Topic 2: Linear Equations and Regression Polynomial Functions

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.

Explore… 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… 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… 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… 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… f) After 2 seconds, what is the volume of water in the glass? You can get these values by simply looking at the graph, too!!! Explore… f) After 2 seconds, what is the volume of water in the glass? g) At what time is there 250 mL of water remaining

Explore… h) Determine an appropriate domain and range for the question.

Information 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.

Example 1 Drawing a line of best fit Nathan wonders whether he can predict the size of a person’s hand span based on the person’s height. His math class investigated this relationship and recorded measurements from 15 students in the tables below. 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. A possible domain could be {h|150<h<180, h∈R} and a possible range could be {s|15<s<25, s∈R}.

Example 1 Drawing a line of best fit c) The data was plotted on the grid below. Describe the relationship between the variables. As the height increases, the hand span is larger! d) Use a ruler to draw a line that approximates the trend in your scatter plot.

Information So far we’ve 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.

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].

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.

Note: Once your equation is in Y=, your calculator can interpolate and extrapolate for you! 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. 2006 is an x-value. Press 2nd 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.

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 2nd Trace 5:Intersect and then press enter 3 times.] A distance of 82 km is reached in the year 2001.

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.

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.

You’re ready! Try the homework from this section. 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. You’re ready! Try the homework from this section.