Tutoring Worksheet for Graphing Functions and Scaling Axes 1. The American life span has been increasing over the past century. Advancements in medical services have been a significant contributor to this increase. Americans have also been armed with better information regarding nutrition, exercise and the effects of smoking. The table below shows life expectancies at birth for Americans in various years. Year of Birth 1975 1977 1980 1982 1985 1987 1993 Life Expectancy (years) 72.6 73.3 73.7 74.5 74.7 74.9 75.5
A. Which is the independent variable? Which is the dependent variable? Label appropriate columns with x and y. B. What is the range of values on the horizontal axis? What is the range of values on the vertical axis? C. What is an appropriate title for your graph? D. Use the graph below to draw in the axes, label them appropriately, and determine the best scaling for each axis. Write in the scaling once you determine it.
E. Graph the points and draw a smooth curve (line of best fit). F. Predict the life expectancy from the trend in the data for someone born in the year 2000.
G. Predict the year someone was born if his/her life expectancy is 78 years.
2. Sketch a graph of the following data, which models the number of Internet users worldwide. Year 1995 1996 1998 1999 2001 Number of Users (millions) 34 54 149 230 457
A. Which is the independent variable? Which is the dependent variable? Label appropriate columns with x and y. B. What is the range of values on the horizontal axis? What is the range of values on the vertical axis? C. What is an appropriate title for your graph? D. Use the graph below to draw in the axes, label them appropriately, and determine the best scaling for each axis. Write in the scaling once you determine it.
E. Is the data best modeled with a straight line? Explain.
F. Using the graph, predict how many users will be using the Internet in 2000.
G. Using the graph, predict when there was 90 million users.
3. A tennis ball is tossed upwards and its height and the elapsed time are recorded. The data from the experiment are given in the table below: Time (in seconds) 0.00 0.02 0.04 0.06 0.10 0.12 0.14 Height (in feet) 1.6097 1.8960 2.0886 2.2363 2.2435 2.1138 1.9734
A. Which is the independent variable? Which is the dependent variable? Label appropriate columns with x and y. B. What is the range of values on the horizontal axis? What is the range of values on the vertical axis? C. What is an appropriate title for your graph? D. Do you notice anything particularly special about the data that is different than the previous two examples? Explain your observations. E. Use the graph below to draw in the axes, label them appropriately, and determine the best scaling for each axis. Write in the scaling once you determine it.
F. Is the data best modeled with a straight line? How do you think the line drawn relates to the data in the table and your observations in (D)? Explain.
G. Looking at the graph, estimate the highest or lowest point. What does this point mean in terms of the variables measured? H. Predict from the graph the height at 0.05 seconds.
I. Predict from the graph the time elapsed for a height of 1.7000 feet.
Answers to Graphing Functions and Scaling Axes Problem 1: A. Independent variable: Year of Birth; Dependent variable: Life Expectancy B. Range on horizontal axis: 18 years; Range on vertical axis: 2.9 years C. Answers may vary – should include variable names. D. Appropriate scaling: vertical: 72, 72.5, 73, 73.5, 74, 74.5, etc… Appropriate scaling: horizontal: 1974, 1976, 1978, etc… (answers could vary but should be uniform and fit all data) E. Line of best fit should be approximately linear. F. Around 76.9 years G. Around 2006 Problem 2: A. Independent variable: Year; dependent variable: Number of Users. B. The range on the horizontal axis: 6; the range on the vertical axis: 423 C. Answers may vary – should include variable names. D. Appropriate scaling for horizontal axis: every year starting at 1994 – show even years every other square Appropriate scaling for vertical axis: every 50, or 100 for every two squares E. The data is exponential – dramatic increase. F. Around 311 users. G. Around 1997.
Problem 3: A. Independent variable: Time; dependent variable: Height B. The range on the horizontal axis is 0.14. The range on the vertical axis is 0.6638. C. Answers may vary – should include variable names. D. The height data increases and then decreases as time increases. E. Appropriate scaling for horizontal axis: use 0.02 every 2 squares. Appropriate scaling for vertical axis: start at 1.6 and increment every 0.1 up to 2.3 F. The data is best modeled by a quadratic funciton. The line shows an increase to a maximum and then a decrease. G. Around 2.24 feet in 0.10 seconds is the maximum. H. Around 2.1 feet. I. Around 0.01 seconds.