Transformations and Properties of Exponential Functions

The video tutorial covers exponential functions, focusing on the constant e and its application in continuous compounding. It explains the general form of exponential functions and how transformations affect their graphs. Techniques for graphing, including understanding asymptotes, are discussed. The tutorial also covers domain, range, and finding additional points for graphing.

What is the general form of an exponential function with transformations?

• f(x) = a * sin(bx + c) + d

• f(x) = a * b^(x - h) + k

• f(x) = a * x^2 + bx + c

• f(x) = a * log_b(x - h) + k

In the parent graph of an exponential function, what is the typical y-intercept?

• (1, 0)

• (0, 0)

• (0, 1)

• (1, 1)

How does the parameter 'h' affect the graph of an exponential function?

• Stretches or compresses the graph

• Reflects the graph over the x-axis

• Shifts the graph up or down

• Shifts the graph left or right

What happens to the graph of y = e^x when it is multiplied by 2?

• The graph is compressed vertically

• The graph is stretched vertically

• The graph is shifted to the right

• The graph is shifted upwards

When graphing an exponential function, why is it useful to create an xy table?

• To calculate the slope of the tangent

• To identify the maximum and minimum points

• To find the equation of the asymptote

• To determine key points on the graph

What is the effect of the transformation f(x) = e^(x - 2) + 4 on the graph of y = e^x?

• Shift right 2 units and up 4 units

• Shift left 2 units and down 4 units

• Shift right 4 units and up 2 units

• Shift left 4 units and down 2 units

What is the domain of an exponential function?

• All integers

• All negative numbers

• All positive numbers

• All real numbers

How is the range of a transformed exponential function determined?

• By the vertical shift of the graph

• By the x-intercept of the graph

• By the horizontal shift of the graph

• By the slope of the tangent line

What is an asymptote in the context of exponential functions?

• A line that the graph crosses at infinity

• A point where the graph intersects the y-axis

• A point where the graph intersects the x-axis

• A line that the graph approaches but never touches

If the asymptote of a parent graph is y = 0, what is the new asymptote after a vertical shift of 4 units up?

• y = 8

• y = 0

• y = -4

• y = 4