Functions and Direct Proportions Packet
Name______
We will be learning the following standards using this packet. On ______, you will have a summative assessment on topics in this packet.
MCC6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
MCC6.RP.3a Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
MCC6.RP.3b Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
VOCABULARY Some of these(*) terms are in your book and others will be filled in as we complete the packet.
*Rate:______
*Ratio table:______
*Proportion:______
*Coefficient:______
*Variable:______
*Constant:______
*Origin:______
*Function:______
*Function table:______
Direct Proportion:______
Constant of proportionality:______
Functions
A function is a relationship of input and output pairs. We can compare the relationship in a function ______using the variables x and y. The input value (x) is also called the ______variable since it is the number we choose, and the output (y) is considered the ______variable since it relies on the value of the chosen input. We can organize the information on a function ______and then use the input/output pairs to create a graph.
EXAMPLE Let’s try an example to help this all make sense!
Penny is buying some gummy bears at $2.00 per pound and a container to put them in for $4.00.
- Write an equation (rule) that relates the number of pounds of gummy bears, x, to the total amount she spends, y.
y = _____x + _____ Note: x and y are the variables, _____ is a coefficient, and _____ is a constant.
- Make a function table to display the x and y values.
X / Y
1
2
3.5
5
0.5
- Use the (x,y) pairs to make a graph of the function. These are also called ordered pairs.
Functions, continued…
PRACTICE: Complete each function table and use the ordered pairs to make a graph:
1.)y = 0.5x2.) y = 3x -5 3.) y = x + 6
1
X / y0
1
4
5
X / y
2
3
4
5
X / y
0
3
2
7
1
HOMEWORK: book p. 499-502 # 1-7, 10, 11-26
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Writing Function Rules
We can also work backward to use ordered pairs to write the function rule. This takes a little bit more thinking and some guess and check! Try to see if you can establish a pattern and use it to write an equation with x and y:
1.2.
Equation ______Equation ______
3.4.
Equation ______Equation ______
5. 6.
Equation ______Equation ______
HOMEWORK: book p. 516-518 # 11-21
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Direct Proportions
(Direct variation)
A direct proportion is a special type of function that shows a direct relationship between x and y with a special ratio we call “k”. We use the format y = kx
Let’s look at a simple example of a direct proportion:
Everyday, Maria walks 2 miles.
We can represent the relationship of days and total miles walked with the equation y=2x.
The number of miles walked depends on the number of days walked. In other words they are directly proportionate. The more days pass, the more miles walked.
y is ______, x is______, and k is______.
Let’s graph the direct proportion:
ADD AXIS LABELS!!
FIVE things must be true for a function to be a direct proportion (direct variation)…
1.
2.
3.
4.
5.
Are the following examples of direct variations? Explain how you know.
10.) 11.) y= 7x-4
X / y1 / 2
3 / 4
8 / 9
20 / 21
12.) 13.) (1,3), (5, 15), (8, 24)
It’s all about that k!!!
BIG CONCEPT SINCE y=kx, what is k??? Use inverse operations to find an expression for k!!!
k =
This is useful in finding the constant of proportionalitywhen given only a graph or points on a direct proportion. The constant of proportionality is thevalue of the ratio of two proportional quantities x and y. Try to find k and then write an equation using information from the following direct proportions:
14.) 15.) x y16.) (14,7) and (8,4)
8 20
4 6
10 25
k=____, so the rule is: ______k=____, so the rule is: ______k=____, so the rule is: ______
17.) ( 2, 10) and (4, 20)18.)
Direct Proportion Practice:
1.)What is the constant of proportionality for each point from the graph on the left(You may use the formula )?______. Use the k towrite an equation for the direct
proportion ______
2.)List 5 ways you know the graph is an example of direct proportion:
3.) The equation y=4x represents how Jaclyn eats 4 doughnuts every day.
What part of the story is represented by y?______
What about the x?______What do we call the coefficient 4 in the equation? ______
4.)Anna knits the same amount of socks every day! On the 4th day she had completed 17 socks. On the 7th day, she had completed 26 socks. Is this an example of direct proportion? Why or why not?
5.)William can type 55 words per minute. Write an equation that represents how many words he would have typed after a given amount of minutes. ______
6.)Do the points (7,16) and (5, 13) vary directly? ______
7.)Jay makes $52.00 for working 8 hours and $130.00 for working 20 hours. Write his pay as a direct proportion. ______How much would he earn for working 5 hours?______
8.)What is the k value for the graph on the right?______Use it to write the
equation for the line______
1