Section 1.2 -
Non-Linear Relationships and
Functions
Specific Curriculum Outcomes covered
C1
Model real world phenomena using quadratic functions
C3
Sketch graphs from descriptions, tables and collected data.
F1
Analyze scatter plots, and determine and apply the equations for curves of
best fit, using appropriate technology
C8
Describe and translate between graphical, tabular, written and symbolic
representations of quadratic relationships
C29
Analyze tables and graphs to distinguish between linear, quadratic and
exponential relationships.
A7
Describe and interpret domains and ranges using set notation
Assumed Prior Knowledge
Using technology to graph linear and non-linear functions and to find equations of the
curve of best fit
Using symbols to represent or model real-world relationships
Using common differences to classify non-linear data
INVESTIGATION 3 - Analyzing Non-Linear Data
Read Procedures A and B on p. 15 of your text.
Complete the Think About question in the margin of p. 15 in the following space:
Instructions for Using the CBR (Calculator Based RangerTM) with the TI83 Plus
1.
MAKE SURE YOUR CALCULATOR IS TURNED OFF. Obtain a CBR and a
link cable and plug into the appropriate slots on the calculator and CBR. The
calculator should automatically turn on. If not, then turn it on.
2.
Press the APPS button and select 2:CBL/CBR, then press ENTER.
3.
Follow the on screen instructions until you get to a screen where you can
choose 3:RANGER.
4.
Follow the on screen instructions until you get to the MAIN MENU. Select
1:SETUP/SAMPLE and press ENTER.
5.
Use the following settings. If any need to be changed, use the ARROW keys to
select it and press ENTER.
REALTIME:
TIME (S):
DISPLAY:
BEGIN ON:
SMOOTHING:
UNITS:
YES
15
DIST
ENTER
NONE
METERS
6.
Use the UP ARROW key to go to START NOW at the top of the screen.
Press ENTER.
7.
Follow the on screen instructions to carry out your experiment.
8.
Once the experiment has been completed and the CBR draws a graph of the
data for you, press ENTER and select 1:SHOW PLOT, then press ENTER to
see a better plot of the graph.
Sample original Plot
9.
Better plot of same experiment
We will only be interested in considering the portion of the graph for which
the cart was rolling up and down the ramp i.e. only the parabola shaped
portion. To select only this portion for further study, press ENTER and select
2:SELECT DOMAIN.
Your plot will appear again but with a blinking cursor and a message asking
LEFT BOUND?. Use the ARROW keys appropriately to position the cursor at
the beginning of the parabola and press ENTER. Similarly, when RIGHT
BOUND? appears, position the cursor NEAR where the parabola ends and
press ENTER. Wait a moment for the calculator to produce your new plot.
10.
Once your new plot appears, press ENTER and select QUIT.
The calculator has collected the data and placed it in lists for you. The values
for TIME will be stored in L1 and DISTANCE in L2 .
To see this data, press STAT, select EDIT and press ENTER.
Copy the first ten values for time and distance into a table of values and plot
the points.
11.
C)
A)
What type of regression should be performed on the data - linear,
quadratic, or cubic?
B)
(MATH 3205 ONLY) Use the data to ALGEBRAICALLY find the
equation of the curve of best fit.
Set up a scatter plot for this data using 2nd STAT PLOT on the calculator.
When the plot is set up, don’t forget to press ZOOM and select
9:ZOOMSTAT to have the calculator automatically set up a scale for you.
Use the calculator to find the equation of the curve of best fit.
12.
Suppose the experiment were repeated using the same force but on a much
steeper ramp than in the investigation. Predict the shape of the graph of time
versus distance traveled by a ball or car as it rolled up and down a steeper
ramp. Sketch your graph below.
13.
A)
Use the data in the table from step 10. Find the starting distance from
the CBR when time t = 0 seconds and call this value d0. Complete the
following table:
Time (s)
Distance from CBR
Distance from Start
Position
(d0 - Distance from CBR)
Plot a graph of Time and Distance from Start Position below.
B)
Compare your graph to the one you sketched in the Think About
question, part b).
C)
Use the graphing calculator to find the equation of the curve of best
fit.
What you should have learned from this Investigation:
‘
‘
How to analyze real world data
How to fit a quadratic function to data from an accelerating/decelerating
object
Focus B - Comparing Linear and Non-Linear
Relationships
Read the Focus.
You will need graph paper.
You will have to draw 5 scatter plots:
•
•
•
•
•
Speed and Braking Distance on Snow/ice
Speed and Braking Distance on Light Frost
Speed and Braking Distance on Gravel
Speed and Braking Distance on Asphalt
Speed and Perception-Reaction Distance
Ensure you can easily identify any linear and quadratic relationships
Complete Focus Questions #10-16, p.19
What you should have learned from this Focus:
‘
This Focus was mainly an opportunity to review and apply previously learned
techniques for finding linear and quadratic functions.
Investigation 4 - Finding Maximum and
Minimum Values
Using the Graphing Calculator
Problem:
A gardener has 50m of fencing to create a rectangular garden. She will
use her house as one side of the garden and the fence for the other
three sides. What dimensions (length and width) should she use to make
the garden with the largest area?
Procedure A.
Think about some possible widths. We will place some of these in a list. To do this:
Enter the following data in L1 :
L1
L2
L3
5
6
7
8
9
10
11
Go to the header for L2 . Press ENTER.
Type in “50 - 2×L1" then press ENTER.
(note: you will find the quotation marks in green above the + key).
Explain why this formula represents the possible lengths.
Add 12 as another data value in L1 and see what happens (The quotation marks link
the lists so that L2 is updated whenever a new value is added to L1). Keep adding in
as many possible widths as you can that make sense.
Use L1 and L2 to calculate the possible areas in L3 by entering a formula that uses
only L1 as the independent variable.
Procedure B
Create a scatter plot of width (L1) and area (L3). What type of
function should be used to model this data? Explain.
Procedure C
Find the equation of the curve of best fit.
#18, p. 20 in the space below. You can use the TRACE or 2ND CALC (select
4:maximum)feature on the graphing calculator to help answer this question.
#19, p. 20
A)
If w represents the width of the garden, what
formula would allow you to calculate the length, l ?
B)
If A is the area, and A = lw, use your formula in
part A) to replace l so that A is now expressed only
in terms of w.
C)
What type of function did you create in part B)?
D)
Show that this formula is now the same as the equation of the curve of
best fit you found in Procedure C.
What you should have learned from this Investigation:
G
G
How to find the maximum or minimum point of a quadratic function using
technology
How to model problems using quadratic functions to maximize or
minimize a quantity such as area
Extra Practice
Maximum/Minimum Problems
Note: To answer these problems, you can use the TRACE or the 2nd CALC
4:maximum (or 3:minimum) feature of your graphing calculator.
1.
The height, h of an object above the ground t seconds after it was fired from
the top of a 40m high building can be approximated by the function h = -5t2 +
50t + 40.
a)
How long does it take for the object to reach its maximum height?
b)
What is the maximum height reached by the object?
2.
On one bounce of a basketball, the height, h, of the ball after t seconds is
modeled by the function h = -4.9t2 + 9.8t.
a)
What is the maximum height reached by the ball and how long does it
take to reach this height?
b)
If the ball bounces to 60% of its previous maximum height on each
successive bounce, what will be the maximum height two bounces later?
3.
Jenilee just got her license because her driving instructor took pity on her.
The other day she hit a fire hydrant and sent it flying through the air in a
parabolic path described by the function h = -4.9t2 + 19.6t , where t is
the time in seconds and h is the height of the hydrant above the ground in
meters. What was the maximum height reached by the hydrant?
4.
John has 24m of fencing to make a rectangular flower garden. He plans to use
his house as one side of the garden. What dimensions will give the maximum
area?
5.
A lifeguard marks off a rectangular swimming area using 100m of rope. If he
uses the beach as one side of the swimming area, what are the dimensions that
give the maximum swimming area?
6.
At the local swimming pool, a swimming instructor plans to create a
rectangular swimming area for the beginners using one corner of the pool for
two sides and 12m of rope for the other two sides. What dimensions should she
use if she wants to obtain the maximum area?
7.
Find two numbers which differ by 18 and whose product is a minimum. (hint:
why can this problem be modeled by the function p=x(x+18) ?)
8.
Find two numbers that differ by 20 and whose product is a minimum.
9.
Two numbers have a sum of 36. If their product is to be a maximum, what are
the numbers?
10.
Two numbers differ by 16. If the sum of their squares is to be a minimum,
what are the numbers? What is the minimum product?
11.
A flower garden is to be made in the shape of a right triangle. The legs are to
have a total length of 28m. What is the maximum area that can be enclosed in
the garden?
12.
A rectangular piece of sheet metal is to be folded to make a trough with a
rectangular cross sectional area. The sheet metal is 150m wide. What
dimensions will maximize the area of the cross section?
13.
A number is added to its own square. What is the number if this sum is to be a
minimum? What is the sum?
14.
A long piece of sheet metal 30cm wide is bent to form an eavestrough with a
rectangular cross section. To attach the trough to the house, one of the
vertical sides must be 6cm longer than the other. What are the dimensions of
the trough that will yield a maximum cross sectional area?
(MATH 3205 ONLY)
The Newfoundland Theatre by the Sea group sells all 400 tickets each night
15.
if it charges $4 per ticket. The manager estimates that the ticket sales would
decrease by 25 for each $1.00 increase in the ticket price. What price would
result in the greatest revenue for the group?
16.
Each side of a square is 20cm. Four points on the sides are joined to form an
inner square. What is the minimum area of the inner
square?
17.
A rectangle has one vertex at (0,0), and the opposite vertex on the line
x+y=10. The other two vertices are on the positive x-axis and the positive yaxis. What is the maximum area of the rectangle?
18.
An ice cream store sells 1200 cones per day when the price is 90¢. Market
research indicates that for every 5¢ increase in price there will be 100 fewer
sales. What price, to the nearest nickel, will produce the highest revenue from
sales of ice cream cones?
Answers
#1. a)
5s
#3.
b) 165m
#2. a)
4.9m in 1 second
19.6m
#4.
6m by 12m
#7.
-9 and 9
#8.
-10 and 10
#11.
98m2
#12.
75cm by 37.5cm
#13.
Number is -½, Min. Sum is -¼
#15.
They increase the price by $6 to get $10 for the new ticket price
#16.
200 cm2
#17.
.1.8m
#5.
#9.
#14.
25 units 2
b) 1.764m
50m by 25m
18 and 18
#10.
6cm deep, 12cm wide
#18.
75¢
#6. 6m by 6m