Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 68827
Graphing a Quadratic Function
Students are asked to graph a quadratic function and answer questions about the intercepts, maximum, and minimum.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, quadratic, graphing, maximum, minimum, zeros of a function
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_GraphingaQuadraticFunction_Worksheet.docx
MFAS_GraphingaQuadraticFunction_Worksheet.pdf
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problems on the Graphing a Quadratic Function worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to correctly graph the quadratic function.
Examples of Student Work at this Level
The student:
Graphs
translated three units down.
page 1 of 4 Attempts to calculate the coordinates of the vertex and other points but makes significant errors.
Questions Eliciting Thinking
What kind of equation is this? What is the graph of this kind of equation called?
Can you tell from the equation whether the parabola will open up or down?
What is the vertex of a parabola? How is it found?
How did you find the other points you graphed?
Instructional Implications
Review linear functions and their various forms: ax + by = c (standard form), y = mx + b (slope-intercept form), and
student that the graph of a linear function is a line. Introduce the student to the standard form of a quadratic function,
(point-slope form). Remind the
, and explain that the graph of a
quadratic function is a parabola. Assist the student in understanding how to identify a quadratic function from its equation. Provide the student with equations of a variety
of linear and quadratic functions and ask the student to identify each as either linear or quadratic.
Provide an opportunity for the student to explore the graphs of a variety of linear and quadratic functions by using a graphing calculator or other graphing utility. In each
case, ask the student to identify any x- and y-intercepts. For the graphs of the quadratic functions, ask the student to also describe the orientation of the parabola, identify
its vertex, and state whether the vertex is a maximum or minimum.
Provide instruction on graphing quadratic functions. Guide the student to find the vertex and any intercepts along with any additional points needed to sketch the graph.
Be sure the student understands to select x-coordinates of points on either side of the vertex and then use the equation to find the associated y-coordinates. Remind the
student of the symmetry of the graph. Encourage the student to use symmetry to assist in locating additional points. For example, if the vertex is at (-1, -4) and the yintercept is at (0, -3), then the point symmetric to the y-intercept about the line of symmetry is (-2, -3).
Provide additional opportunities to graph quadratic functions and describe features of their graphs.
Moving Forward
Misconception/Error
The student can graph the quadratic function but cannot interpret features of the graph.
Examples of Student Work at this Level
The student shows no understanding of the zeros of the function or the maximum or minimum of the function.
page 2 of 4 Questions Eliciting Thinking
What are zeros of a function?
What do you think is meant by a minimum or maximum?
Which way will this parabola open? Should it have a maximum or should it have a minimum?
Instructional Implications
Explain that the zeros of a function are the values of the independent variable that make the dependent variable equal to zero. Guide the student to substitute zero for the
dependent variable and then solve the resulting equation. Relate finding zeros to finding the x-intercepts of the graph of the function. Model factoring the function to find
its zeros. Allow the student to explore the connections between the equation and the graph of a quadratic function using a graphing calculator or a website such as Open
Math reference (http://www.mathopenref.com/quadraticexplorer.html). Guide the student to recognize that when the parabola opens upward (or has a positive coefficient
on the variable squared term), it will have a minimum at the vertex and when the parabola opens downward (or has a negative coefficient on the variable squared term), it
will have a maximum at the vertex.
Provide additional opportunities to graph quadratic functions and describe features of their graphs.
Almost There
Misconception/Error
The student makes a minor error interpreting a feature of the graph.
Examples of Student Work at this Level
The student:
Identifies all three intercepts as zeros of the function.
Describes the minimum by giving only its x-coordinate.
Questions Eliciting Thinking
What is meant by the zero of a function?
Can you be more specific about the minimum of this function? Can you describe this point with an ordered pair?
Instructional Implications
Provide feedback to the student regarding any error and allow the student to revise his or her work. Clarify the meaning of zeros of a function and ask the student to
describe maximum and minimum using y-coordinates or with an ordered pair. Provide additional opportunities to graph quadratic functions and describe features of their
graphs.
Consider implementing the MFAS task Zeros of a Quadratic (A-APR.2.3).
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 first finds the vertex of the graph, (-1, -4), by using the formula
to find the x-coordinate and then calculating f(-1) using the equation. The student then
graphs the vertex and finds two or more points to plot, at least one on each side of the vertex. The student may use a table to display these values, then graphs these
points, and sketches the parabola. The student may also find the x-intercepts, the y-intercept, and the point symmetric to the y-intercept to use to sketch the graph. In
addition, the student:
Describes the zeros as (1,0) and (-3,0) and indicates they are located on the x-axis,
States that this function does not have a maximum, and
States that this function does have a minimum, at the vertex (-1,-4).
Questions Eliciting Thinking
How can you tell from the equation whether the parabola will open up or open down?
Are there other points you could have found and used to graph this parabola after finding the vertex?
Is there a way you could have found the x-coordinate of the vertex, other than using the formula
?
What is the equation of the line of symmetry for this graph?
Instructional Implications
Ask the student to graph quadratic functions that have no zeros. Challenge the student to describe, in general, the graphs of quadratic functions that have this quality
(e.g., a parabola that opens upward and whose vertex is in the first or second quadrant).
Consider introducing the student to completing the square and implementing the MFAS tasks Complete the Square - 1 (A-REI.2.4), Complete the Square - 2 (A-REI.2.4),
and Complete the Square - 3 (A-REI.2.4).
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Graphing a Quadratic Function 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.3.7:
Description
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases. ★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value
functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
d. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing
end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude, and using phase shift.
page 4 of 4