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Slide 1
Polynomial Functions Topic 2: Linear Equations and
Regression
Slide 2
I can graph data, and determine the linear function that best
approximates the data. I can interpret the graph of a linear
function that models a situation, and explain the reasoning. I can
solve, using technology, a contextual problem that involves data
that is best represented by graphs of linear functions, and explain
the reasoning.
Slide 3
Aya quenches her thirst after a soccer game by drinking a large
glass of water at a constant rate. a) What is the independent
variable? b) What is the dependent variable? The independent
variable is the time (s). The dependent variable is the volume (mL)
since the volume of water in the glass is dependent on how much
time has passed. Explore
Slide 4
c) What is the y-intercept and what does it represent in the
context of this question? The y-intercept is 600mL. It represents
the volume of water in the glass when Aya starts to drink (t=0).
Explore
Slide 5
d) A linear equation that represents this situation is V =
-50t+600, where t is the number of seconds elapsed and V is the
volume of water that is in the glass. Find each of the following,
and explain what each means in the context of this question. i.
lead coefficient and a description of the trend ii. constant iii.
t-intercept The graph decreasing at a constant rate. The lead
coefficient represents the amount of water consumed every second
(50mL decrease every second). The constant is the y-intercept so it
is equal to the amount of water at the beginning (600mL). The
t-intercept is the amount of time it takes for the glass volume to
reach 0 mL (12 seconds). Explore
Slide 6
e)Use the equation to interpolate the x-value for a y-value of
300. Does it make sense in the context of this question? When the
V-value is 300 the t-value is about 6 seconds. We can also use the
equation: Keep in mind that using the equation will make your
answer more accurate. Explore
Slide 7
f) After 2 seconds, what is the volume of water in the glass?
g) At what time is there 250 mL of water remaining You can get
these values by simply looking at the graph, too!!! Explore
Slide 8
h) Determine an appropriate domain and range for the question.
Explore
Slide 9
A set of data can be represented by ordered pairs. The
independent variable is the variable being manipulated. The
dependent variable is the variable that is being observed. The
ordered pairs can be plotted on a grid. A scatter plot is a set of
points on a grid, used to visualize a possible trend in the data. A
line of best fit is a line that best approximates the trend in a
scatter plot. A linear regression function is the equation of a
line of best fit. Technology uses linear regression to determine
the equation that balances the points in the scatter plot on both
sides of the line. Information
Slide 10
Example 1 Nathan wonders whether he can predict the size of a
persons hand span based on the persons height. His math class
investigated this relationship and recorded measurements from 15
students in the tables below. Drawing a line of best fit a) Choose
the dependent variable. Explain your choice. b) What is a
reasonable domain and range for the relationship in the data? The
dependent variable is the hand span, since Nathan is checking to
see if it is determined by height.
Slide 11
Example 1 c) The data was plotted on the grid below. Describe
the relationship between the variables. Drawing a line of best fit
d) Use a ruler to draw a line that approximates the trend in your
scatter plot. As the height increases, the hand span is
larger!
Slide 12
So far weve looked at how to interpret a graph. Now we will
look at how to create a graph based on a list of information. First
we create a scatterplot on our calculator. Then we find the
equation of best fit using the regression function on our
calculator. Once we have graphed the line of best fit we can answer
a variety of questions. The next two slides outline the steps for
performing a regression. Information
Slide 13
Slide 14
Slide 15
Example 2 Using linear regression to solve a problem A bicycle
event occurs once a year on the same day. The winnder is determined
by the person who travels the furthest in one hour, The table below
shows the winning distances and the year in which they were
accomplished. a) Determine an appropriate window. Since the years
start in 2003 and end in 2009, an appropriate window for x might be
X: [2000, 2010, 1]. Since the distance goes from 83.72 to 90.60,
and appropriate window for y might be Y: [80, 95, 1].
Slide 16
Example 2 Using linear regression to solve a problem b) Use
technology to create a scatter plot and draw a sketch. c) Determine
the equation for the linear regression function that models the
data. Add this line to your scatter plot sketch. Use the calculator
steps at the end of your workbook to draw the scatterplot. Perform
the linear regression and enter the equation into Y= to display the
graph.
Slide 17
Example 2 Using linear regression to solve a problem d)
Interpolate what a world-record distance might have been in the
year 2006, to the nearest hundredth of a kilometre. e) Compare your
estimate with the actual world-record distance of 85.99 km in 2006.
Note: Once your equation is in Y=, your calculator can interpolate
and extrapolate for you! 2006 is an x-value. Press 2 nd Trace 1:
Value and enter the x-value of 2006. In 2006, the world-record for
distance travelled in one hours is 86.287 km. 86.286567 85.99 =
0.297. The record is 0.297 km less than my interpolated value.
Slide 18
Example 4 Using linear regression to solve a problem f) Use
your equation to extrapolate the x-value for a y-value of 82. 82 is
a y-value. Enter 82 into Y=. Then solve for the x-value of the
intersection point. [Press 2 nd Trace 5:Intersect and then press
enter 3 times.] A distance of 82 km is reached in the year
2001.
Slide 19
Need to Know A scatter plot is a set of points on a grid used
to visualize the data. If there seems to be a trend, there may be a
relationship between the independent and dependent variable. The
independent variable is the variable being manipulated. The
dependent variable is the variable that is being observed.
Slide 20
Need to Know If there appears to be a trend, then a line of
best fit can be determined. A line of best fit is a line that best
approximates the trend in a scatter plot. A linear regression
function is the equation of a line of best fit. Technology uses
linear regression to determine the equation that balances the
points in the scatter plot on both sides of the line.
Slide 21
Need to Know A line of best fit can be used to predict values
that are not recorded or plotted. Predictions can be made by
reading values from the line of best fit on a scatter plot or by
using the equation of the line of best fit. Interpolation is the
process used to estimate a value within the domain of a set of
data, based on a trend. Extrapolation is the process used to
estimate a value outside the domain of a set of data, based on a
trend. Youre ready! Try the homework from this section.