressing STAT > EDIT. Use the up arrow to place the
cursor on the name of the list you want to erase. Press CLEAR, then arrow back
down into the list.
To make a graphical display:
All the statistical graphs are under 2nd STATPLOT. You must turn on the graph
you want, select the appropriate type of graph and enter the correct list. Then
press either GRAPH or ZOOM 9: Zoom stat to see your graph.
What if you get an error? Check that the graph is turned on, make sure no other
graphs are on (including on the Y = screen), make sure you’ve entered the correct
list, in the case of a scatter plot make sure both the X and Y lists have the same
number of values, try hitting ZOOM 9 because sometimes it just needs to be
recentered. Otherwise rest your calculator by pressing 2nd , +, 712
To find the correlation coefficient:
First, your calculator must be set up to display the correlation. (You only have to
set it up once, so if you’ve done it in class, skip this part. Sometimes if you
change batteries you have to do it again.) Hit 2nd CATALOG (this is over the 0
button). Go down to DiagnosticOn, hit ENTER then ENTER again. It is now
set up to display correlation with the regression line.
Go to STAT>CALC 4:LinReg (a+bx) and hit ENTER. It is now pasted to the
home screen. You must input the names of the list containing the X values
followed by a comma then the list containing the Y values followed by a
comma, y1 For example, if my X values are in L1 and Y values are in L2, I would
enter
LinReg(a+bx) L1,L2, y1
Remember to hit ENTER, then you will see
Y = ax + b
a = ……..
b = ……..
r2 = ……..
r = …….
“r” correlation coefficient measures the strength of the correlation -1 < r < 1
“r2” Don’t worry about this. It is called coefficient of determination. It measures how
well the regression line represents the data. It is between 0 and 1
QUADRATIC MODELS (2-6)
y  ax2  bx  c (a  0)
If a  0 , the parabola opens downward and has a maximum and if a  0 , the
A model based on quadratic functions of the form:
Recall:
parabola
opens upward and has a minimum.
This max. or min. point is the vertex. To find the x-coordinate of the vertex:
x  b
2a
.
Once you have x, plug it into the equation to get the y-coordinate.
x-intercepts: (when y  0 )
(quadratic formula)
2
x   b  b  4ac
2a
y-intercept: (when x  0 ) y  c
Example
f (x)  2x2  9x  3
1) Consider the function
a. Find the x- and y-intercepts.
b. Sketch the graph.
Quadratic relation between height and time of an object
thrown upward: h   1 gt 2  v t  h
2
0
0
g = acceleration due to gravity = 32 ft/sec2 (or 9.8 m/sec2)
h = height after time t, v0=initial velocity, and h0=initial height.
Example
2) A projectile is shot from a tower 10 feet high with an upward velocity of 100 ft per
second.
a. Write an equation for the relationship between height h (in feet) and time t (in
seconds) after the
projectile is shot.
b. How long will the projectile be in the air?
c. How high will the object be after 4 seconds?
d. When will the object be 160 feet high?
ATM
Finding Quadratic Models (2-7)
To test the hypothesis that underinflated or overinflated tires can increase tire wear, new
tires of the same type were tested for wear at different pressures. The results are shown
in the table below.
x
Pressure
(in psi)
29
30
31
32
33
y
Mileage
(in thousands)
25
30
33
35
36
34
35
36
35
32
27
1. Make a scatterplot of these data.
2 a. If you were given data for only
x = 29 to x = 33 what kind of model
would best fit the data?
b. Choose two points and find a linear
model that fits these points. Graph it in
your calculator with the scatterplot.
3 a. If you were given data for only x = 33 to x = 36 what kind of model would best fit the data?
b. Choose two points and find a linear model that fits these points. Graph it in your calculator
with the scatterplot.
4. Use your calculator to find the line of best fit for all the data.
5. Do any of these lines fit the data well?
6. Choose three points and find a quadratic model.
7. Use your calculator and find the quadratic model of best fit.
8. Which model (linear or quadratic) fits better?
9. Over what domain would you expect your quadratic model to hold?