Graphing Exponential Functions With e, Transformations, Domain and Range, Asymptotes, Precalculus

Summary

This video teaches how to graph exponential functions, including those with base 'e.' The process starts by explaining the concept of horizontal asymptotes and how they shift based on constants. Key steps include making tables, selecting appropriate values for x, and plotting corresponding points to visualize the graph. The video explores different examples, explaining domain, range, and shifts in the function. It demonstrates transformations, such as shifting, reflecting, and altering the horizontal asymptote. By the end, viewers will understand how to graph exponential functions with various transformations and accurately determine their domain and range.

Takeaways

• 😀 Exponential functions can be graphed by plotting points and using a table to calculate values for x.
• 😀 Horizontal asymptotes (ASM tootes) are crucial for determining the behavior of the graph, and they represent the line the graph approaches but never crosses.
• 😀 If there is a vertical shift in the function, the horizontal asymptote will shift accordingly (e.g., y = 3 for a shift up).
• 😀 For any exponential function like y = 2^x, the domain is always (-∞, ∞), meaning all real x-values are valid.
• 😀 The range of an exponential function depends on its horizontal asymptote: if there’s no vertical shift, the range is (0, ∞).
• 😀 If a number is added or subtracted from the exponential term (e.g., x + 1 or x - 4), the graph will shift horizontally, and this will affect the x-values you plug into the table.
• 😀 When graphing shifted exponential functions, adjusting your chosen x-values (e.g., plugging in 1 instead of 0) helps accommodate for the horizontal shift.
• 😀 The function y = e^x behaves the same as other exponential functions, but it uses the base e, which is approximately 2.718.
• 😀 Exponential functions like y = 3 - e^(x-2) with a negative sign in front of the exponential term reflect the graph over the horizontal asymptote.
• 😀 The range of a function depends on the location of the horizontal asymptote. For example, with y = 3 - e^(x-2), the range is (-∞, 3) but does not include 3.

Q & A

What is the first step to graphing an exponential function?

The first step is to create a table of values and plug in appropriate points. A common choice is to use x-values such as 0 and 1.

What is the horizontal asymptote of the function y = 2^x?

The horizontal asymptote of y = 2^x is the x-axis, or y = 0, because there is no vertical shift in this function.

How does adding a constant to the exponent, such as in y = 2^(x - 4), affect the graph?

Adding a constant to the exponent, like -4, shifts the horizontal asymptote to y = 4. The graph is translated vertically based on this constant.

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

The range of y = 2^x is from 0 to infinity, excluding 0. The function never reaches 0 but approaches it as x decreases.

How does the horizontal asymptote change when we have a vertical shift in the function?

The horizontal asymptote shifts vertically by the constant added or subtracted from the function. For example, in y = 2^x + 3, the horizontal asymptote would shift to y = 3.

In the function y = 3^x + 1 - 2, how does the transformation affect the graph?

The function y = 3^x + 1 - 2 is shifted down 2 units due to the -2, and the horizontal asymptote is y = -2.

What is the range of the function y = 3^x + 1 - 2?

The range of this function is from -2 to infinity, not including -2, as it starts from the horizontal asymptote and increases.

What happens to the graph of an exponential function when the base is e?

The graph behaves similarly to other exponential functions, but with base e, which is approximately 2.718. The function is just more specific in terms of rate of growth.

How do you graph the function y = e^x - 1?

For y = e^x - 1, the horizontal asymptote is y = -1. You would plot points for x = 0 and x = 1, then sketch the curve based on the asymptote.

What is the effect of the negative sign in front of the exponential function, as seen in y = 3 - e^(x - 2)?

The negative sign reflects the graph over the horizontal asymptote, inverting the direction of the curve. The graph is now below the horizontal asymptote y = 3.