Introduction

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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. 1 4.2.2: Solving Problems Given Functions Fitted to Data

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Page 1: Introduction

IntroductionData 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.

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Page 2: Introduction

Introduction, continuedGraphing 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.

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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.

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Key Concepts, continued• 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.

• 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.

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Page 5: Introduction

Key Concepts, continued• 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.

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Page 6: Introduction

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

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Page 7: Introduction

Guided PracticeExample 1Andrew 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.

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4.2.2: Solving Problems Given Functions Fitted to Data

Gallons Miles15 31317 34018 40119 42318 39217 37920 40819 43716 36620 416

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Guided Practice: Example 1, continuedCreate 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?

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Page 9: Introduction

Guided Practice: Example 1, continued1. Plot each point on the coordinate plane.

Let the x-axis represent gallons and the y-axis represent miles.

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4.2.2: Solving Problems Given Functions Fitted to Data

Mile

s

Gallons

Page 10: Introduction

Guided Practice: Example 1, continued2. 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 = 10xy = 10(0) = 0 Substitute 0 for x.y = 10(10) = 100 Substitute 10 for x.

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Guided Practice: Example 1, continuedTwo points on the line are (0, 0) and (10, 100).

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4.2.2: Solving Problems Given Functions Fitted to Data

Mile

s

Gallons

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Guided Practice: Example 1, continued3. 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.

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Guided Practice: Example 1, continuedTwo points on the line are (0, 0) and (0, 220).

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Mile

s

Gallons

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Guided Practice: Example 1, continued4. 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.

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Guided Practice: Example 1, continued5. 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.

In this example, y is miles and x is gallons. The

slope is .

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Guided Practice: Example 1, continuedFor the equation y = 22x, the slope of 22 is equal

to .

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.

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4.2.2: Solving Problems Given Functions Fitted to Data

Guided Practice: Example 1, continued

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Page 18: Introduction

Guided PracticeExample 2The principal at Park High School records the total number of students each year. The table to the right shows the number of students for each of the last 8 years.

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4.2.2: Solving Problems Given Functions Fitted to Data

Year Number of students

1 6302 6553 6904 7315 7526 8007 8448 930

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Guided Practice: Example 2, continuedCreate 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?

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Guided Practice: Example 2, continued1. Plot each point on the coordinate plane.

Let the x-axis represent years and the y-axis represent the number of students.

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Num

ber o

f stu

dent

s

Year

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Guided Practice: Example 2, continued2. 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 from the data table.

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x y0 600(1.05)0 = 6001 600(1.05)1 = 6303 600(1.05)3 = 694.5755 600(1.05)5 = 765.7697 600(1.05)7 = 855.260

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Guided Practice: Example 2, continuedPlot these points on the same coordinate plane. Connect the points with a curve.

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Num

ber o

f stu

dent

s

Year

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Guided Practice: Example 2, continued3. 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.

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Guided Practice: Example 2, continued4. 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 = 930.797The 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.

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4.2.2: Solving Problems Given Functions Fitted to Data

Guided Practice: Example 2, continued

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