Linear Functions
Answer Key
Vocabulary: coordinate plane, coordinates, equation, function, input, linear function,
mapping diagram, ordered pair, output, relation
Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
[Note: The purpose of these questions is to activate prior knowledge and get students thinking.
Students who do not already know the answers will benefit from the class discussion.]
In Musical Chairs six players circle five chairs as music plays. When
the music stops, players rush to sit in a chair. The player left standing
is eliminated. A chair is removed and play continues until only two
players and one chair are left. The winner is the last seated player.
1. How many chairs are needed for 8 people to play? 7
2. In general, if x people want to play, how many chairs are needed? Explain.
There would be (x – 1) chairs needed. You always need to start with one less than the total
number of people playing.
Gizmo Warm-up
In the Linear Functions Gizmo™, you can create relations. A relation is a set of
(input, output) or (x, y) ordered pairs. To make a relation in the Gizmo, either
drag points onto the graph to create (x, y) points, or click-and-drag arrows from
input values to output values in the mapping diagram.
A sample mapping diagram is shown to the right. The relation mapped here
contains the ordered pairs (3, 3), (5, 9), and (6, 5).
1. Click-and-drag an arrow in the Gizmo’s mapping diagram to map any input
value to any output value.
A. What (input, output) ordered pair did you create? Answers will vary.
B. What happened in the graph? A point was graphed matching the (input, output) pair
you mapped.
2. Click Clear. Now, in the Gizmo, drag a point onto the graph.
A. What ordered pair did you create? Answers will vary.
B. What happened in the mapping diagram? An arrow was drawn from the input value
to its associated output value.
Activity A:
Identifying
functions
Get the Gizmo ready:
 Click Clear.
1. In the Gizmo’s mapping diagram, click and drag arrows to map 4 to 3, 5 to 4, and 6 to 6.
A. Select Show linear function test. Describe the analysis of this relation you created.
Does the Gizmo say this is a function?
The Gizmo says that a straight line cannot be drawn through all 3 points, but there is
exactly one output value for each input value. This is a function, but it is not linear.
B. Slowly drag one of the points around the graph. As you do, watch the numbers under
Input in the mapping diagram. You should see a number being circled occasionally.
What seems to cause an Input value to be circled?
An input value is circled if it has 2 or more arrows coming from it.
C. Be sure that there are still at least 3 points on the graph. Again, slowly drag one of
the points around the graph. This time, watch the Gizmo’s analysis under Show
linear function test. When is a relation NOT a function, according to the Gizmo?
A relation is not a function when there are 2 or more points that have the same
x-value.
2. Click Clear. Using the mapping diagram or points on the graph, create
the relation shown to the right: (2, 7), (1, 3), (3, 1), and (2, 1).
A. Why is this relation not a function?
The x-value 2 has two different y-values (both 7 and 1).
Input
2
1
3
2
Output
7
3
1
1
B. What could you change to turn this relation into a function?
You could delete one of the points with an x-value of 2, or you could just change one
of those x-values to any number other than 1, 2, or 3.
3. In general, when is a relation a function and when is it not?
A relation is a function if every input value (x) has only one output value (y). If there is any
input value that has 2 or more outputs, the relation isn’t a function.
Activity B:
Test for functions
Get the Gizmo ready:
 Click Clear.
1. In the Gizmo’s mapping diagram, click and drag arrows to map 2 to 3, 4 to 5, and 6 to 7.
A. Explain why this relation is a function.
Every input (x) value is paired with exactly one output (y) value.
B. Select Show linear function test. Why is this a linear function?
A straight line can be drawn through all 3 points.
C. Drag three more points onto the grid that lie on the same line segment. How are the
(input, output) or (x, y) values related for every point in this linear function?
The output (y) value is always exactly one greater than the input (x) value.
D. Write an equation in terms of input (x) and output (y) values to describe this relation.
Output = input + 1, or y = x + 1.
2. Click Clear. Use the mapping diagram to plot (1, 3), (3, 7), and (4, 9).
A. Why do these points determine a function?
Every input (x) value is paired with exactly one output (y) value.
B. Select Show linear function test to see why this is a linear function. Write an
equation in terms of input (x) and output (y) to describe the relationship.
Output = 2 * input + 1, or y = 2x + 1.
C. Drag one of the points to a different location so that the relation no longer represents
a function. What are the new coordinates of the point? Answers will vary.
D. Explain why this set of points is not a function now.
If the relation is no longer a function, that means that at least one of the x-values is
paired with, or mapped to, more than one y-value.
Activity C:
Horizontal and
vertical lines
Get the Gizmo ready:
 Click Clear.
1. In the Gizmo’s graph, plot a set of points that all have the same input (x) value.
A. What are the coordinates of the points you plotted? Answers will vary.
B. Examine the mapping diagram and tell why these points do not represent a function.
One x-value is paired with or mapped to more than one y-value.
C. Select Show linear function test. Describe the properties of the graph in the grid.
The graph is a vertical line.
D. Click the Table tab. Describe the relationship between the ordered pairs in the table
and the graph of the segment. The points all have the same x-value. When you
graph those points, they all wind up sitting on a vertical line.
E. What equation describes all points on this segment? Answers will vary.
[x = a number.]
2. Click Clear. Plot two points that have the same output (y) value.
A. What are the coordinates of the points you plotted? Answers will vary.
B. Explain why the relation above is or is not a function. This relation is a function
because each x-value is paired with only one y-value. The fact that the y-values of
the two points are the same does not matter.
C. Click the Table tab. What equation describes the ordered pairs in the table and all
points on this segment? Answers will vary. [y = a number.]
3. Use words or equations and describe the difference between points on a horizontal segment
and points on a vertical segment.
Points on a vertical line all have the same x-value, so they do not form a function, and the
equation of this relation is x = a number. Points on a horizontal line all have the same
y-value, so they do form a function, and the equation of this function is y = a number.