## Lesson 44 Scatter Plots And Lines Of Fit

**Essential Question**

How can you use a scatter plot and a line of fit to make conclusions about data?A scatter plot is a graph that shows the relationship between two data sets. The two data sets are graphed as ordered pairs in a coordinate plane.

**EXPLORATION 1Finding a Line of FitWork with a partner. **A survey was taken of 179 married couples. Each person was asked his or her age. The scatter plot shows the results.a. Draw a line that approximates the data. Write an equation of the line. Explain the method you used.b. What conclusions can you make from the equation you wrote? Explain your reasoning.

**EXPLORATION 2Work with a partner.** The scatter plot shows the median ages of American women at their first marriage for selected years from 1960 through 2010.a. Draw a line that approximates the data. Write an equation of the line. Explain the method you used.b. What conclusions can you make from the equation you wrote?c. Use your equation to predict the median age of American women at their first marriage in the year 2020.

**Communicate Your Answer**

Question 3.How can you use a scatter plot and a line of fit to make conclusions about data?Answer: A scatter plot is a graph that shows the relationship between two data sets. The two data sets are graphed as ordered pairs in a coordinate plane.

**4.4 Lesson**

How many calories are in the smoothie that contains 51 grams of sugar?

Question 2.How many grams of sugar are in the smoothie that contains 250 calories?

Question 3.

Question 4.

Answer:

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## Lesson 43 Writing Equations Of Parallel And Perpendicular Lines

**Essential EquationHow can you recognize lines that are parallel or perpendicular?**

**EXPLORATION 1Recognizing Parallel LinesWork with a partner. **Write each linear equation in slope-intercept form. Then use a graphing calculator to graph the three equations in the same square viewing window. Which two lines appear parallel? How can you tell?

**EXPLORATION 2Recognizing Perpendicular LinesWork with a partner. **Write each linear equation in slope-intercept form. Then use a graphing calculator to graph the three equations in the same square viewing window. Which two lines appear perpendicular? How can you tell?

**Communicate Your Answer**

How can you recognize lines that are parallel or perpendicular?

Answer:If the slopes of two lines are equal, then they are parallel lines.If the slope of one line is the negative reciprocal of the second line, then the lines are perpendicular.

Question 4.Compare the slopes of the lines in Exploration 1. How can you use slope to determine whether two lines are parallel? Explain your reasoning.

Answer:Slopes are \, \The slopes are not equal.So, the lines are not parallel.

Question 5.Compare the slopes of the lines in Exploration 2. How can you use slope to determine whether two lines are perpendicular? Explain your reasoning.

Answer:Slopes are \, \the slopes are not negative reciprocals.So, the lines are not perpendicular.

**4.3 Lesson**

Question 1.Line a passes through and . Line b passes through and . Are the lines parallel? Explain.

**Monitoring Progress**

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## Writing Linear Functions Chapter Review

Question 1.Write an equation of the line in slope-intercept form.

Answer:y = \x + 1

Explanation:m = \ = \y + 1 = \y + 1 = \x + 2y = \x + 1

Question 2.Write an equation in point-slope form of the line that passes through the point and has a slope of -1.

Answer:

f = \ + \

Explanation:The points are and The point-slope form of the line is = mm = \ = \y 6 = \y = \ + \ + 6y = \ + \

Question 6.Line a passes through and .Line b passes through and .Line c passes through and .

Answer:Line a: y = \ + 4Line b: y = \ + 1Lines a, b are parallelLine c: y = 2

Explanation:Line a: = \y = \ + 4Line b: = \y = \ + 1Line c: y 0 = \y = 2

Line a: 2x 7y = 14Line b: y = \x 8Line c: 2x + 7y = -21

Answer:Lines b and c are perpendicular

Explanation:Line a: 2x 14 = 7yy = \ 2Line c: 7y = -21 2xy = \ 3

Question 8.Write an equation of the line that passes through and is parallel to the line y = -4x + 2.

Answer:

y = -4x + 9

Question 9.Write an equation of the line that passes through and is perpendicular to the line y = -2x 3

Answer:y = \x 4

Explanation:The slope of the line m = \y + 3 = \y = \x 4

Question 10.What is the roasting time for a 12-pound turkey?

Answer:The roasting time for a 12-pound turkey is 4.0 hours.

Question 11.Write an equation that models the roasting time as a function of the weight of a turkey. Interpret the slope and y-intercept of the line of fit.

Answer: and = \y 4 = \x 2.25y = \x + 1.75The y-intercept is 1.75 and slope is \

Question 23.y = 2 | x + 2 | 3

## Big Ideas Math Book Algebra 1 Solutions For Chapter 4 Writing Linear Functions

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## Piecewise Functions 47 Exercises

**Vocabulary and Core Concept Check**

Question 1.Compare piecewise functions and step functions.Answer:

Question 2.**WRITING**Use a graph to explain why you can write the absolute value function y = | x | as a piecewise function.Answer:

**Monitoring Progress and Modeling with Mathematics****In Exercises 312, evaluate the function.**

Question 3.

Question 35.**MODELING WITH MATHEMATICS**The cost to join an intramural sports league is $180 per team and includes the first five team members. For each additional team member, there is a $30 fee. You plan to have nine people on your team. Write and graph a step function that represents the relationship between the number p of people on your team and the total cost of joining the league.Answer:

Question 36.**MODELING WITH MATHEMATICS**The rates for a parking garage are shown. Write and graph a step function that represents the relationship between the number x of hours a car is parked in the garage and the total cost of parking in the garage for 1 day.Answer:f = {4, if 0< x18, if 1< x212,if 2< x3

**In Exercises 3746, write the absolute value function as a piecewise function.**

Question 37.

y = 7| x + 1 | 5

Answer:g = -7 5, if 7 5 < 07 5, if 7 5 0g = {-7x 13, if x < -2/77x + 2 , if x -2/7

**THOUGHT PROVOKING**Explain whydoes not represent a function. How can you redefine y so that it does represent a function?Answer:

**Maintaining Mathematical Proficiency****Write the sentence as an inequality. Graph the inequality.**

f = x h = 4x + 3

Answer:

## Analyzing Lines Of Fit 45 Exercises

**Vocabulary and Core Concept Check**

Question 1.When is a residual positive? When is it negative?Answer:

Question 2.**WRITING**Explain how you can use residuals to determine how well a line of fit models a data set.

Answer:A residual is a measurement to determine how well a scatter plots data fits its trend line. The actual value is represented by the dot on the scatter plot. The predicted value is given by the trend line or line of best fit. This is they-value that the trend line guesses our data point will appear at.

Question 3.

The points are evenly dispersed about the horizontal axis. So the equation is a good fit.

Question 9.**ANALYZING RESIDUALS**The table shows the growth y of an elks antlers during week x. The equation y = -0.7x + 6.8 models the data. Is the model a good fit? Explain.Answer:

Question 10.**ANALYZING RESIDUALS**The table shows the approximate numbers y of movie tickets sold from January to June for a theater. In the table, x = 1 represents January. The equation y = 1.3x + 27 models the data. Is the model a good fit? Explain.

Answer:The equation y = 1.3x + 27 does not model the data well.

Explanation:

34.8 | 35 34.8 = 0.2 |

Most of the residuals are below the horizontal axis. So, the equation does not model the data well.

**In Exercises 1114, use a graphing calculator to find an equation of the line of best fit for the data. Identify and interpret the correlation coefficient.**

Question 11.

Question 12.

Question 13.

Question 14.

Question 15.

Day | |
---|---|

79 | 90 |

Question 32.

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## Writing Equations In Slope

Question 1.A linear function that models a real-life situation is called a __________.Answer:

Question 2.**WRITING**Explain how you can use slope-intercept form to write an equation of a line given its slope and y-intercept.

Answer:Substitute the given value for the slope and the y-intercept in the slope intercept form to get the equation of the line.

Explanation:Writing the equation of a line with the slope and y-intercepty = mx + bSubstitute the given value for the slope and y-intercept in the above form to get the equation of the line.

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 38, write an equation of the line with the given slope and y-intercept.**

Question 3.

Describe and correct the error in writing an equation of the line shown.Answer:Slope = \ = \y = \x + 4

Question 29.**MODELING WITH MATHEMATICS**In 1960, the world record for the mens mile was 3.91 minutes. In 1980, the record time was 3.81 minutes.a. Write a linear model that represents the world record for the mens mile as a function of the number of years since 1960.b. Use the model to estimate the record time in 2000 and predict the record time in 2020.Answer:

Question 31.**WRITING**A line passes through the points and . Is it possible to write an equation of the line in slope-intercept form? Justify your answer.Answer:

Question 35.**ANALYZING A GRAPH**Line is a reflection in the x-axis of line k. Write an equation that represents line k.Answer:

**Solve the equation. **

Question 38.

**Communicate Your Answer**

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## Writing Equations In Point

**Vocabulary and Core Concept Check**

Question 1.**USING STRUCTURE**Without simplifying, identify the slope of the line given by the equation y 5 = -2. Then identify one point on the line.Answer:

Question 2.**WRITING**Explain how you can use the slope formula to write an equation of the line that passes through and has a slope of 4.

Answer:The equation of a line passes through a point and having slope m is = m = 4y = 4x 14

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 310, write an equation in point-slope form of the line that passes through the given point and has the given slope.**

Question 3.

The equation is y = \x \

Explanation:Rewrite f = 10 as , f = -2 as m = \ = \ = my + 2 = \y = \x \ 2y = \x \

**In Exercises 2730, tell whether the data in the table can be modeled by a linear equation. Explain. If possible, write a linear equation that represents y as a function of x.**

Question 27.

Question 28.

Answer:Because the y values increase at a constant rate, the data can be modeled by a linear equation. A linear model is y = -3x + 7

Explanation:\ = -3, \ = -3\ = -3\ = -3y 16 = -3

Question 30.

Answer:Because the y values are not changing at a constant rate, the data cannot be modeled by a linear equation.

Explanation:\ = -3\ = \\ = \Because the y values are not changing at a constant rate, the data cannot be modeled by a linear equation.

Answer:m = \ = \y 2 = \

Question 35.Describe two ways to graph the equation y 1 = \.Answer:

Answer:f = \x \

Answer:

## Ncert Solutions For Class 9 Maths Chapter 4 Linear Equations In Two Variables

In this chapter, the knowledge of linear equations in one variable is recalled and extended to that of two variables. Any equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables. The Maths NCERT Solutions of Class 9 offers chapter-wise solutions with precise explanations of the exercises provided in the textbook. Students can easily understand the concept of linear equations of Algebra with the help of easy examples provided in these **NCERT Solutions**.

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## Summary Of Ncert Solutions For Class 9 Maths Chapter 4 Linear Equations In Two Variables

Linear Equations In Two Variables is the 4th chapter of the Class 9 Maths **NCERT Textbook**, and it falls under Unit 2 Algebra. For the board exams, from the unit Algebra, you usually get 7 questions that are 1 multiple choice question of 1 mark, 2 short answers with reasoning for a total of 4 marks, 3 short answer questions of a total of 9 marks and 1 long answer question of 6 marks. Thus, the total weightage for the unit is 20 marks. Thus, the total weightage for the unit is 20 marks. Topics covered under this chapter are listed below.

Chapter 4 |

Equations of Lines Parallel to the x-axis and y-axis |

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## Lesson 45 Analyzing Lines Of Fit

**Essential QuestionHow can you analytically find a line of best fit for a scatter plot?**

**EXPLORATION 1Finding a Line of Best FitWork with a partner.**The scatter plot shows the median ages of American women at their first marriage for selected years from 1960 through 2010. In Exploration 2 in Section 4.4, you approximated a line of fit graphically. To find the line of best fit, you can use a computer, spreadsheet, or graphing calculator that has a linear regression feature.a. The data from the scatter plot is shown in the table. Note that 0, 5, 10, and so on represent the numbers of years since 1960. What does the ordered pair represent?b. Use the linear regression feature to find an equation of the line of best fit. You should obtain results such as those shown below.c. Write an equation of the line of best fit. Compare your result with the equation you obtained in Exploration 2 in Section 4.4.

**Communicate Your Answer**

How can you analytically find a line of best fit for a scatter plot?

Question 3.The data set relates the number of chirps per second for striped ground crickets and the outside temperature in degrees Fahrenheit. Make a scatter plot of the data. Then find an equation of the line of best fit. Use your result to estimate the outside temperature when there are 19 chirps per second.Answer:Equation of the best fit line is y = 3.275x + 24.984x = 19 in y = 3.275x + 24.984y = 3.275 + 24.984

761.8 | 760 761.8 = -1.8 |

Answer:

If x = 12, y = -9.8 + 850 = 732.4

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