Forgetting the order of transformations. Remember to apply horizontal shifts and stretches/compressions before vertical shifts and stretches/compressions. This affects the position of the asymptote and key points on the graph.
The horizontal asymptote is usually y = 0 for the basic exponential function. Transformations, especially vertical shifts, will change the position of the horizontal asymptote. Consider the limit of the function as x approaches positive or negative infinity.
Misinterpreting the decay factor. If the function is of the form y = A(1 - r)^x, then r represents the rate of decay, and (1-r) is the decay factor. Ensure that you correctly identify and interpret ‘r’ in the context of the problem.
Choose a range of x-values, including negative values, zero, and positive values. Calculate the corresponding y-values using the exponential function. Plot these points carefully on the graph, paying attention to the scale.
Use graphing software or online tools to visualize the graphs. Experiment with different parameters and transformations to see how they affect the shape and position of the graph. This can help build intuition and understanding.
The base determines whether the function represents exponential growth (base > 1) or decay (0 < base < 1). A base of 1 results in a constant function, not an exponential one. Understanding the base helps in predicting the graphs behavior.