Download Activity 5

Activity 2 - 5
Predicting Population
5-Minute Check on Activity 2-4
Identify each of the following with the type of shift involved:
y = 3x – 5
y = |x|
y = 3(x + 2) – 5
y = |x| + 1
H-Shift left 2
V-Shift up 1
y = 3x – 5 – 4
V-Shift down 4
y = 3(x – 3) – 5
H-Shift right 3
y = 3x – 5 + 6
V-Shift up 6
y = |x – 9|
H-Shift left 9
y = |x + 8|
H-Shift left 8
y = |x| – 7
V-Shift down 7
Click the mouse button or press the Space Bar to display the answers.
Objectives
• Write an equation for a linear function give its slope
and y-intercept
• Write linear functions in slope-intercept form,
y = mx + b
• Determine the relative error in a measurement or
prediction using a linear model
• Interpret the slope and y-intercept of linear functions
in contextual situations
• Use the slope-intercept form of linear equations to
solve problems
Vocabulary
• Error – is the difference between the actual value and
the predicted value; error = actual - predicted
• Observed Value – also known as the actual value
• Expected Value – also known as the predicted value
• Relative Error – is the ratio of the error to the observed
value
Activity
According to the US Bureau of the Census, the
population of the United States was approximately 132
million in 1940 and 151 in 1950. Assume that the rate of
change of the population with respect to time is a
constant value over the decade from 1940 to 1950.
P
Write the data as two ordered pairs 200
(1940, 132M) and (1950, 151M)
150
Plot the two data points and draw
a line through them
125
100
t
1940
1950
Activity cont
What is the average rate of change of the population from
1940 to 1950 (time t = 0 to t = 10)?
151 – 132
19
------------- = ----- = 1.9 million per year
10 – 0
10
What is the slope of the line?
m = 1.9
What is the practical meaning of the slope in the model?
During that decade the population increased by an average of
1.9 million people each year
What is the P-intercept of this line?
With 1940 as t=0, 132 million is the P-intercept
What is the practical meaning of the intercept?
The population in 1940 was 132 million
Activity cont 2
What is the slope-intercept form of the population model
line?
p = 1.9 (t) + 132
or
p = 1.9 ( t – 1940) + 132
Assume that the average rate of change holds true
through 1960. Use this equation to predict the 1960 US
population.
p = 1.9 (20) + 132
or
p = 1.9 ( 1960 – 1940) + 132
p = 38 + 132 = 170
or
p = 1.9(20) + 132 = 170
Error and Relative Error
•
The error in a prediction is the difference between the
observed value (actual value that was measured) and
the predicted value from the model. The Relative
error is the ratio of the error to the observed value.
Observed value – predicted value
Relative error = -------------------------------------------------Observed value
=
•
error
------------------------Observed value
It is normally expressed as a percentage
Activity cont 3
The actual population for the US in 1960 was
approximately 179 million. What is the relative error in
the prediction we made?
179 – 170 = 9 (error)
9 / 170 = 5.29% (relative error)
What do you think was the cause of the model’s error?
Population growth was not a linear function
Why would that be true (History question!)?
Baby-boom generation after WWII
Population Model
Lets look at more recent data. The US population was
approximately 249 million in 1990 and 281 million in 2000.
Determine the slope of this line
y
281 – 249
32
------------- = ----- = 3.2 million per year
10 – 0
10
Compare it with the slope of the
1940 to 1950 model
x
Which decade had the greatest population increase?
The 90’s had 32 million compared to 19 million in the 40’s
Population Model cont
Determine the P-intercept in the new line
If t = 0 in 1990, then P-intercept is 249
Write the slope-intercept form of the equation of the line
p = 3.2t + 249
Predict the population in 2010
p(20) = 3.2(20) + 249 = 313 million (Current Estimate: 310,354,711)
What assumption did we have to make in the prediction?
That population growth is a linear function
Population Model cont
According to the linear model, P(t) = 3.2 t + 249, in what
year will the population be 350 million?
350 = 3.2 t + 249
101 = 3.2 t
31.5625 = t
So approximately July of 2021
Summary and Homework
• Summary
– Error is the difference between actual and predicted
values
– Relative error is the ratio of the error to the observed
value
– Relative error is usually reported as a percentage
• Homework
– Pg 221-4; 1 - 6