Understanding Transformations and Properties of Exponential Functions

The video tutorial explains how to transform the graph of f(x) = e^x to g(x) = e^(-x) + 3. It involves reflecting the graph across the y-axis and shifting it up by three units. The tutorial also covers how to determine the domain and range of the transformed function, highlighting the horizontal asymptote at y = 3. The domain is all real numbers, while the range is greater than three.

What is the first step in transforming the graph of g(x) from f(x)?

• Reflect the graph across the y-axis

• Reflect the graph across the x-axis

• Shift the graph down three units

• Shift the graph up three units

When reflecting the graph of y = e^x, which axis is used?

• Y-axis

• Z-axis

• X-axis

• No reflection is needed

What happens to the graph when b is changed to -1?

• It reflects across the x-axis

• It reflects across the y-axis

• It shifts down

• It shifts up

How does changing d to 3 affect the graph?

• It shifts the graph left

• It shifts the graph up

• It shifts the graph down

• It shifts the graph right

What is the new function after reflecting and shifting the graph?

• y = e^(-x) - 3

• y = e^(-x) + 3

• y = e^x - 3

• y = e^x + 3

What is the domain of the function y = e^(-x) + 3?

• All real numbers

• x = 0

• x > 0

• x < 0

What is the horizontal asymptote of the function y = e^(-x) + 3?

• y = 0

• y = 1

• y = 3

• y = -3

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

• y > 3

• y < 3

• y = 3

• y ≤ 3

Why is the number 3 not included in the range of y = e^(-x) + 3?

• Because the graph starts at 3

• Because the graph never reaches 3

• Because the graph is below 3

• Because the graph is above 3

In interval notation, how is the range of y = e^(-x) + 3 expressed?

• [3, ∞]

• (3, ∞]

• [3, ∞)

• (3, ∞)