Grade 9: Functions
March 2013
Grade 9
Functions and Relationships
Goals:
□ Determine input values, output values, and formulae for functions using:
o Flow diagrams
o Tables
o Formulae
□ Find equivalent forms for functions
□ Present and interpret relationships using:
o words
o flow diagrams
o tables
o formulae
o equations
o graphs in the Cartesian plane
Terminology






Function (Relationship)
Rule
Input (Domain)
Output (Range)
𝑥 and 𝑦
Formula
1
Grade 9: Functions
Function Representations
A function (or relationship) assigns input numbers (also called the domain) to output
numbers (also called the range) using an algebraic rule. The following diagram illustrates the
concept:
𝑥
INPUT
RULE
𝑦
OUTPUT
Here’s a basic example of a relationship, this time using integers:
10
−1
𝑥
19
30
𝑦 = 𝑥 + 11
1
12
𝑦
−9
−20
11,5
0,5
In the above example…
What are the input numbers?
What are the output numbers?
What is the rule for the relationship?
If the number 60 was an INPUT, what would the OUTPUT be?
What INPUT would be needed for an OUTPUT of 111?
NOTE: Functions are similar to patterns and are often represented in very similar ways. The
main distinction is that functions do not have the same ordering as patterns do. We won’t
talk about “term 1, 2, 3, etc…” or “consecutive terms.” Instead, we can use any real
numbers for our input values (not just natural numbers).
NOTE: We often use the letter 𝑥 for the INPUT values and the letter 𝑦 for the OUTPUT
values. This will become most important as we write formulas for our functions.
Function Representations – As with the patterns we explored in the last section,
relationships can be represented in a number of ways. Often the challenge in working with
relationships is moving from one representation to another. Here’s a list of representations,
each of which demonstrate the same relationship differently.
1. Flow Diagram:
𝑥
31
10
4,25
0
___
𝑥
13,75
𝑦 = 3𝑥+1
___
−8
2
3𝑥 + 1
𝑦
Grade 9: Functions
2. Table:
Input: (𝑥)
Output: (𝑦)
−2
−5
−1
−2
0
1
4
2,5
10
301
3. Formula:
𝑦 = 3𝑥 + 1
4. Cartesian Plane:
(NOTE: We’ll learn lots about
Cartesian graphs in term 3.)
5. Words:
Take the INPUT, multiply it by 3 and
then add 1 to get the OUTPUT.
Exercise 1
1. Copy the following flow diagrams and then fill in the blanks:
1.1.
10
1
___
−2
___
𝑥
1.2.
𝑥
1.3.
𝑥
10
6
−4
2
___
0
3
−1
10
___
21
3
11
___
___
𝑦 = 2𝑥 + 1
𝑦
5
3
___
1
10
𝑦=
𝑦
2
8
0
___
202
𝑦 = (𝑥 + ___ ) × ____
𝑦
1.4.
𝑥
40
8
24
−2
___
𝑦 = (𝑥 ÷ ___) + ____
11
3
__
0
2
3
𝑦
𝑥
3𝑥 + 1
Grade 9: Functions
2. For each of the following rules, determine what the OUTPUT 𝑦 will be for each of the 5
following inputs 𝑥: {−2, −1, 0, 1, 2}
𝑥 = −2
Rule
2.1
𝑦 = 2𝑥
2.2
𝑦 =𝑥+2
2.3
𝑦 = 5𝑥
2.4
𝑦 = 10 − 𝑥
2.5
𝑦 =𝑥−3
2.6
𝑦 =𝑥÷2
2.7
𝑦 =𝑥+4
2.8
𝑦 = 3𝑥
2.9
𝑦 = 𝑥 −1
2.10
𝑦 = 𝑥2
𝑥 = −1
𝑥=0
𝑥=1
3. Represent each of the following 8 relationships as a flow chart, using 𝑥 =
{−2, −1, 0, 1, 2} as input values.
3.1. 𝑦 = 𝑥 + 1
3.5. 𝑦 = 2 − 𝑥
3.2. 𝑦 = 2𝑥
3.6. 𝑦 = 2𝑥
3.3. 𝑦 = 𝑥 + 7
3.7. 𝑦 = 𝑥 2
3.4. 𝑦 = 10𝑥
3.8. 𝑦 = 𝑥 0
4. Give the correct INPUT value for each of the following OUTPUTS and functions:
4.1. OUTPUT: 𝑦 = 3;
FUNCTION: 𝑦 = 3𝑥;
INPUT: 𝑥 = _____
4.2. OUTPUT: 𝑦 = 12;
FUNCTION: 𝑦 = 𝑥 ÷ 2; INPUT: 𝑥 = _____
4.3. OUTPUT: 𝑦 = 99;
FUNCTION: 𝑦 = 𝑥 + 1; INPUT: 𝑥 = _____
4.4. OUTPUT: 𝑦 = −3; FUNCTION: 𝑦 = 𝑥 − 4; INPUT: 𝑥 = _____
4.5.
4.6.
4.7.
OUTPUT: 𝑦 = 7;
OUTPUT: 𝑦 = 100;
OUTPUT: 𝑦 = 8;
4.8. OUTPUT: 𝑦 = −3;
4.9. OUTPUT: 𝑦 = 1/2;
4.10. OUTPUT: 𝑦 = −3;
FUNCTION: 𝑦 = √𝑥;
FUNCTION: 𝑦 = 𝑥 2 ;
FUNCTION: 𝑦 = 𝑥 3 ;
3
INPUT: 𝑥 = _____
INPUT: 𝑥 = _____
INPUT: 𝑥 = _____
FUNCTION: 𝑦 = √𝑥;
INPUT: 𝑥 = _____
FUNCTION: 𝑦 = 𝑥 ÷ 2; INPUT: 𝑥 = _____
FUNCTION: 𝑦 = 2𝑥;
INPUT: 𝑥 = _____
4
𝑥=2
Grade 9: Integers
Function Formulae
When writing down a formula for a relationship, we often use the letter 𝑥 to represent the
input values and the letter 𝑦 to represent the output values. For example, consider the
following flow chart:
32
10
14
4
𝑦 = 3𝑥 + 2
2
0
−7
−3
3𝑥 + 2
𝑥
𝑦
𝑥
Exercise 2
1. For each of the following relationships, write down a correct formula using 𝑥 and 𝑦. For
each relationship the 𝑥 and 𝑦 values are given for when 𝑥 = {−2, −1, 0, 1, 2, 3}
Rule
𝒙 = −𝟐
𝒙 = −𝟏
𝒙=𝟎
𝒙=𝟏
𝒙=𝟐
𝒙=𝟑
1.1
𝑦=
𝑦 = −3
𝑦 = −2
𝑦 = −1
𝑦=0
𝑦=1
𝑦=2
1.2
𝑦=
𝑦=2
𝑦=1
𝑦=0
𝑦 = −1
𝑦 = −2
𝑦 = −3
1.3
𝑦=
𝑦 = −4
𝑦 = −2
𝑦=0
𝑦=2
𝑦=4
𝑦=6
1.4
𝑦=
𝑦 = −1
𝑦 = −0,5
𝑦=0
𝑦 = 0,5
𝑦=1
𝑦 = 1,5
1.5
𝑦=
𝑦=4
𝑦=1
𝑦=0
𝑦=1
𝑦=4
𝑦=9
1.6
𝑦=
𝑦 = 12
𝑦 = 13
𝑦 = 14
𝑦 = 15
𝑦 = 16
𝑦 = 17
𝑦
2. For each of the following
relationships, write down
a correct formula using 𝑥
and 𝑦.
−20
𝑥
−2
−10
−1
2.1 𝑦 =
0
0
10
1
20
2
0
𝑥
0
−1
1
2.2 𝑦 =
25
−5
100
10
4
2
5
𝑦
Grade 9: Integers
0
𝑥
−1
22
46
2.3 𝑦 =
−8
−5
𝑥
100
4
14
6
10
0
2
2.4 𝑦 =
1
1
7
49
0
0
6
2
𝑦
𝑦
Grade 9: Integers
Graphs
+𝒚
The Cartesian Coordinate System
An interesting, visual way to represent functions is
on Cartesian graphs. A Cartesian graph (a.k.a. 𝑥𝑦axis) represents 𝑥 values on the horizontal axis and 𝑦
values on the vertical axis. Points on an 𝑥𝑦-axis are
often written in the form (𝑥, 𝑦). For example, the
point A = (−2,3) means go 2 units left in the
−𝒙
negative 𝑥 direction and 3 units up in the 𝑦
direction. The point A is shown on the graph to the
right:
𝟑
−𝟐
+𝒙
−𝒚
Exercise 3
1. Identify the points P, Q, R, S, T, U, and V on the graph at
right using (𝑥, 𝑦) coordinates.
2. The following (𝑥, 𝑦) coordinates all belong to the
function 𝑦 = 2 − 𝑥: (−3,5); (−2,4);
(−1,3); (0, −2); (2,0); (3, −1).
2.1. Graph the above coordinates.
2.2. Connect the points you plotted in 2.1. What do
you notice about the result?
2.3. Does the coordinate (1,1) fall along the shape
you drew in 2.2? What does this mean for 𝑥 = 1,
𝑦 = 1 in the function 𝑦 = 2 − 𝑥?
3. The following (𝑥, 𝑦) coordinates all belong to the function 𝑦 = −2𝑥:
(−3,6); (−2,4); (−1,2); (0,0); (1, −2); (2, −4); (3, −6).
3.1. Graph the above coordinates.
3.2. Connect the points you plotted in 3.1. What do you notice about the result?
3.3. Does the coordinate (2,2) fall along the shape you drew in 2.2? What does this
mean for 𝑥 = 2, 𝑦 = 2 in the function 𝑦 = −2𝑥?
4. The following (𝑥, 𝑦) coordinates all belong to the function 𝑦 = 𝑥 2 :
(−3,9); (−2,4); (−1,1); (0,0); (1,1); (2,4); (3,9).
4.1. Graph the above coordinates.
4.2. Connect the points you plotted in 4.1. What do you notice about the result?
7
Grade 9: Integers
Function Problem Solving
Exercise 4
1. Identify three differences and three similarities between a relationship and a pattern.
2. What does a toaster have in common with a mathematical relationship? Name one
other household appliance which is similar to a mathematical relationship and describe
the input, function, and output.
3. What similarities and/or differences are there between human relationships and
mathematical relationships?
4. Create your own relationship and represent it using a formula, a table, and a flow chart.
Use at least 5 input and 5 output values.
5. Inverse Relationships – Four relationships are given below. Draw a flow chart for each of
them. What happens to the relationship if the INPUT and OUTPUT are reversed? What is
the new rule? (Note: This is called an inverse relationship.)
5.1. 𝑦 = 𝑥 + 2
5.2. 𝑦 = 𝑥 − 10
5.3. 𝑦 = 2𝑥
5.4. 𝑦 = 𝑥 ÷ 3
6. Write a formula for the each of the functions represented in the graphs below.
6.1
6.2
8
Grade 9: Integers
7. Domain Problems
Consider the relationship 𝑦 =
7.1.
2𝑥−4
6
.
7.1.1. Complete the table below:
𝒙 = −𝟏
Rule
1.1
𝒙=𝟎
𝒙=𝟏
𝒙=𝟐
𝒙=𝟑
𝒙=𝟒
𝑦 = (2𝑥 − 4) ÷ 6
7.1.2. For which value(s) of 𝑥 is there a “problem” for 𝑦? Why?
7.2. Consider the relationship 𝑦 = √𝑥.
7.2.1. Complete the table below:
𝒙 = −𝟏
Rule
1.1
𝒙=𝟎
𝒙=𝟏
𝒙=𝟒
𝒙=𝟗
𝒙 = 𝟏𝟔
𝑦 = √𝑥
7.2.2. For which value(s) of 𝑥 is there a “problem” for 𝑦? Why?
8. Working Backwards: Given an OUTPUT value and a relationship rule, describe in at least
2 sentences how to work backwards to get the INPUT value. Use the example below to
guide
you:
𝑥
?
+2
÷3
9
6
𝑦
Grade 9: Integers
10