Getting Started 
Misconception/Error The student does not understand the concept of a zero. 
Examples of Student Work at this Level The student:
 Draws a line on the coordinate plane.
 Does not indicate the zeros and draws a graph similar to the graph of a cubic function.
 Attempts to rewrite the function by multiplying the factors without providing a graph.
 Draws a parabola with point A as the vertex.
 Indicates a zero of two and attempts to draw a graph.

Questions Eliciting Thinking What is the definition of a zero of a polynomial?
How can you find the zeros of a polynomial? 
Instructional Implications Review the definition of a zero (i.e., a value of x that makes y equal to zero) and the process of finding the zeros of a polynomial. Remind the student that finding zeros is equivalent to finding xintercepts. Make clear that the process involves substituting zero for the dependent variable, y, and solving for the independent variable, x. Provide the student with additional linear, quadratic, and simple cubic functions, and ask the student to find the zeros of each. Allow the student to selfcheck by using graphing technology to find the zeros (which will also reinforce the relationship between zeros and xintercepts). 
Moving Forward 
Misconception/Error The student attempts to find the zeros but makes errors. 
Examples of Student Work at this Level The student:
 Finds zeros x = 2 and x = 1 but neglects x = 0 and draws two lines for the graph.
 Writes 2, 1 and interprets it as a point (2, 1) on the graph between points A and B.
 Makes a sign error on one of the zeros.

Questions Eliciting Thinking How many zeros should this function have? How can you tell?
What do the zeros tell you about the graph of the function?
There is an error in your work. Can you find it? 
Instructional Implications Remind the student that each factor is associated with a zero. Guide the student to observe there are three different factors, so there should be three different zeros. Review the relationship between the zeros and the graph of a function.
Allow the student to use graphing technology to graph a variety of cubic functions. Assist the student in observing the basic form of a cubic function as well as the number of relative extrema and the end behavior.
Consider using MFAS task Zeros of a Quadratic (AAPR.2.3). 
Almost There 
Misconception/Error The student does not understand how to use the zeros to graph a cubic function. 
Examples of Student Work at this Level The student correctly finds the zeros of the function. However, the student is unable to use the zeros to sketch the graph of P between points A and B. The student:
 Graphs the zeros on the yaxis.
 Correctly graphs the zeros but sketches a graph that is not cubic in form.

Questions Eliciting Thinking What does the graph of a cubic function look like?
Can you write the zeros as ordered pairs? Where would the zeros be located if you graphed them? 
Instructional Implications Ask the student to write the zeros as ordered pairs and to graph the function clearly showing the location of the zeros. Explain that the zeros coincide with the xintercepts and emphasize that the processes of finding zeros and xintercepts are the same.
Allow the student to use graphing technology to graph a variety of cubic functions. Assist the student in observing the basic form of a cubic function as well as the number of relative extrema and the end behavior.
Consider using 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 determines that the zeros of the function are x = 0, x = 1, and x = 2.
The student uses the zeros to correctly sketch the graph of P between points A and B.

Questions Eliciting Thinking What if a polynomial function had four zeros? What do you think its graph might look like?
How can you determine the number of zeros a polynomial function will have?
How can you determine the number of times the graph of a polynomial function will change direction?
Does the graph of this polynomial have endpoints A and B? Why or why not?
What happens to the graph if the leading coefficient is changed from positive to negative? 
Instructional Implications Consider using MFAS task Zeros of a Quadratic (AAPR.2.3).
Challenge the student to identify the zeros of other cubic functions. Consider using MFAS task Zeros of a Cubic (AAPR.2.3).
Challenge the student to find zeros in context. Consider using MFAS task Jumping Dolphin (ASSE.2.3). 