Getting Started 
Misconception/Error The student is unable to correctly graph the quadratic function. 
Examples of Student Work at this Level The student:
 Graphs Â translated three units down.
 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 (slopeintercept form), and Â (pointslope form). Remind the student that the graph of a linear function is a line. Introduce the student to the standard form of a quadratic function,Â , 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 yintercepts. 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 xcoordinates of points on either side of the vertex and then use the equation to find the associated ycoordinates. 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 yintercept 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.

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 xintercepts 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 xcoordinate.

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 ycoordinates 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 (AAPR.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 The student first finds the vertex of the graph, (1, 4), by using the formula Â to find the xcoordinate 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 xintercepts, the yintercept, and the point symmetric to the yintercept 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 xaxis,
 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 theÂ xcoordinate of the vertex, other than using the formulaÂ ?
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  1Â (AREI.2.4), Complete the Square  2Â (AREI.2.4), and Complete the Square  3 (AREI.2.4). 