Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 55307
Identifying the Graphs of Functions
Students are given four graphs and asked to identify which represent functions and to justify their choices.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, functions, graphs, domain, range
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_IdentifyingTheGraphsOfFunctions_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with a small group, or with the whole class.
1. The teacher asks the student to complete the problems on the Identifying the Graphs of Functions worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student does not demonstrate an understanding of the definition of a function.
Examples of Student Work at this Level
The student is unable to correctly identify the graphs of functions and provides explanations that are unrelated to the definition of a function. For example, the explanation
might be based upon:
Whether the graph is linear or curved.
page 1 of 4 The idea of “having slope.”
Whether the graph is “constant” or not.
Whether or not “there are ordered pairs.”
Symmetry or continuity.
Passing the vertical line test, but the student does not demonstrate an understanding of this test and how it relates to the definition of a function.
Questions Eliciting Thinking
What is a function?
What is the vertical line test? How can you use the vertical line test to determine whether or not the graph represents a function? Can you show me how you are using
the vertical line test?
Does a graph have to be a straight line to represent a function?
Does a graph have to be continuous to represent a function?
How did you determine that this graph did/did not represent a function?
Instructional Implications
Review the definition of a function emphasizing that each input value can have only one output value. Expose the student to a variety of functions, both linear and
nonlinear, presented algebraically, graphically, and in tables, or given by verbal descriptions. Ask the student to explain why each represents a function. Provide feedback as
needed.
Explain the rationale behind the vertical line test and assist the student in using the test to identify the graphs of functions. Present the student with additional examples
and nonexamples of graphed functions. Expose the student to a variety of graphs to include linear, quadratic, cubic, rational, trigonometric, absolute value, logarithmic,
exponential, step and piecewise. Include both horizontal and vertical lines. Have the student indicate whether or not each graph represents a function and justify his or her
answers.
Moving Forward
Misconception/Error
The student demonstrates a minor misconception by identifying one of the graphs incorrectly as being a function or not a function.
Examples of Student Work at this Level
The student:
Is confused by the graph of the equation x = 3 and indicates that it represents a function.
Does not understand the end behavior of the graph of the cubic function thinking that it becomes vertical and therefore, incorrectly indicates that it does not represent
a function.
Does not understand the difference between open and closed dots and indicates that the fourth graph is not a function.
page 2 of 4 Does not realize that functions are not always continuous and indicates that the fourth graph is not a function.
Additionally, the student may not clearly explain or justify all answers.
Questions Eliciting Thinking
Why did you indicate this graph was/was not a function?
What happens to this graph (pointing to the cubic function) as x gets progressively larger (or smaller)?
Could you tell that the point at (-2, 4) is represented by an open dot (in the fourth option)? Do you know what an open dot signifies?
You said this graph (pointing to the fourth option) is not a function because it has multiple parts. Can you explain this in terms of the definition of a function?
Instructional Implications
Ask the student to list several points from the first graph on the worksheet in a table of values and to determine whether or not this table of values represents a function.
Have the student justify his or her answer.
For the second graph, have the student draw a vertical line on the graph that intersects the graph in more than one point. Then have the student circle the points of
intersection and list those points in a table of values. Ask the student to determine whether or not this table of values represents a function. Have the student justify his or
her answer.
Discuss the graph of the cubic function with the student. Use a graphing calculator to graph a cubic function such as y =
. Then ask the student to use the trace
function to investigate the graph at its extremes. Have the student observe the coordinates of points at the extremes, so it becomes apparent that the graph is not
vertical. Reinforce this by asking the student to consider what will happen when y-coordinates are calculated for two different values of x (e.g., ask the student to
determine if
=
when
).
Discuss the fourth graph with the student and explore why it represents a function. If needed, explain the difference between an open and a closed dot. Expose the
student to additional examples of graphs that contain discontinuities, including some that are functions and some that are not functions. Ask the student to identify the
ones that are functions and justify his or her choices.
Almost There
Misconception/Error
The student accurately identifies all of the graphs that represent functions but is unable to explain or justify his or her choices.
Examples of Student Work at this Level
The student correctly distinguishes between the graphs that represent functions and those that do not. However, when justifying his or her decisions, the student
provides an explanation that is unclear or incomplete.
Questions Eliciting Thinking
How can the vertical line test be used to determine whether or not a graph represents a function?
Why must the vertical line only cross the graph at one point at a time? (Explanation should relate directly to the definition of a function.)
What are the implications of the vertical line test crossing the graph at more than one point?
Instructional Implications
Review the definition of a function emphasizing that for every input value there can be only one output value. Help the student relate the definition of a function to the
vertical line test. Model explaining how the vertical line test can be used to determine whether or not a graph represents a function. Have the student identify points on
the graph that indicate an element in the domain is paired with more than one element in the range. Ask the student how the vertical line test enables one to identify such
points.
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
page 3 of 4 The student correctly determines that the graphs in a) and b) do not represent functions while the graphs in c) and d) represent functions. The student explains that a)
and b) are not functions because either:
The same x-value is paired with more than one y-value; or
The graph fails the vertical line test.
The student explains that c) and d) are functions because either:
Each x-value is paired with exactly one y-value; or
The graph passes the vertical line test.
If the student justifies his or her choices by using the vertical line test, the student is able to explain why this test works. For example, the student says that if a vertical line
intersects a graph in more than one point, then the points of intersection represent ordered pairs with the same x-coordinate but different y-coordinates. Consequently,
the same x-value is paired with more than one y-value.
Questions Eliciting Thinking
Does a horizontal line represent a function?
What do you think happens to the third graph as the values of x get progressively larger (or smaller)?
Instructional Implications
Introduce the student to the idea of one-to-one functions. Encourage the student to go back through the task to identify the graphs of functions that are one-to-one.
Have the student sketch other examples of graphs that represent one-to-one functions. Challenge the student to find a line test that can be used to identify one-to-one
functions.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Identifying the Graphs of Functions worksheet
SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.F-IF.1.1:
Description
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).
page 4 of 4