Getting Started 
Misconception/Error The student does not understand the relationship between the parameters of an exponential function and the features of its graph. 
Examples of Student Work at this Level The student:
 Is unable to graph the functions with or without technology.
 Compares the functions algebraically instead of comparing them graphically.
 Describes the functions as growth or decay.

Questions Eliciting Thinking What type of functions are you graphing? What do you know about exponential functions?
Can you show me how you graphed each function?
What is the basic form of an exponential function? What does exponential growth look like? What does exponential decay look like? 
Instructional Implications Provide the student with instruction on graphing exponential functions with technology. Review how to use the GRAPH and TABLE functions on the graphing calculator. Provide the student with additional opportunities to graph exponential functions using technology.
Provide the student with instruction on graphing basic exponential functions by hand by creating a table of values. Guide the student to observe the basic forms of the graphs of exponential growth and decay. Encourage the student to identify functions as exponential growth or exponential decay before choosing input values. Allow the student to check his or her graph using technology.
Be sure the student has a basic understanding of the parameters of an exponential function. Using graphing technology, ask the student to graph the function along with a number of other functions of the form . 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 vertical units. Make explicit the relationship between the sign of c and the direction of the translation.
Next, ask the student to graph a number of functions of the form Â where b > 1. Guide the student to observe that as b increases, the rate of change increases.
Then, ask the student to graph a number of functions of the form Â where 0 < b < 1. Guide the student to observe that as b gets closer to zero, the rate of change increases.
Have the student predict the effect of changing the parameters of an exponential function on its graph and then check the prediction using a graphing calculator or an interactive website such as Hot Math: http://hotmath.com/learning_activities/interactivities/exp_2.swf. 
Making Progress 
Misconception/Error The student cannot provide a complete description of the effects that changing the parameters have on the graphs. 
Examples of Student Work at this Level The student correctly graphs the functions using technology but cannot specifically describe how the value of c and/or the value of b change the graph. The student states:
 c moves the graph up or down.
 b changes the slope of the graph.
 b changes the steepness of the graph.
 b changes the direction of the graph.
 The graph shows exponential decay when Â instead of stating .

Questions Eliciting Thinking Can you explain more specifically how c changes the graph? How does the graph change when c is positive? How does the graph change when c is negative?
How is the yintercept related to c?
Can you explain more specifically how b changes the graph? When will the graph show exponential growth? When will the graph show exponential decay?
You said the graph shows exponential decay when b < 1, but what if b = 0 or b = 1? 
Instructional Implications Be sure the student has a basic understanding of the parameters of an exponential function. Using graphing technology, ask the student to graph the function Â along with a number of other functions of the form . 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 vertical units. Make explicit the relationship between the sign of c and the direction of the translation.
Next, ask the student to graph a number of functions of the form Â where b > 1. Guide the student to observe that as b increases, the rate of change increases.
Next, ask the student to graph a number of functions of the form Â where 0 < b < 1. Guide the student to observe that as b gets closer to zero, the rate of change increases.
Have the student predict the effect of changing the parameters of an exponential function on its graph and then check the prediction using a graphing calculator or an interactive website such as Hot Math: http://hotmath.com/learning_activities/interactivities/exp_2.swf. 
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 first three functions using technology. The student describes the effect of c on the graph of Â in terms of a vertical translation of the graph of . The student may also explain that the yintercept is at (0, c) and the horizontal asymptote shifts up or down since it has the equation y = c.
The student correctly graphs the next three functions using technology. The student explains that the graph ofÂ Â changes depending whether b > 0Â or 0 < b < 1. When b > 0,Â the graph is increasing and represents exponential growth, and as the value of b increases, the rate of change increases. When 0 < b < 1, the graph is decreasing and represents exponential decay. The student may also explain that as b gets closer to 0, the rate of decay increases. 
Questions Eliciting Thinking What would happen to the values of f(x) if Â in the functionÂ ?
Can you predict the effect of a on the graph of the function ? 
Instructional Implications Ask the student to explore exponential functions of the formÂ Â using different values of a and then to describe how the graph is affected by changing the value of a.
Consider using MFAS tasks Writing an Exponential Function From Its Graph (FLE.1.2), Writing an Exponential Function From a Description (FLE.1.2), and Writing an Exponential Function From a Table (FLE.1.2). 