Getting Started 
Misconception/Error The student does not understand how to use the graphs of f and g to find solutions of the equation f(x) =Â g(x). 
Examples of Student Work at this Level The student graphs functions f and g but is unable to use the graphs to clearly identify the solutions of the equation f(x) = g(x). The student:
 Does not identify any solutions.
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 Identifies one or both points of intersection but does not understand how they relate to solutions of the equation f(x) = g(x).
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 Identifies values that are not solutions.
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Questions Eliciting Thinking What does it mean for f(x) to equal g(x)? Can you write out the equation f(x) = g(x) using the functions in this problem?
If you were trying to solve the system of equations that consist of f and g, where would you find the solutions?
How can you use the coordinates of the points of intersection to find the solutions of the equation f(x) = g(x)? 
Instructional Implications Relate solving the equation f(x) = g(x) to solving a system of equations. Review the graphical representation of the solution of a system of linear equations in two variables. Explain that since the point of intersection of the graphs is a point on each equationâ€™s graph, it represents a solution of each equation in the system. Since it represents a solution of each equation in the system, it is (by definition) a solution of the system. Consequently, the xcoordinates of the points of intersection are solutions of the equation f(x) = g(x).
Using the functions given in this task, model using the graph to find the solutions of the equation f(x) = g(x). Make clear that the solutions of the equation f(x) = g(x) are the xcoordinates of the points of intersection on the graphs. Assist the student in estimating the two solutions.
Next, ask the student to use each function to calculate the functional values of the estimated solutions. For each solution a, guide the student to compare f(a) to g(a) to determine if these values are reasonably close. Emphasize that a is an estimate of a real number, so one would not expect that f(a) is exactly equal to g(a).
Consider implementing MFAS tasks Finding Solutions (AREI.4.10) and What Is the Point? (AREI.4.10). 
Making Progress 
Misconception/Error The student does not understand how to use the functions to check estimates of solutions. 
Examples of Student Work at this Level The student understands that the solutions of the equation f(x) = g(x) can be found at the points where the graphs of f and g intersect. The student graphs the functions and estimates the two solutions as, for example, Â xÂ 3.9 and xÂ 2.8. However, the student does not understand how to use the functions to check these estimates. The student:
 Determines the estimates are incorrect because they do not satisfy the equation f(x) = g(x), orÂ .
 Checks coordinates of points of intersection in only one of function f or g.
 Calculates functional values ofÂ each estimate using one of the functions, and compares these functional values to the corresponding ycoordinates of the points of intersection as indicated by the technology used.
 Says to â€śplug them into the equationâ€ť and is unable to further explain or demonstrate when asked.
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Questions Eliciting Thinking What makes a value of x a solution of the equation f(x) = g(x)?
How can you use the functions to check each estimate? 
Instructional Implications Review what it means for an ordered pair to be a solution of a system of equations. Explain that any solution of the system must satisfy every equation in the system. Guide the student to use each function to calculate functional values of the estimated solutions. For each solution a, ask the student to compare f(a) to g(a) to determine if these values are reasonably close. Emphasize that a is an estimate of a real number, so one would not expect that f(a) is exactly equal to g(a).
Challenge the student to use technology to find more precise estimates that result in closer values of f(a) and g(a).
Consider implementing a similar task for additional practice (e.g., Population and Food SupplyÂ https://www.illustrativemathematics.org/illustrations/645). 
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 uses technology to graph the functions and estimates the two solutions by identifying the xcoordinates of the two points of intersection. For example, the student estimates the solutions as xÂ 3.9 and xÂ 2.8. The student checks these estimates by calculating f(3.9) and g(3.9) and then compares these two values and determines that they are reasonably close. The student repeats this process for f(2.8) and g(2.8). 
Questions Eliciting Thinking Do you think it is possible there are more than two solutions? What would have to be true of the graphs in order for there to be more than two solutions?
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Instructional Implications Challenge the student to use technology to find more precise estimates. Ask the student to evaluate the precision of each estimate a, by comparing f(a) to g(a). 