Transcript Lesson_3-6 Comparing Models

Comparing models
Lines, quadratics, and exponential equations
• This table of values shows the depreciation of the value of a car
from year to year, where x is the number of years after it was
purchased, which was in 2000.
Years
Value of car
• Calculate the rate of change in
value from 2000 to 2003.
• What units should be used to
describe the rate of change?
• Describe the rate of change.
Minds on
after
($)
purchase
0
1
2
3
4
23 000
21 620
20 323
19 103
17 957
• Rate of change is calculated by finding:
Rate of Change = Change in y
Change in x
• aka Change in y / Change in x
• So the units work the same way!
• Unitsfor Rate of Change = Unitsfor y / Unitsfor x
Units for Rate of Change
• In a distance vs. time graph, the units for the rate of
change could be…
• In a table showing earnings over time, the units for the
rate of change could be…
• In measuring the class average against the number of
students added to a class, the units would be…
Examples
• I can distinguish between linear,
quadratic, and exponential models
• I can compare pairs of relations
Learning Goals
LINEAR MODELS
Characteristics of graphs, equations, rates of
change, and first differences
• A linear model represents quantities that increase
or decrease by a constant amount over equal
intervals
• In a table of values, the first differences are equal
• The graph is a straight line
• The equation can be written in the form y = m x
+ b, where m = slope, and b = y-intercept
• The rate of change is constant
Linear Models
500-W power setting
1000-W power setting
A cup of coffee is reheated in a microwave. There are 2 power
settings. The temperature of the coffee, C degrees Celsius,
after t seconds in the microwave can be modeled by the above.
How do these situations compare?
• This table shows the median age of Canada’s population from 1975
to 2000.
• A) Determine the equation of the line of best fit
• B) Predict the median age of Canada’s population in 2020.
• With the graphing calculator
• Without the graphing calculator
YEAR
MEDIAN AGE
(years)
1975 1980 1985 1990 1995 2000
27.4 29.1 31.0 32.9 34.8 36.8
Fitting a Linear Model to Data
YEAR
MEDIAN AGE
(years)
1975 1980 1985 1990 1995 2000
27.4 29.1 31.0 32.9 34.8 36.8
http://www.meta-calculator.com/online/
Fitting a Linear Model to Data
• Pg. 293 # 1-3, 7, 10, 14
• Homework Quiz on MONDAY!
Homework
GRAPHICAL MODEL
INVESTIGATION
Complete in pairs using the graphing calculators
QUADRATIC MODELS
How are they different from linear models?
• I can describe quadratic models in terms of their:
•
•
•
•
Visual graphs
Equations
Rates of change
First and second differences
Learning Goals
• http://www.thirteen.org/get-the-math/the-challenges/math-inbasketball/introduction/181/
Math in Basketball
• In a table of values, the second differences are
equal, and not zero
• The graph is a curve called a parabola
• The equation can be written in the form
ax2+bx+c =0 where a is not zero. WHY???
• The rate of change is always changing  either
from increasing to decreasing or from decreasing
to increasing
Characteristics of
Quadratic Models
EXAMPLE 1 Stopping Distances
The graph shows the stopping distance (in metres)
and speed of a car (in kilometres per hour)
(a) Describe the relationship between stopping
distance and speed
As speed increases, the stopping distance
also increases
(b) Use the graph to estimate the stopping distance
at
(i) 50 km / h
At 50 km/h, stopping distance is
approximately 10 metres
(ii) 100 km / h
At 100 km/h, stopping distance is
approximately 40 metres
EXAMPLE 1 Stopping Distances
(c) How many times further is the stopping distance
at 100 km/h compared to the stopping distance at 50
km/h?
Stopping dis tan ce at 100 km / h

Stopping dis tan ce at 50 km / h

40 m
10 m
 4 times
The stopping distance at
100 km/h is 4 times
further than the stopping
distance at 50 km/h
(d)Consider the rate of change of stopping distance
with respect to speed. What are the appropriate
units for this rate of change?
Stopping dis tan ce The appropriate units
Rate of change 
Speed
are metres per
kilometre per hour
m
Rate of change 
km / h
EXAMPLE 1 Stopping Distances
(e) Is the rate of change of stopping distance with
respect to speed increasing, constant or decreasing?
Explain.
The rate of change is increasing
 Graph is curving up and is gradually getting
steeper
EXAMPLE 2
Analysing a Free Fall
+ 0.2
The table shows distance and time ++0.2
0.2
data for the Drop Zone ride at
+ 0.2
Canada’s Wonderland.
+ 0.2
+ 0.2
(a) Does the time column show equal+ 0.2
+ 0.2
time intervals?
+ 0.2
Yes
 Each time interval is going up+ 0.2
+ 0.2
by 0.2 seconds
+ 0.2
+ 0.2
EXAMPLE 2
Analysing a Free Fall
+ 0.2
(b) Calculate the 1st Differences and++0.2
0.2
record it in the table.
+ 0.2
+ 0.2
(i)Does this model a linear
+ 0.2
relationship? Explain.
+ 0.2
+ 0.2
No
+ 0.2
 1st Differences are not
+ 0.2
constant
+ 0.2
+ 0.2
(ii) Do the 1st Differences imply an
+ 0.2
increasing, constant or decreasing rate
of change of distance with respect to
time?
Since 1st differences are
becoming more positive, the
rate of change is increasing
0.2 – 0.0 = + 0.2
0.8 – 0.2 = + 0.6
1.8 – 0.8 = + 1.0
+ 1.4
+ 1.8
+ 2.2
+ 2.6
+ 3.0
+ 3.4
+ 3.8
+ 4.2
+ 4.6
+ 5.0
EXAMPLE 2
Analysing a Free Fall
2nd
(c) Calculate and record the
Differences. Do the 2nd Differences
imply a quadratic model? Explain.
Yes
 2nd Differences are constant
+ 0.2
+ 0.6
+ 1.0
+ 1.4
+ 1.8
+ 2.2
+ 2.6
+ 3.0
+ 3.4
+ 3.8
+ 4.2
+ 4.6
+ 5.0
0.6 – 0.2 = + 0.4
1.0 – 0.6 = + 0.4
1.4 – 1.0 = + 0.4
+ 0.4
+ 0.4
+ 0.4
+ 0.4
+ 0.4
+ 0.4
+ 0.4
+ 0.4
+ 0.4
PREVIOUS
EXAMPLE 2
Analysing a Free Fall
(d) Create a scatterplot with time on the horizontal axis and distance on the vertical
axis.
Time vs. Distance
Distance
(metres)
Time (seconds)
EXAMPLE 2
Analysing a Free Fall
(e) Does the graph imply a
linear or non-linear
relation?
NON-LINEAR
 Points form a
smooth curve
(f) Compare the table and
the graph. Does the rate of
change of distance with
respect to time appear to be
increasing, constant or
decreasing.
INCREASING
- curve on graph is
going up and
becoming steeper
Time vs. Distance
Distance
(metres)
Time (seconds)
How can I tell if I’m looking at quadratic
data?
•
•
•
•
First differences are NOT equal
Second differences are equal
Graph looks like a curve – specifically parabolic
Equation has a squared term in it: usually written as x2
Consolidate
• I can describe quadratic models in terms of their:
•
•
•
•
Visual graphs
Equations
Rates of change
First and second differences
Homework:
Pg. 303
# 1-3, 5-8, 10
Learning Goals
• I can describe exponential models in terms of
their:
•
•
•
•
Visual graphs
Equations
Rates of change
First and second differences
Learning Goals
KEY CONCEPTS
For all exponential models, the first and second differences are
always non-constant
The ratios can be calculated by DIVIDING each y-value by
the y-value that comes BEFORE it.
When RATIOS are CONSTANT, the relation is exponential.
In an exponential relation, there is a constant percent increase
over equal intervals
EXAMPLE 1
Bacteria Growth
(b) Use the graph to estimate the number of bacteria
after each time period
(i) 20 minutes
20
________
(iii) 60 minutes
80
________
40
(ii) 40 minutes ________
(c) Calculate the ratios
Divide the number of bacteria after 40 min by the
number after 20 min
40
# of Bacteria at 40 min

2
# of Bacteria at 20 min
20
Divide the number of bacteria after 60 min by the
number after 40 min
# of Bacteria at 60 min 80
2

40
# of Bacteria at 40 min
EXAMPLE 1
Bacteria Growth
Divide the number of bacteria after 40 min by the
number after 20 min
40
# of Bacteria at 40 min

2
# of Bacteria at 20 min
20
Divide the number of bacteria after 60 min by the
number after 40 min
# of Bacteria at 60 min 80
2

40
# of Bacteria at 40 min
(d) What happens to the number of bacteria every 20
minutes?
The number of bacteria doubles
EXAMPLE 1
Bacteria Growth
(e) Consider the rate of change of number of bacteria
with respect to time. What are suitable units for this rate
of change?
Rate of change 
# of Bacteria
Time
Rate of change 
Bacteria
min ute
The units for rate
of change is
bacteria per
minute
(f) Is the rate of change of number of bacteria with
respect to time increasing, constant or decreasing.
Explain.
INCREASING
 Curve is going upwards
 Curve is getting steeper
EXAMPLE 2
Smoke Detectors
Americium-241 (Am-241) is a
manufactured element. It is a
silvery radioactive metal, which is
used in smoke detectors.
Household smoke detectors
contain about 200 micrograms
( g) of Am-241. The amount of
Am-241 present in the detector
decreases or decays over time.
The table shows the mass of Am241 remaining, in micrograms,
over 1000 years.
(a) Does the Years column show
equal time intervals?
Yes
 Going up by 100 years
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
1st Differences
Not constant
Not Linear!
EXAMPLE 2
Smoke Detectors
(b) Calculate the 1st Differences
and 2nd Differences. Is the
relationship linear, quadratic or
neither? Explain.
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
170 - 200 = – 30
145 - 170 = – 25
124 – 145= – 21
– 19
– 15
– 14
– 11
– 10
–8
–7
PREV
1st Differences
Not constant
Not Linear!
EXAMPLE 2
Smoke Detectors
(b) Calculate the 1st Differences
and 2nd Differences. Is the
relationship linear, quadratic or
neither? Explain.
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
+ 100
– 30
– 25
– 21
– 19
– 15
– 14
– 11
– 10
–8
–7
2nd Differences
Not constant
Not Quadratic!
– 25 – (– 30)
= – 25 + 30
=+5
+5
+4
+2
+4
+1
+3
+1
+2
+1
– 21 – (– 25)
= – 21 + 25
=+4
– 19 – (– 21)
= – 19 + 21
=+2
1st Differences
Not constant
Not Linear!
EXAMPLE 2
Smoke Detectors
(c) Calculate the ratios. Divide
the second mass by the first, the
third mass by the second, etc.
How do you know that the relation
is exponential?
2nd Differences
Not constant
Not Quadratic!
Ratios
Constant!!!
EXPONENTIAL!
+ 0.85
+ 0.85
+ 0.85
+ 0.85
+ 0.85
+ 0.85
+ 0.85
+ 0.85
+ 0.85
+ 0.85
Divide successive y values!
170 / 200
= + 0.85
124 / 145
= + 0.85
145 / 170
= + 0.85
EXAMPLE 2
Smoke Detectors
(d) Draw a graph with Years on the horizontal axis and Mass remaining on the
vertical axis.
(e) How do you know that
this graph is:
Mass ( g)
(i) Not linear?
Not a straight line
(ii) Not quadratic?
Unable to tell (would
need to look beyond
given data)
Time (Years)
EXAMPLE 2
Smoke Detectors
(d) Draw a graph with Years on the horizontal axis and Mass remaining on the
vertical axis.
Mass ( g)
Time (Years)
(f) Compare the table
and graph. Does the rate
of change of mass
remaining with respect to
years appear to be
constant, increasing or
decreasing? Explain.
Look at 1st Differences
They are increasing
BUT...
Graph is decreasing
 Curve is
becoming less
steep
EXAMPLE 2
Smoke Detectors
(g) What are the suitable units for the rate of change of mass remaining with respect
to years?
Mass remaining
Rate of change 
Time
Rate of change 
g
Year
The units for the rate of change
of mass remaining with respect
to years is micrograms per year
• I can describe exponential models in terms of
their:
•
•
•
•
Visual graphs
Equations
Rates of change
First and second differences
Learning Goals
EXPONENTIAL MODELS
With technology
EXAMPLE
Draw Cards
Sandor drew cards from a standard deck of 52 playing cards until he drew a heart.
He repeated this experiment many times. Each time, Sandor recorded the number of
cards he drew before drawing a heart. For example, if he drew a heart on the first
draw, then he drew zero cards before he drew a heart. This happened 50 times.
(a) Calculate the 1st Difference, 2nd Differences and Ratios. Does the relationship
between frequency and number of cards appear to be linear, quadratic or
exponential. Explain.
1st Differences
Not constant
Not Linear!
38 – 50 = – 12
28 – 38 = – 10
21 – 28 = – 7
–5
–4
–3
EXAMPLE
Draw Cards
Sandor drew cards from a standard deck of 52 playing cards until he drew a heart.
He repeated this experiment many times. Each time, Sandor recorded the number of
cards he drew before drawing a heart. For example, if he drew a heart on the first
draw, then he drew zero cards before he drew a heart. This happened 50 times.
(a) Calculate the 1st Difference, 2nd Differences and Ratios. Does the relationship
between frequency and number of cards appear to be linear, quadratic or
exponential. Explain.
1st Differences
Not constant
Not Linear!
2nd Differences
Not constant
Not Quadratic!
– 12
– 10
–7
–5
–4
–3
– 10 – (– 12)
– 10 + 12
=+2
+2
+3
+2
+1
– 7 – (– 10)
– 7 + 10
=+3
+1
– 5 – (– 7)
–5+7
=+2
EXAMPLE
Draw Cards
Sandor drew cards from a standard deck of 52 playing cards until he drew a heart.
He repeated this experiment many times. Each time, Sandor recorded the number of
cards he drew before drawing a heart. For example, if he drew a heart on the first
draw, then he drew zero cards before he drew a heart. This happened 50 times.
(a) Calculate the 1st Difference, 2nd Differences and Ratios. Does the relationship
between frequency and number of cards appear to be linear, quadratic or
exponential. Explain.
1st Differences
38 / 50
= + 0.76 Not constant
Not Linear!
+ 0.76
+ 0.74
+ 0.75
+ 0.76
+ 0.75
+ 0.75
28 / 38
= + 0.74
2nd Differences
Not constant
Not Quadratic!
21 / 28
= + 0.75 Ratios
Relatively CONSTANT
EXPONENTIAL!
EXAMPLE
Draw Cards
EXPONENTIAL REGRESSION
1. You need to set-up your calculator so it
can perform exponential regression.
Press 2nd
0 (zero)
x-1
Use the DOWN cursor key until you reach
Diagnostic On. Once you reach this
command, press ENTER and ENTER
2. Clear the data table by pressing STAT
4:ClrList
2nd and 1 then “,”
2nd and 2 (* your screen should look like
the one on the bottom right)
ENTER
ClrList L1, L2
EXAMPLE
Draw Cards
EXPONENTIAL REGRESSION
3. Enter the data into the lists by pressing
STAT and 1:Edit. Enter the “# OF
CARDS” data in L1 and “FREQUENCY”
data in L2.
4. Create a scatter plot by pressing 2nd,
Y=, 1 and ENTER. Make sure that the
cursor is on “On” when you press Enter.
By default, your Xlist should be L1 and
Ylist should be L2.
5. To display your graph, press ZOOM and
9.
EXAMPLE
Draw Cards
EXPONENTIAL REGRESSION
3. Enter the data into the lists by pressing
STAT and 1:Edit. Enter the “# OF
CARDS” data in L1 and “FREQUENCY”
data in L2.
4. Create a scatter plot by pressing 2nd,
Y=, 1 and ENTER. Make sure that the
cursor is on “On” when you press Enter.
By default, your Xlist should be L1 and
Ylist should be L2.
5. To display your graph, press ZOOM and
9.
Sketch the scatterplot below. Label all
axes
Frequency
# of Cards
EXAMPLE
Draw Cards
REGRESSION ANALYSIS and EXPONENTIAL EQUATION OF BEST FIT
1. To perform regression analysis, press
STAT
Move the right cursor over to CALC
Press 0:ExpReg (this activates the exponential regression function)
2. You must tell the calculator which data to perform the quadratic regression on. We
do this by entering the lists from our data table. You do this by pressing
2nd, 1 (this will tell it to pick L1) then “,” (comma key)
2nd, 2 (this will tell it to pick L2) then “,” (comma key)
EXAMPLE
Draw Cards
REGRESSION ANALYSIS and
EXPONENTIAL EQUATION OF BEST
FIT
3. We will store the information for our line
of best fit in a variable called Y1. To do
this, press:
VARS
Right cursor key to Y-VARS
1:Function
1:Y1
Your screen should look like this:
ExpReg L1, L2,Y1
4. Press ENTER to execute. Fill the
information that is given to you:
ExpReg
y = a*b^x
a=
50.028
b=
0.751
r2 =
0.999
r=
0.999
(a) Write the exponential equation of
y = 50.028(0.751)x
best fit _________________________
(b) Press GRAPH to view the
scatterplot with the exponential curve of
best fit. Draw the curve on your sketch
in #5 above.
EXAMPLE
Draw Cards
4. Press ENTER to execute. Fill the
information that is given to you:
ExpReg
y = a*b^x
Frequency
# of Cards
a=
50.028
b=
0.751
r2 =
0.999
r=
0.999
(a) Write the exponential equation of
y = 50.028(0.751)x
best fit _________________________
(b) Press GRAPH to view the
scatterplot with the exponential curve of
best fit. Draw the curve on your sketch
in #5 above.
EXAMPLE
Draw Cards
(c) Use the equation from 4(a) to answer
to predict the frequency of drawing eight
(8) cards before drawing a heart
Substitute x = 8 into the equation
y = 50.028(0.751)x
y = 50.028(0.751)8
y = 50.028(0.1012)
y=5
The frequency of drawing 8 cards before
drawing a heart is 5
4. Press ENTER to execute. Fill the
information that is given to you:
ExpReg
y = a*b^x
a=
50.028
b=
0.751
r2 =
0.999
r=
0.999
(a) Write the exponential equation of
y = 50.028(0.751)x
best fit _________________________
(b) Press GRAPH to view the
scatterplot with the exponential curve of
best fit. Draw the curve on your sketch
in #5 above.
• http://www.thirteen.org/get-the-math/thechallenges/math-in-restaurants/introduction/179/
What’s wrong with this picture?
LINEAR GRAPHS
QUADRATIC GRAPHS
EXPONENTIAL GRAPHS
y=
y=
y=
m>0
m<0
a>0
a<0
a >0 & b > 1
a >0 & 0 < b < 1
___________________ differences
___________________ differences
___________________ factors are
are ___________________
are ___________________
___________________
MINDS ON: Now to our friendly
unit study guide…
• Quadratic and Exponential Regression  Graphing
Calculators!
Using Technology
• In a science experiment, students punched a hole near the bottom of a
2-L pop bottle. They filled the bottle with water and measured how the
water level changed over time. The results are shown in the table.
TIME (s)
WATER
LEVEL
(cm)
0
25 50 75 100
30.0 22.3 16.1 11.2 7.8
Using the graphing
calculators, perform
linear, quadratic, and
exponential
regression to
determine the best
model to represent
this data.
Use Technology to Select
the Best Model
Manny throws a
baseball
Gloria throws a
baseball
h = -0.05x2 + 0.45x + 4.5 h = -0.03x2 + 0.54x + 6.24
MINDS ON: How do these
two situations compare?
Simple Interest
A = 250 +(0.07 ✕ 250)x
Compound Interest
A = 250(1.07x)
How do these two
situations compare?
• I can distinguish between linear, quadratic, and
exponential models
• I can compare pairs of relations
Learning Goals
• I can select an appropriate mathematical model to
represent a situation
Learning Goals
Cup X cooled at a constant rate. The tempe
Sect ion
one-half every 20 min.
Pract ise
5.4
c) Ball X rolled down a ramp. Ball Y was thro
1. Match each situation with its graph.
a) The number of bacteria in Colony X remained
A the same over time.
Colony Y started with 50 bacteria and doubled every half-hour.
Date:
b) Two cups of water were cooled in different controlled environments.
Cup X cooled at a constant rate. The temperature of Cup Y decreased by
Sect ion
Pract ise
one-half every 20 min.
5.4
1. Match each situation with its graph.
c) Ball X rolled down a ramp. Ball Y was thrown from a point above the ground.
a) The number of bacteria in Colony X remained the same over time.
B
Colony Y started with 50 bacteria and doubledAevery half-hour.
b) Two cups of water were cooled in different controlled environments.
Cup X cooled at a constant rate. The temperature of Cup Y decreased by
one-half every 20 min.
c) Ball X rolled down a ramp. Ball Y was thrown from a point above the ground.
A
B
B
C
C
2. The population of Town X started at 90 000 an
The population of Town Y started at 4000 and
is true?
A The population of Town X is always greate
C
B The rate of change of the population of Tow
C The rate of change of the population of Tow
2. The population of Town X started at 90 000 and increased by 25 000 every year.
D The population
Townstatement
Y is greater than th
The population of Town Y started at 4000 and doubled
every year.of
Which
• PROBLEM 1
• The population of Town
X started at 90 000 and
increased by 25 000 every
year.
• The population of Town
Y started at 4000 and
doubled every year.
• PROBLEM 2
• Ing has the choice of two
payment options for her
new job.
• Option A: Starting salary
of $48 000, with a $1000
raise every following year
• Option B: Starting salary
of $45 000, with a 2.5%
raise every following year
Based on what you know about LINES, QUADRATICS and EXPONENTIAL
FUNCTIONS, make a prediction about which graph would suit each situation.
Make a prediction
• Come up with an equation to represent the following
situations. For each, write “let” statements i.e. “Let x
represent…. Let y represent….” before you write your
equation:
• The number of people who hear a rumour, started by one
person, triples every day.
• The value of my computer depreciates by 12% each year,
and I bought it for $875
• When Bob and Jeanette got married, the only piece of
jewelry she owned was her wedding ring. Bob likes shiny
things, so he buys her a piece of jewelry every year on her
birthday, their anniversary, and Valentine’s day.
On your own
• I can select an appropriate mathematical model to
represent a situation
• “Seatwork Handout” due at the end of class
Learning Goals