Graphing Exponential Functions: Key Concepts and Techniques

The video tutorial focuses on graphing exponential functions, including those with base e. It begins with a simple function y = 2^x, explaining how to graph it using a table of values and identifying its domain and range. The tutorial then provides examples of graphing functions like y = 3^(x+1) - 2 and y = e^x + 1, discussing horizontal asymptotes, shifts, and reflections. Each example includes a step-by-step guide to plotting points and determining the domain and range of the function.

What is the first step in graphing the function y = 2^x?

• Identify the horizontal asymptote

• Create a table of values

• Determine the domain

• Plot the points directly

How does the graph of y = 2^x behave as x approaches negative infinity?

• Approaches zero

• Approaches infinity

• Becomes undefined

• Remains constant

What does a horizontal asymptote at y = 0 indicate about the function?

• The function has no x-intercepts

• The function has a maximum at y = 0

• The function decreases without bound

• The function approaches zero but never touches or crosses the x-axis

What is the range of the function y = 2^x?

• (-Infinity, 0)

• (0, Infinity)

• (-Infinity, Infinity)

• (0, 1)

What is the domain of any exponential function?

• 1 to Infinity

• Negative Infinity to Infinity

• Negative Infinity to 0

• 0 to Infinity

If a function is shifted down by 2 units, what is the new horizontal asymptote?

• y = -2

• y = 0

• y = 2

• y = 1

For the function y = 3^(x+1) - 2, what is the correct transformation description?

• Shifted down by 1 and right by 2

• Shifted up by 2 and right by 1

• Shifted down by 2 and left by 1

• Shifted up by 1 and left by 2

What happens to the graph of an exponential function if the exponent has a negative sign?

• It shifts to the left

• It shifts to the right

• It reflects over the horizontal asymptote

• It becomes linear

What is the effect of a base e in exponential functions compared to other bases?

• Slower decay

• Faster growth

• More rapid oscillations

• No difference in behavior

What is the range of the function y = 3 - e^(x-2)?

• (-Infinity, 3]

• (-Infinity, 3)

• (0, 3]

• (0, 3)