Getting Started 
Misconception/Error The student does not understand the relationship between the parameters of a linear function and its graph. 
Examples of Student Work at this Level The student:
 Describes the graphs of the four functions as parallel, perpendicular and/or intersecting the graph of f(x) = x.
 Is unable to compare the graphs of the four functions to the graph of f(x) = x without graphing each function.
 Incorrectly describes each graph as a horizontal translation of the graph of f(x) = x.
 Compares each function algebraically to the function f(x) = x rather than compares their graphs.

Questions Eliciting Thinking What is slopeintercept form? What does m represent? What does slope mean? What does b represent? What is aÂ yintercept?
What do you know about the graph of g(x) = x + 3? Can you determine its slope? What is its yintercept? How do they compare to the slope and yintercept of f(x) = x? 
Instructional Implications Be sure the student has a basic understanding of the two parameters of a linear function (i.e., rate of change and initial value) and how these parameters are related to the slope and yintercept of the graph. Using a graphing utility, ask the student to graph f(x) = x along with a number of other functions of the form g(x) = x + b. Guide the student to compare the graph of g to the graph of f by describing g as a translation of f a specific number of units horizontally or vertically. Make explicit the relationship between the sign of b and the direction of the translation as well as the magnitude of b and the length of the translation. Next, ask the student to graph a number of functions of the form g(x) = mx. Guide the student to use slopes to compare the steepness of the graphs and to describe the graph of g as a reflection of the graph of fÂ when m < 0.
Have the student predict the effect of changing parameters of a linear function on its graph and then check the predictions using a graphing calculator or an interactive website such as Math Open Reference: Linear Function Explorer (http://www.mathopenref.com/linearexplorermxb.html). 
Making Progress 
Misconception/Error The student cannot provide a complete description of the transformations of the graphs. 
Examples of Student Work at this Level The student:
 Uses terminology that is unclear when comparing the graphs (e.g., the student says the graph would have â€śa positive relationshipâ€ť).
 Describes the graphs as having different yintercepts and/or different slopes but cannot specifically describe the transformations.

Questions Eliciting Thinking What do you mean by a â€śpositive relationshipâ€ť? How does this affect the graph?
Can you describe specifically how the yintercept is different? How would that difference appear in the graph?
Can you describe specifically how the slope is different? How would that difference appear in the graph? 
Instructional Implications Be sure the student has a basic understanding of the two parameters of a linear function (i.e., rate of change and initial value) and how these parameters are related to the slope and yintercept of the graph. Review with the student the terminology of transformations (e.g., translations and reflections). Guide the student to compare graphs of linear functions by describing one as a translation of the other or as a reflection across an axis. Remind the student that the slope is a rate of change, so differences in slope can be described as differences in how fast the graph is rising or in the steepness of the graph.
Give the student descriptions of graphs of linear functions compared to the graph of f(x) = xÂ (e.g., function h is a reflection across the yaxis followed by a vertical translation 4 units up of function fÂ )Â and ask the student to write the equation of function h. 
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 explains that the graph of:
 g(x) = x + 3 is a translation of f(x) = x three units up or three units to the left.
 h(x) = x â€“ 5 is a translation of f(x) = x five units down or five units to the right.
 j(x) = 2x is a reflection of f(x) = x across the yaxis and rises more quickly than f(x) = x, so its graph appears steeper.
 k(x) = 3x + 8 is a translation of f(x) = x eight units up and rises more quickly than f(x) = x, so its graph appears steeper.

Questions Eliciting Thinking Imagine a line that is parallel to f(x) = x but is translated 7 units up. Can you describe its equation?
Imagine a line that is perpendicular to f(x) = x. Can you describe its equation? 
Instructional Implications Ask the student to consider why the graph of g(x) = x + 3 can be described as a translation of f(x) = x three units up or three units to the left while the graph of k(x) = 3x + 8 can only be described as a translation of f(x) = x eight units up.
Give the student descriptions of graphs compared to the graph of a given function [e.g., function h is a vertical translation 5 units down of f(x) = x + 2] and ask the student to write the equation of h. 