Getting Started 
Misconception/Error The student is unable to correctly graph the functions. 
Examples of Student Work at this Level The student incorrectly finds or graphs solutions.
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Questions Eliciting Thinking Did you make a table of values for each function?
What values for x make sense to choose? Why?
If you substitute four for x in the first function, what is the corresponding value of h(x)?
According to your graph, when the x = 3, y = 3. Is (3, 3) a solution of the equation?
What is any root of zero? What happens when you try to find the square root of a negative number?
What happens when you take the cube root of a negative number? 
Instructional Implications Provide instruction on graphing square root and cube root functions. Explain that a good way to graph a function with which one is unfamiliar is to use the equation to make a table of values. Encourage the student to find as many solutions as needed to feel confident that he or she understands the basic shape of the graph. Guide the student to pick integer values of x whose ycoordinates are also integers. However, emphasize that the domain consists of all real numbers (or a subset of them) and that solutions found for purposes of graphing are chosen for convenience. After finding a set of solutions, have the student graph the points and sketch the graph. Discuss with the student why the domain of function h is the set of nonnegative real numbers while the domain of function g is the set of all real numbers. Explain that limitations on the domain of a function have implications for its graph.
Provide additional root functions to graph such as Â or . Encourage the student to consider the domain before making a table of values. Guide the student to observe the general shape of functions involving Â and . Explain to the student that one is less likely to make errors when graphing functions whose general shapes are familiar. 
Moving Forward 
Misconception/Error The studentâ€™s graph contains some errors. 
Examples of Student Work at this Level The student finds solutions of each equation but does not produce a completely correct graph. For example, the student:
 Only graphs points with integer xcoordinates for one or both graphs.
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 Shows the graph as finite or as a line rather than a curve.
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 Graphs only the nonnegative part of the cube root function or extends the graph beyond the domain of the square root function.
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Questions Eliciting Thinking What happens to the function between x = 1 and x = 4? Is it defined for these values of x?
Is this graph really linear? What is g(1)? What happens to the graph when you plot (1, 1)?
Is there any reason why you cannot substitute negative values of x into function h? Into function g? 
Instructional Implications Discuss with the student why the domain of function h is the set of nonnegative real numbers while the domain of function g is the set of all real numbers. Explain that limitations on the domain of a function have implications for its graph. Emphasize that the domain consists of real numbers and that solutions found and used for graphing are chosen for convenience.
Guide the student to correct his or her specific graphing error. Then provide the student with additional opportunities to graph root functions. 
Almost There 
Misconception/Error The student makes errors in describing features of the graph. 
Examples of Student Work at this Level The student correctly graphs each function but makes an error when describing:
 A domain (e.g., lists only integer values in the domain).
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 A maximum (e.g., says the first function has a maximum).
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 The interval over which the function is increasing (e.g., does not understand what it means to describe this interval).
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Questions Eliciting Thinking Can x = 0 for either of these functions? Should zero be in the domain?
What does the minimum or maximum of the function mean? How can you describe a minimum or maximum?
What happens to the function values of the cube root function in the third quadrant? As x increases, what happens to g(x)? 
Instructional Implications Review:
 How to describe the domain of a function. Explain that the domain is generally described as a set or subset of a specific number system (e.g., all real numbers greater than or equal to zero).
 The meaning of maxima and minima and how to describe them.
 What it means for a function to increase or decrease over an interval. Guide the student to describe the interval using appropriate notation.
Provide the student with additional opportunities to graph functions and interpret features of the graphs. Include linear, quadratic, absolute value, and exponential functions and intervals of the domain that contain the absolute maximum or minimum of the function.
Consider implementing MFAS tasks Graphing a Quadratic Function (FIF.3.7), Graphing a Step Function (FIF.3.7), Graphing a Rational Function (FIF.3.7), and Graphing Exponential Functions (FIF.3.7). 
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 correctly graphs the functionÂ Â andÂ states:
 The domain of the function is all real numbers greater than or equal to zero.
 The function does not have a maximum but has a minimum at (0, 0).
 The graph is increasing over the interval , and the graph does not decrease.
The student correctly graphs the functionÂ Â andÂ states:
 The domain of the function is all real numbers.
 As the values of x get larger, the values of g(x) also increase.
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Questions Eliciting Thinking Can you explain why your graph of Â is limited to just the first quadrant?
Can you explain why your graph of Â is in both the first and third quadrants? 
Instructional Implications Provide additional opportunities for the student to graph equations and interpret features of the graphs.
Consider implementing MFAS tasks Graphing a Quadratic Function (FIF.3.7), Graphing a Step Function (FIF.3.7), Graphing a Rational Function (FIF.3.7), and Graphing an Exponential FunctionÂ (FIF.3.7). 