1. BUNDLE: Functions and Relations With Lab Activity
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2. Understand Input and Output Identify Functions and Relations Today’s
Goals:
Graph a Linear
Function
Use the
Vertical Line
Test
Write a function
In Functional
Notation
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3. By definition, a FUNCTION is a rule that establishes a relationship between two quantities. These two quantities are called an input and an output; in a function, for each input, there is exactly one output.
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4. A RELATION is ANY set of ordered pairs.
A relation can be a function, or it cannot be a function. Let’s take a closer look at this…

5. This is an example of a RELATION:
{(1,3), (2,4), (2,5)}
Notice that two of the input values are the number 2. Since a relation is any set of ordered pairs, this is, most assuredly, a relation.
However, it is NOT a function. This is because all the inputs (or x values) must be different. As you can see, in this set of ordered pairs all x values are not different; two are the same.
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6. Let’s look at a few more examples:
Determine whether or not each of the following sets of ordered pairs is or is not a function:
{(-3,3), (0,3), (2,0)}
{(2,0), (0,2), (2,2)}
{(4,5), (-4,4), (2,3)}
{(3,-3), (2,2), (3,3)}
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7. Answers to examples:
Determine whether or not each of the following sets of ordered pairs is a function:
{(-3,3), (0,3), (2,0)} Yes
{(2,0), (0,2), (2,2)} No
{(4,5), (-4,4), (2,3)} Yes
{(3,-3), (2,2), (3,3)} No
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8. There are many ways to represent a Relation:
As a set, using brackets and parentheses
With a table
With an input/output chart
With a graph
With an equation using functional notation
The domain is the set of all input values, or x values; the range is the set of all output values, or y values
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9. A relation can be shown as a set using brackets and parentheses
{(1,2), (0,1), (-1,0)}
Is this a function?
What is the domain?
What is the range?
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10. As a set, using brackets and parentheses:
{(1,2), (0,1), (-1,0)}
Is this a function? yes
What is the domain?
Set of all x values: (1, 0, -1)
What is the range?
Set of all y values: (2,1,0)
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11. As a set, using brackets and parentheses:
{(1,-3), (1,1), (-4,0)}
Is this a function?
What is the domain?
What is the range?
Try
One!
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12. {(1,-3), (1,1), (-4,0)} Set of all x values: (1, 1, -4)
As a set, using brackets and parentheses:
{(1,-3), (1,1), (-4,0)}
Is this a function? No
What is the domain?
Set of all x values: (1, 1, -4)
You really only need to write the duplicating numbers once: (1, -4)
What is the range?
Set of all y values: (-3,1,0)
Check
Your
answers
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13. A relation can be shown using a table:
Input
(x) 2 -2 3
Output
(y) -5
Is this a function?
Name the Domain
Name the Range
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14. A relation can be shown using a table:
Input
(x) 2 -2 3
Output
(y) -5
Is this a function? Yes
Name the Domain (0, 2, -2, 3)
Name the Range (-5, 2, 0)
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15. A relation can be shown using an input/output chart:y x 3 -1 5 -2 4 6
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16. A relation can be shown using an input/output chart:
Yours To
try y x 6 1 4 -2 2 -1 3
Is this a function?
Name the domain.
Name the range.
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17. A relation can be shown using an input/output chart:
Answers y x
Is this a function? Yes
Name the domain.
(6, 1, 4, -2)
Name the range.
(2, -1, 0, 3) 6 1 4 -2 2 -1 3
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18. A relation can also be shown by plotting the ordered pairs on a coordinate plane:
(3,3)
(2,1) x
(-3, 0)
(-1, -1)
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19. Is this a function? Name the domain. Name the range.y
(0,1) x
(-1, 0)
(-1, -1)
(2,-2)
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20. Is this a function? No Name the domain. (0, -1, 2)
Name the range. (1, 0, -1, -2) y
Answers
(0,1) x
(-1, 0)
(-1, -1)
(2,-2)
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21. There is a test you can use to determine whether or not a graph of plotted points represents a function; it is called the VERTICAL LINE TEST.
Here is how it works:
To be a function, no vertical line can intersect the graph of multiple ordered pairs more than once.
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22. No vertical line can intersect the graph of multiple ordered pairs more than once if it is a function.
Now, drawing a vertical line through the graph, does it intersect the graph once, or more than once?
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23. No vertical line can intersect the graph of multiple ordered pairs more than once if it is a function.
More than once!
This is not a function!
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24. Does this represent a function?
No vertical line can intersect the graph of multiple ordered pairs more than once if it is a function.
Does this represent a function?
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25. No vertical line can intersect the graph of multiple ordered pairs more than once.
Does this represent a function? Yes, a vertical line passes through the graph only one time; it passes the vertical line test!
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26. f (x) instead of y in slope-intercept form of an equation.
Functional Notation is a convenient way to define a function. To write a function using functional notation, you simply write
f (x) instead of y in slope-intercept form of an equation.
“f (x)” is read as “f of x”
Here is an equation: y = 2x + 1
In functional notation, it is written:
f (x) = 2x + 1
Their meanings are the same.
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27. Evaluating a function: Example: Evaluate f (x)=x-3 when x is 2
Therefore, when x is 2, y is -1
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28. Try this! Any volunteers?
f (x)= 2x + 3 when x = 4
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29. Check your answer! f (x)= 2x + 3 when x = 4 f (x) = 2(4) + 3
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30. Graphing a Linear Function Graph f (x) = 2x -3
First, re-write it in equation form: y=2x -3
Next, make a table of values (select random values for x), figure out the value of y to make a true statement, then plot the points: X 2 3 -1 y 1 -5
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31. Graphing a Linear Function
Graph f (x) = -2x + 2
Give it
a try!
Hint: select small values for x, like 1, -1, or 0 X y
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32. Use the model to estimate the distance traveled after 28 days.
Monarch butterflies migrate from Northern United States to Mexico, distance of 2000 miles in approximately 40 days.
Write a linear function that models the distance traveled by monarch butterflies. Recall that d=rt
Use the model to estimate the distance traveled after 28 days.
Graph your model and label the point representing the distance traveled after 28 days.
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33. Use the model to estimate the distance traveled after 28 days.
Write a linear function that models the distance traveled by monarch butterflies. Recall that d=rt
r = 2000/40 = 50
D = 50t
Use the model to estimate the distance traveled after 28 days.
D = 50 (28) = 1400 miles
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34. Graph your model and label the point that the butterflies will be after 20 days.
d=50t
2000
1800
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800
600
400
200
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35. Use the model to estimate the distance traveled after 12 days.
Try this!
It’s easy!!
Monarch butterflies migrate from Northern United States to Mexico, distance of 2000 miles in approximately 50 days.
Write a linear function that models the distance traveled by monarch butterflies. Recall that d=rt
Use the model to estimate the distance traveled after 12 days.
Graph your model and label the point representing the distance traveled after 12 days.
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36. Use the model to estimate the distance traveled after 12 days.
Write a linear function that models the distance traveled by monarch butterflies. Recall that d=rt
r = 2000/50 = 40
D = 40t
Use the model to estimate the distance traveled after 12 days.
D = 40 (12) = 480 miles
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37. Graph your model and label the point that the butterflies will be after 12 days.
Volunteer
To graph
This?
d=40t
2000
1800
1600
1400
1200
1000
800
600
400
200
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38. Graph your model and label the point that the butterflies will be after 12 days.
This is how
Your
Graph
Should
Look…
d=40t
2000
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1600
1400
1200
1000
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200
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39. Thank You!!! Kindly Pass the Math
Thank you for your dedication to children…
Thank you for your welcomed encouragement…
Thank you for your trust in my TPT product…
Butterfly clip art by:
jimmiet_Monarch_Butterfly.png
And
monarch_butterfly_clipart_png_
Mathematical designs by,
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40. Math Lab Is it a Function or NOT?
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41. The following activity Functions and Relations.
reinforces
Functions and Relations.
Students may work independently or with a partner, whatever your pleasure!
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42. A pair of die (each a different color)
Materials Needed:
A pair of die (each a different color)
A ruler
Graph paper (included in slides)
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43. Directions:
Roll a pair of die (each die a different color) five times to complete the chart on the next slide. Let one color die represent the x coordinate and the other color die represent the y coordinate. Write the results of each toss as an ordered pair with x on the left and y on the right (x, y). Record each of the ordered pairs thrown, unless you throw the same pair twice; then, roll again.
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44. Die #1 (x) Die #2 (y) As an Ordered Pair (x, y)
Toss #1_________________________________________
Toss
#2__________________________________________
#3__________________________________________
#4__________________________________________
#5_________________________________________
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45. Plot your ordered pairs on the graph provided:x
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46. (this means use brackets { }).
Analysis Questions
Directions: In the spaces provided below, write (or type) the question in one color, and answer it in another color using full and complete sentences.
1. Write the ordered pairs as a set of ordered pairs
(this means use brackets { }).
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47. What is the DOMAIN of your set of ordered pairs
What is the DOMAIN of your set of ordered pairs? Write the DOMAIN as a set of numbers. Again, use brackets { }.
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48. Does the domain show that your set of ordered pairs is a function
Does the domain show that your set of ordered pairs is a function? What is it that indicates whether or not your relation is a function? Explain.
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49. 4. What is the RANGE of your set of ordered pairs
4. What is the RANGE of your set of ordered pairs? Write the RANGE as a set of numbers Again, use brackets
{ }.
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50. For example, if you rolled (2,6), the inverse would be (6,2).
What is the inverse of your relation? Write the inverse as a set of ordered pairs Again, use brackets { }.
For example, if you rolled (2,6), the inverse would be (6,2).
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51. Which, if any, of your ordered pairs is a solution to 2x – y = 3
Which, if any, of your ordered pairs is a solution to 2x – y = 3? Test each ordered pair you rolled in this equation. There are an infinite number of possibilities!
Yes/no
2( ) - ( ) = 3
(x , y)
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52. Ordered Pair: Unique Equation: a) b) c) d) e)
Write a unique linear equation for each of the ordered pairs you rolled.! For example, if you rolled (2,4), you could write 2x + y = 8
or y – x = 2. There are an infinite number of possibilities. First write your ordered pair, and then write your unique equation.
Ordered Pair: Unique Equation: a) b) c) d) e)
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53. 8. In your own words, describe the difference between a relation and a function. Be neat in your work! Please…! No scribbles or lunch residue smudges! In other words, and I repeat, be neat in your work. I am looking for quality work.
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54. The VERTICAL LINE TEST for a function states that if any vertical line passes through no more than one point of the graphed relation, the relation is a function. Does the vertical line test on the next page indicate that your relation is a function? To find out, plot your ordered pairs, connect them from left to right, and then draw a vertical line through the graph. Does the vertical line pass through the graphed ordered pairs in more than one point? If so, your ordered pairs does not indicate that it is a function. Briefly explain the situation with your set of ordered pairs here; is it a function, or is it not?:
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55. Plot your ordered pairs, connect them from left to right, and then apply the
Vertical Line Test: y x
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56. Thank You!!! Kindly Pass the Math
Thank you for your dedication to children…
Thank you for your welcomed support…
Thank you for your trust in my TPT products…
This product has been originally written By
Kindly Pass the Math
Please remember to rate this product and follow my store to receive future products and freebies!
@ all rights reserved by Kindly Pass the Math 2016