Getting Started 
Misconception/Error The student is unable to correctly graph the exponential function. 
Examples of Student Work at this Level The student:
 Graphs the function g(x) = 3x.
 Only graphs the points in the first quadrant.
 Correctly graphs the points in the first quadrant but then reflects the curve over the yaxis to create a parabola.
 Reverses the x and y coordinates in the table.

Questions Eliciting Thinking What kind of function is this? What do the graphs of these equations look like? What type of equation did you graph?
How did you determine your graph? How did you get the points for your graph?
Did you use any negative values for x?
If x = 1Â what is g(x)? Where would this point be on the graph? If x = 2, what is g(x)? Where would this point be on the graph? 
Instructional Implications Review the basic form of an exponential function emphasizing the role of its two parameters: the initial amount and the growth/decay factor. Provide opportunities for the student to explore and investigate examples of exponential functions, both growth and decay. Have the student make a table of values for each example. Then guide the student to use the table of values to graph each function. Assist the student in observing the general form of the graph of an exponential function. Provide instruction on how to differentiate between exponential growth and exponential decay using the equation. Explain the difference between the growth/decay factor and the rate of growth or decay. Explain that the growth factor is (1 + r) where r is the rate of growth, and the decay factor is (1 â€“ r) where r is the rate of decay. Consequently, given , the function represents growth when b > 1Â and decay when 0 < b < 1.
Consider using an interactive website such as Hot Math: http://hotmath.com/learning_activities/interactivities/exp_2.swf to let the student explore how changes in the parameters change the graph of the function.
Provide additional opportunities for the student to graph exponential functions and describe key features of the graphs. 
Making Progress 
Misconception/Error The student cannot correctly describe features of the graph. 
Examples of Student Work at this Level The student correctly graphs the function but makes an error in describing one or more of its features. For example, the student:
 Says the function represents exponential decay.
 Writes there are no intercepts.
 States the intercept is 1.
 Writes that (0, 1) is the xintercept.
 Says that as x gets larger, g(x) goes â€śup by a value of 3â€ť or â€śgoes down.â€ť

Questions Eliciting Thinking What is the difference between exponential growth and exponential decay? What is happening to g(x) as x gets larger and larger? Is it getting smaller or larger?
Show me on your graph where there is an intercept. Is this an x or yintercept? What are the coordinates of this point?
What do you mean by â€śgoing up by 3â€ť? 
Instructional Implications Provide the student with instruction on how to differentiate between exponential growth and exponential decay using both the equation and the graph.
Review with the student how to find intercepts from a graph and algebraically.
Provide feedback to the student with regard to any incorrect descriptions of the behavior of g(x) as x increases. Use the coordinates of three points, such as (1, 3), (2, 9), and (3, 27), to demonstrate that as x increases, g(x) also increases.
Provide additional opportunities for the student to graph exponential functions and describe key features of the graphs. 
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 function and identifies it as exponential growth. The student writes there are no xintercepts and the yintercept is (0, 1). The student answers that as x gets larger and larger, g(x) also gets larger and larger.

Questions Eliciting Thinking How did you know this function represents exponential growth?
Why does the graph of this function not have an xintercept? Is this true for all exponential functions?
What is the domain of this function? What is the range of this function?
Is the rate at which g(x) increases staying the same or changing? How can you tell? 
Instructional Implications Ask the student to determine the average rate of change over several increasing intervals and to describe the differences in the rate of change.
Consider implementing the MFAS task Comparing Functions â€“ Exponential (FBF.2.3) if not previously used. 