Common Core Math I: Unit 3 - Functions
Day 3 – Class Notes – Sequences and Writing Function Rules
Definitions:
Sequence-Recursive Function-
Arithmetic Sequence-Explicit Function-
Geometric Sequence-Common Ratio-
I.Identifying the Arithmetic Sequence
Example: Determine if each sequence is Arithmetic. If so, justify….
1)1, 2, 4, 8…….2)½, ¼, 0, -1/4, ……… 3)24, 16, 8, 0…. 4)3, 6, 12, 24…..
Problems:
5) 7,6,5,4… 6) 10,12,15,18… 7) 9,5,-1,-5… 8) -15,-11,-7,-3… 9) -0.3 ,0.2 ,0.7 ,1.2… 10) 2.1, 4.2, 8.4, 17.6…
II.Identifying the Geometric Sequence.
Example: Determine if each sequence is Geometric. If so, state the common ratio.
1) 1, 2, 4, 8… 2) ½, ¼, 0, -1/4, … 3) 4, -8, 16, -32… 4) 0, 5, 10… 5) 1, 5, 25, 125… 6) 60, 72, 86.4…
III.Writing recursive functions from tables and comparing graphs.
A.Write a NOW-NEXT equation for the following data.
x / 0 / 1 / 2 / 3 / 4y / 10 / 15 / 20 / 25 / 30
1)2)
x / 0 / 1 / 2 / 3 / 4y / 5 / 10 / 20 / 40 / 80
NEXT= starting at NEXT= starting at
3)4)
x / 0 / 1 / 2 / 3 / 4y / 3 / 6 / 12 / 24 / 48
x / 0 / 1 / 2 / 3 / 4
y / 5 / 10 / 15 / 20 / 25
NEXT= starting at NEXT= starting at
B.Write the explicit equationfor the data. (Note that the data tables are the same as the ones in part A.)
1)2)
x / 0 / 1 / 2 / 3 / 4y / 5 / 10 / 20 / 40 / 80
x / 0 / 1 / 2 / 3 / 4
y / 10 / 15 / 20 / 25 / 30
3)4)
x / 0 / 1 / 2 / 3 / 4y / 3 / 6 / 12 / 24 / 48
x / 0 / 1 / 2 / 3 / 4
y / 5 / 10 / 15 / 20 / 25
C.Put it all together. (New data tables). For each set of data, (a) Find the NOW-NEXT equation, (b) Sketch a scatterplot and determine what type of data it is (linear or exponential), and (c) find the explicit equation for the data, and (d) solve for the indicated value of y.
5)6)
x / 0 / 1 / 2 / 3 / 4y / 2 / 6 / 18 / 54 / 162
x / 0 / 1 / 2 / 3 / 4
y / 7 / 11 / 15 / 19 / 23
(a) NEXT= starting at (a)NEXT= starting at
(c)(c)
(d) Find y if .(d) Find y if .
IV.Applications
Example:
The table below show the average amount of gas Jack’s truck uses depending on how many miles he drives.
a) Predict the average number of miles driven for 6, 7, or 15 gallons.
b) Write an equation to represent the function between gas and miles driven.
Problems:
1)Imagine folding a square piece of paper in half, then in half again, and then in half again, and so on. The fold makes at each stage divide the original square into a number of smaller regions.
(a) Complete the table showing the number of regions for n folds.
n folds / 1 / 2 / 3 / 4 / 5R regions
(b)What are the independent and dependent variables?
(c)Write a NOW-NEXT equation showing how to calculate the number of smaller regions at any stage of the folding process.
(d)Describe the pattern in the values.
(e)Write an explicit equation for this process (a equation).
(f)Find the number of regions for 8 folds.
2)The Farm Bureau Feed Company tested a new weight-gain program for pigs. The company found that the weight of an average pig increased by 0.7 kg per day after going on the weight-gain program. Grunt is a pig that weights 25 kg and is put on the weight-gain program.
(a) Make a table of Grunt’s weight for each day of the first week of his weight-gain program.
Day Number / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7Weight (kg) / 25
(b) What are the independent and dependent variables?
(c) Write a NOW-NEXT equation showing how to estimate Grunt’s weight on a given day.
(d) Describe the pattern of change in weight as number of days increases.
(e) Write an explicit equation for this process (a equation).
(f) Find Grunt’s weight on Day 15.
(g) A pig must weight 75 kg before it is sold. How long will Grunt be on the weight-gain program before he is sold?
3)According to the US Census Bureau, the population of the United States in July of 2006 was approximately 299,000,000, and growing at a rate of 1.05% per year.
(a) Estimate the U.S. population in July of 2007, 2008, and 2009.
(b) What are the independent and dependent variables?
(c) Write a NOW-NEXT equation that can be used to project the U.S. population any number of years into the
future.
(e) Write an explicit equation for this process (a equation).
(f) What will be population be in 2015?
(g)When will the population first exceed 350,000,000?
V.Homework
For each set of data, (a) Find the NOW-NEXT equation, (b) Sketch a scatter plot and determine what type of data it is (linear or exponential), and (c) find the explicit equation for the data, and (d) solve for the indicated value of y.
1)2)
x / 0 / 1 / 2 / 3 / 4y / 2 / 6 / 10 / 14 / 18
x / 0 / 1 / 2 / 3 / 4
y / 2 / 8 / 32 / 128 / 512
(a) NEXT= starting at (a)NEXT= starting at
(c)(c)
(d) Find y if .(d) Find y if .
3)Your older sister has just been hired to work at an accounting firm in your town. Her beginning pay is $12 per hour. When she was hired, she agreed that for the first five years, she would be given a 5% raise at the end of each year.
(a) Complete the table below showing how hourly pay p in dollars depends on raises received n.
Raises Received n / 0 / 1 / 2 / 3 / 4 / 5Hourly Pay p (in $) / 12 / 12.60
(b)What are the independent and dependent variables?
(c)Write a NOW-NEXT equation showing how to calculate her pay in each year.
(d) Describe the pattern in the values.
(e) Write an explicit equation for this process (a equation).
(f) If the pattern were to continue beyond 5 years, how much would she make after her 12th raise? When will
she be earning at least $20 per hour?
4)The figure below shows growth in a pattern of geometric figures.
(a)Complete the table below to indicate the number of square tiles that would be needed to make each figure.
Figure Number / 0 / 1 / 2 / 3Num. of Square Tiles
(b)What are the independent and dependent variables?
(c)Write a NOW-NEXT equation showing how the number of tiles changes from one figure to the next.
(d)Describe the pattern in the values.
(e) Write an explicit equation for this process (a equation).
(f)Find the number of tiles in Figure 14.