MCC9-12
CCGPS
Coordinate Algebra
Unit 3A: Functions
Unit 3A Table of Contents
Functions (MCC9-12.A.IF.1,2,5 and MCC9-12.A.REI.10)
Functions Task……………………………………………………………………………………….p. 1-3
Function Notation Task…………………………………………………………………..…………..p. 5-9
Functions Practice…………………………………………………………………………………...….p. 4
Function Notation Practice…………………………………………………………………………p. 10-11
Characteristics of Functions (MCC9-12.A.IF.4)
Increasing and Decreasing Task………………………………………………………………..…...p. 12-13
Walk the Line Task………………………………………………………………….……………...p. 14-15
Domain and Range Task…………………………………………………………………………....p. 16-17
Increasing/Decreasing and Domain/Range Practice………………………………………………..p. 18-24
Average Rate of Change (MCC9-12.A.IF.6)
Average Rate of Change Task………………………………………………………………………p. 25-27
Average Rate of Change Practice………………………………………………………………..…p. 28-29
Task: Functions?
MCC9-12.F.IF.1 – Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Part 1:
1. A certain business keeps a database of information about its customers.
|Customer Name |Home Phone Number |
|Heather Baker |(310) 510-0091 |
|Mike London |(310) 520-0256 |
|Sue Green |(323) 413-2598 |
|Bruce Swift |(323) 413-2598 |
|Michelle Metz |(213) 806-1124 |
a. Let C be the rule which assigns to each customer shown in the table his or her home phone number. Is C a function? Explain your reasoning.
b. Let P be the rule which assigns to each phone number in the table above, the customer name(s) associated with it. Is P a function? Explain your reasoning.
c. Explain why a business would want to use a person’s social security number as a way to identify a particular customer instead of their phone number.
2. Tell whether each relationship of the form (input, output) represents a function. Explain how you know.
a. (person, height) b. (person, first name)
c. (president, country) d. (age, person)
3. Determine whether or not each of the following is a function or not. Write function or not a function and explain why or why not.
|X |Y |
|1 |1 |
|1 |3 |
|1 |5 |
|2 |3 |
|2 |5 |
|3 |3 |
|3 |5 |
|X |Y |
|1 |1 |
|2 |3 |
|3 |5 |
|X |Y |
|1 |1 |
|1 |2 |
|1 |3 |
|1 |4 |
|2 |1 |
|2 |4 |
|3 |1 |
a. b. c.
d. (5, 6) (7, 8) (9, 10) (9, 11) e. (1, 9) (9, 1) (8, 2) (2, 8)
Part 2:
Determine whether or not each of the following is a function or not. Write function or not a function and explain why or why not.
Relation
1. 2.
. [pic]
3. 4. (student’s name, shirt color)
[pic]
Part 3:
Sometimes when you are given a graph of a function on a coordinate plane, it can be helpful to use a vertical line test to determine if the graph is a function.
Given the following determine whether they are a function or not by using the vertical line test.
1. 2.
3. 4.
5. 6.
Practice - Functions
1. Identify whether or not the relationship represents a function. Justify your answer.
|x |Function? __________ |
|y | |
| |Justify: |
|-2 | |
|10 | |
| | |
|0 | |
|12 | |
| | |
|2 | |
|14 | |
| | |
|4 | |
|16 | |
| | |
|6 | |
|18 | |
| | |
|[pic] |Function? __________ |
| | |
| |Justify: |
| | |
| | |
c. (car tag number, owner) d. (grade point average, student name)
Function? Function?
Justify: Justify:
Task: Function Notation
MCC9-12.F.IF.2 – Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Part 1:
While visiting her grandmother, Fiona found markings on the inside of a closet door showing the heights of her mother, Julia, and Julia’s brothers and sisters on their birthdays growing up. From the markings in the closet, Fiona wrote down her mother’s height each year from ages 2 to 16. Her grandmother found the measurements at birth and one year by looking in her mother’s baby book. The data is provided in the table below, with heights rounded to the nearest inch.
| Age (yrs.) |x |
|At age 2, Julia was 35 inches tall. |Natural language |
|When x is 2, y is 35. |Statement about variables |
|When the input is 2, the output is 35. |Input-output statement |
|h(2) = 35. |Function notation |
As you can see, function notation provides shorthand for talking about relationships between variables. With function notation, it is easy to indicate simultaneously the values of both the independent and dependent variables. The notation h(x) is typically read “h of x,” though it is helpful to think “h at x,” so that h(2) can be interpreted as “height at age 2,” for example.
Note: Function notation looks like a multiplication calculation, but the meaning is very different. To avoid misinterpretation, be sure you know which letters represent functions. For example, if g represents a function, then g(4) is not multiplication but rather the value of “g at 4,” that is, the output value of the function g when the input value is 4.
Answer the next set of questions using this data from Part 1.
|Age (yrs.)|x |0 |
|0 |0 |0 |
|9 |120 |120 |
|20 |168 |213 |
|31 |287 |287 |
1. Which runner has a faster average speed for the first 9 seconds? Explain how you know.
2. Which runner has a faster average speed for 9 to 20 seconds? Explain how this is shown in the table and the graph.
3. What is Runner A’s average speed for 9 to 20 seconds? Find the slope of the secant line from t = 9 to t = 20. Show your calculations.
4. What is Runner B’s average speed for 9 to 20 seconds? Show your calculations.
5. Which runner has a faster average speed for 20 to 31 seconds? Find each runner’s average rate of change from t = 20 to t = 31. Show your calculations.
6. Which runner has a faster average speed for 9 to 31 seconds? Show your calculations.
7. Which runner wins the race? How do you know?
Part 2:
For each of the following, draw the secant line between the two points. Write the coordinates of the two points then calculate the slope of the secant line.
1. f(x) = 2x + 1 2. f(x) = 0.5(2)x + 1 3. f(x) = 3-x+2
Secant line slope: Secant line slope: Secant line slope:
Part 3:
Suppose 25 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles in size every week. The equation [pic] can be used to determine the number of beetles after x weeks.
a. Complete the table below, showing the population of beetles at the end of each week.
|Week |0 |1 |2 |3 |4 |5 |
|Population | | | | | | |
b. Calculate the average growth rate between weeks 1 and 3.
c. Calculate the average growth rate for the first five weeks [0, 5].
d. Which average growth rate is higher? Why do you think it is higher?
Practice – Average Rate of Change
1. Calculate the average rate of change for the given intervals using the graph.
a) From x = -6 to x = 1
b) From x = -7 to x = 6
c) From x = 6 to x = 9
2. Find the rate of change for each interval using the table below.
x |-4 |-3 |-2 |-1 |0 |1 |2 |3 |4 | |f(x) |10 |14 |18 |20 |15 |10 |0 |-30 |-90 | |
a) From x = 0 to x = 2 b) From x = -3 to x = -1 c) From x = -4 to x = 4
3. Mr. Throop’s supply of pencils is quickly dwindling as students continue to borrow a pencil and not return it. His supply began at 400 pencils and he has lost half of his pencils each day.
a) Determine the rate of change in pencils over the first 3 days.
b) Determine the rate of change in pencils between the 3rd and 6th days.
c) Determine the rate of change in pencils between the 2nd and 5th days.
4. On a certain winter day, the low temperature is 30 degrees at 4:00 a.m. The high temperature that afternoon is 56 degrees at 2:00 p.m. Determine the average change in temperature as the temperature changed from lowest to highest.
[pic]
5. Find an interval on the graphed function where there is a negative rate of change. Indicate this interval on the graph by drawing a secant line labeling the two intersection points A and B. Estimate its rate of change below.
6. Find an interval on the graphed function where there is a positive rate of change. Indicate this interval on the graph by drawing a secant line labeling the two intersection points C and D. Estimate its rate of change below.
7. Find an interval on the graphed function where the rate of change is zero. Indicate this interval on the graph by drawing a secant line labeling the two intersection points P and Q.
8. Determine an interval on the function that has a rate of change larger than the rate of change between
x = 0 and x = 7. Explain how you know that your rate of change is larger.
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Note: As you have seen above, functions can be described by tables and by graphs. In high school mathematics, functions are often given by formulas, but it is important to remember that not all functions can be described by formulas.
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The vertical line test states that if a vertical line passes through more than one point on the graph of the relationship between two values, then it is not a function
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