UNIT 4 DESCRIPTIVE STATISTICS Lesson 2: Working with Two Categorical and Quantitative Variables Instruction

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Eustacia Blair
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1 Prerequisite Skills This lesson requires the use of the following skills: plotting points on the coordinate plane, given data in a table plotting the graph of a linear function, given an equation plotting the graph of an exponential function, given an equation evaluating a function at a given input value solving a function for x given a y-value interpreting a function in a given context, using the graph or the equation Introduction Data with two quantitative variables can be represented using a scatter plot. A scatter plot is a graph of data in two variables on a coordinate plane, where each data pair is represented by a point. Relationships between the two quantitative variables can be observed on the graph. A function is a relation of two variables where each input is assigned to one and only one output. Functions in two variables can be represented algebraically with an equation, or graphically on the coordinate plane. Graphing a function on the same coordinate plane as a scatter plot for a data set allows us to see if the function is a good estimation of the relationship between the two variables in the data set. The graph and the equation of the function can be used to estimate coordinate pairs that are not included in the data set. Key Concepts Data with two quantitative variables can be represented graphically on a scatter plot. To create a scatter plot, plot each pair of data as a point on a coordinate plane. To compare a data set and a function, plot the function on the same coordinate plane as the scatter plot of a data set. The graph of the function should approximate the shape of the scatter plot. Evaluating or solving a function that has a similar shape as a data set can provide an estimate for data not included in the plotted data set. Solve a function algebraically by substituting a value for y and solving for x. U4-99

2 Solve a function graphically by finding the point on the graph of the function with the known y-value, then finding the corresponding x-value of that point. Evaluate a function algebraically by replacing x with a known value and simplifying the expression to determine y. Evaluate a function graphically by finding the point on the graph of the function with the known x-value, then finding the corresponding y-value of that point. Graph a linear function by plotting two points and drawing a line through those two points. Graph an exponential function by plotting at least five points. Connect the points with a curve. Common Errors/Misconceptions confusing when to evaluate and when to solve a function using a linear function to estimate a relationship between two variables when an exponential function is a better fit using an exponential function to estimate a relationship between two variables when a linear function is a better fit confusing x and y when graphing data points or analyzing a graph U4-100

3 Guided Practice Example 1 Andrew wants to estimate his gas mileage, or miles traveled per gallon of gas used. He records the number of gallons of gas he purchased and the total miles he traveled with that gas. Gallons Miles Create a scatter plot showing the relationship between gallons of gas and miles driven. Which function is a better estimate for the function that relates gallons to miles: y = 10x or y = 22x? How is the equation of the function related to his gas mileage? U4-101

4 1. Plot each point on the coordinate plane. Let the x-axis represent gallons and the y-axis represent miles Miles Gallons U4-102

5 2. Graph the function y = 10x on the coordinate plane. It is a linear function, so only two points are needed to draw the line. Evaluate the function at two values of x, such as 0 and 10, and draw a line through these points on the scatter plot. y = 10x y = 10(0) = 0 Substitute 0 for x. y = 10(10) = 100 Substitute 10 for x. Two points on the line are (0, 0) and (10, 100) Miles y = 10x Gallons U4-103

6 3. Graph the function y = 22x on the same coordinate plane. This is also a linear function, so only two points are needed to draw the line. Evaluate the function at two values of x, such as 0 and 10, and draw a line through these points on the scatter plot. y = 22(0) = 0 Substitute 0 for x. y = 22(10) = 220 Substitute 10 for x. Two points on the line are (0, 0) and (0, 220) Miles y = 22x 160 y = 10x Gallons U4-104

7 4. Look at the graph of the data and the functions. Identify which function comes closer to the data values. This function is the better estimate for the data. The graph of the function y = 22x goes through approximately the center of the points in the scatter plot. The function y = 10x is not steep enough to match the data values. The function y = 22x is a better estimate of the data. 5. Interpret the equation in the context of the problem, using the units of the x- and y-axes. For a linear equation in the form y = mx + b, the slope (m) of the equation is the rate of change of the function, or the change in y over the change in x. The y-intercept (b) of the equation is the initial value. change in miles In this example, y is miles and x is gallons. The slope is change in gallons. For the equation y = 22x, the slope of 22 is equal to 22 miles 1gallon. The gas mileage of Andrew s car is the miles driven per gallon of gas used. The gas mileage is equal to the slope of the line that fits the data. Andrew s car has a gas mileage of approximately 22 miles per gallon. U4-105

8 Example 2 The principal at Park High School records the total number of students each year. The table below shows the number of students for each of the last 8 years. Year Number of students Create a scatter plot showing the relationship between the year and the total number of students. Show that the function y = 600(1.05) x is a good estimate for the relationship between the year and the population. Approximately how many students will attend the high school in year 9? 1. Plot each point on the coordinate plane. Let the x-axis represent years and the y-axis represent the number of students Number of students Year U4-106

9 2. Graph y = 600(1.05) x on the coordinate plane. Calculate the value of y for a few different values of x. Start with x = 0. Calculate the value of the function for at least four more x-values that are in the table of data. x y 0 600(1.05) 0 = (1.05) 1 = (1.05) 3 = (1.05) 5 = (1.05) 7 = Plot these points on the same coordinate plane. Connect the points with a curve Number of students Year U4-107

10 3. Compare the graph of the function to the scatter plot of the data. The graph of the function appears to be very close to the points in the scatter plot. The function y = 600(1.05) x is a good estimate of the data. 4. Use the function to estimate the population in year 9. Evaluate the function y = 600(1.05) x for year 9, when x = 9. y = 600(1.05) 9 = The function y = 600(1.05) x is a good estimate of the population. There will be approximately 931 students in the school in year 9. Example 3 The weights of oranges vary. Maria wants to come up with a way to estimate the number of oranges given a weight. She weighs oranges and records the weights in the table below. Number of oranges Weight in pounds Create a scatter plot showing the relationship between the number of oranges and the weight in pounds. Is the function y = 0.6x 0.5 a good fit for the data? Maria has a bag of oranges that weighs 2 pounds. Approximately how many oranges are in the bag? U4-108

11 1. Plot each point on the coordinate plane. Let the x-axis represent the number of oranges and the y-axis represent the weight in pounds Weight in pounds Number of oranges U4-109

12 2. Graph the function y = 0.6x 0.5 to determine if it is a good estimate for the data set. Find two points on the line by evaluating the function at two values of x. Two easy values to use are 0 and 1. y = 0.6(0) 0.5 = 0.5 Substitute 0 for x. y = 0.6(1) 0.5 = 0.1 Substitute 1 for x. Two points on the line are (0, 0.5) and (1, 0.1). Graph the two points on the same graph as the scatter plot, and then draw a line through the two points Weight in pounds Number of oranges U4-110

3. Look at the relationship between the graph of the function and the graph of the data. Determine if the function closely resembles the graph of the data.

13 3. Look at the relationship between the graph of the function and the graph of the data. Determine if the function closely resembles the graph of the data. If a linear function is a good estimate for a data set, some of the data values will be above the line and some will be below the line. It appears that this equation is a good fit for the data. 4. Use the equation to estimate the number of oranges weighing 2 pounds. In the equation y = 0.6x 0.5, x is the number of oranges and y is the weight of the oranges. Solve the equation for y = 2 to estimate the number of oranges that weigh 2 pounds. 2 = 0.6x 0.5 Set y equal to = 0.6x Add 0.5 to both sides of the equation. 4.2 x Divide both sides by 0.6. Maria can use the equation y = 0.6x 0.5 to estimate how many oranges have a given weight. 4 oranges weigh approximately 2 pounds. U4-111

Name Class Date. Using Graphs to Relate Two Quantities

Name Class Date. Using Graphs to Relate Two Quantities 4-1 Reteaching Using Graphs to Relate Two Quantities An important life skill is to be able to a read graph. When looking at a graph, you should check the title, the labels on the axes, and the general