1. You will find the equation of an exponential function through two points in t
ID: 2851229 • Letter: 1
Question
1.You will find the equation of an exponential function through two points in two different ways.
(points)
(5) a. Find the exponential function of the form that goes through the points
(0, 6) and (2, 8). Show all your steps and use four decimal places. Write a complete answer.
(5) b. Use your graphing calculator to enter your exponential function into Y1 and construct an input/output table for inputs of x = -2, -1, 0, 1, 2.
(5) c. Extend the table by entering into Y2. Provide a printout of this extended table. Label the columns x, y, and ln(y).
(5) d. Do the ordered pairs (x, ln(y)) form a linear or exponential function? Justify your answer.
(5) e. Find a possible formula for ln(y) as a function of x. Show your work.
(5) f. Solve your formula for y.
(5) g. Write your answer in the form showing all steps and compare your answer to part a.
(30)2. You have a total of 860 feet of fencing to enclose a large rectangular area and divide it into four smaller pens of the same dimensions. The fencing used to divide the large pen must be parallel to the same side of the large rectangle, as shown below. Your goal is to maximize the total rectangular area. Give the dimensions as well as the maximum area. Show your steps leading up to the final answer.
3. Choose a country listed on the website:
http://esa.un.org/unpd/wpp/unpp/panel_population.htm
You are going to study its population size and model that population. Explore the following website from the United Nations to look at population sizes.
(2) a. Copy the population numbers counted each five years, as shown in the data base, for the years from 1950 to 2000. Add a column, t, measuring years since 1945.
YEAR
t
POPULATION
1950
5
(5) b. What is the country you selected? In what part of the world is it? What is the magnitude of its population numbers? (100,000’s, millions, hundred millions, billions?) Is it growing or shrinking in population size?
(5) c. Enter years since 1945 and your population numbers into your graphing calculator. Create a scatterplot of the data. Is the data growing or shrinking? Does it appear to be a linear pattern or non-linear? Explain your conclusions.
d. Use your calculator to fit a linear model to the population size.
(2) i. Write the equation of the linear regression and superimpose its graph on your scatterplot.
ii. Use TI-Connect to copy the scatterplot onto your write-up. Make sure to include a scale.
(2) iii. How well does the linear model fit your data?
(2) iv. What is the vertical intercept of the regression model? What does it mean in the context
of the population?
(2) v. What is the slope of the regression model? What does it mean in the context of the
population?
(2) vi. Use the model to predict the population size in the year you were born. Also, use the
model to predict the population size in the year 2007.
e. Next fit an exponential model to your population data.
(2) i. Write the equation of the exponential regression and superimpose its graph on your
scatterplot.
(2) ii. How well does the exponential model fit your data? By looking at the graphs, does it
appear that the exponential model fits better than the linear model?
f. Next fit a power function to your population data.
(2) i. Write the equation of the power regression and superimpose its graph on your
scatterplot.
(2) ii. How well does the power model fit your data? By looking at the graphs, which of the
three models seems to fit the best?
(5) g. Find the linear correlation coefficient for each model and compare them to determine
which model fits the data best.
Explanation / Answer
1. You will find the equation of an exponential function through two points in t
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