Exponential Growth And Decay Worksheet Outlet | www ... - Free Printable

Let's solve and complete the Exponential Functions Notes worksheet step by step, filling in all the blanks with correct information.

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🔹 Definitions Section

Exponential function – a function in the form:
$$
f(x) = b^x \quad \text{where } b > 0
$$

- Asymptote – a line that a graphed function approaches but never touches or crosses.
*(Answer: "line", "touches or crosses")*

- Exponential growth function – an exponential function where $ b $ is greater than 1.
*Example:* $ f(x) = 2^x $ (already given)

- Exponential decay function – an exponential function where $ b $ is between 0 and 1 (or less than 1 and greater than 0).
*Example:* $ f(x) = \left(\frac{1}{2}\right)^x $ (already given)

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🔹 Exponential Growth Function: $ f(x) = 2^x $

Graph Analysis:

- The graph increases rapidly as $ x $ increases.
- It approaches the x-axis as $ x \to -\infty $, but never touches it.

Fill in the blanks:

- Increases from left to right
- Asymptote at $ y = 0 $
- Domain: All real numbers or $ (-\infty, \infty) $
- Range: $ (0, \infty) $ (since output is always positive)
- y-intercept: Plug in $ x = 0 $: $ f(0) = 2^0 = 1 $ → $ (0, 1) $

✔ So:
- Increases from left to right
- Asymptote at $ y = 0 $
- Domain: $ (-\infty, \infty) $
- Range: $ (0, \infty) $
- y-intercept: $ (0, 1) $

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🔹 Exponential Decay Function: $ f(x) = \left(\frac{1}{2}\right)^x $

Graph Analysis:

- The graph decreases as $ x $ increases.
- It also approaches the x-axis as $ x \to \infty $, but never touches it.

Fill in the blanks:

- Decreases from left to right
- Asymptote at $ y = 0 $
- Domain: All real numbers or $ (-\infty, \infty) $
- Range: $ (0, \infty) $ (still always positive)
- y-intercept: $ f(0) = \left(\frac{1}{2}\right)^0 = 1 $ → $ (0, 1) $

✔ So:
- Decreases from left to right
- Asymptote at $ y = 0 $
- Domain: $ (-\infty, \infty) $
- Range: $ (0, \infty) $
- y-intercept: $ (0, 1) $

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✔ Final Completed Worksheet Answers:

#### Definitions:
- asymptote – a line that a graphed function approaches but never touches or crosses
- exponential growth function – where $ b $ is greater than 1
Example: $ f(x) = 2^x $
- exponential decay function – where $ b $ is between 0 and 1
Example: $ f(x) = \left(\frac{1}{2}\right)^x $

#### Exponential Growth Function ($ f(x) = 2^x $):
- Increases from left to right
- Asymptote at $ y = 0 $
- Domain: $ (-\infty, \infty) $
- Range: $ (0, \infty) $
- y-intercept: $ (0, 1) $

#### Exponential Decay Function ($ f(x) = \left(\frac{1}{2}\right)^x $):
- Decreases from left to right
- Asymptote at $ y = 0 $
- Domain: $ (-\infty, \infty) $
- Range: $ (0, \infty) $
- y-intercept: $ (0, 1) $

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📌 Key Takeaways:

- Both exponential growth and decay functions have:
- The same domain: all real numbers
- The same range: positive real numbers only
- The same y-intercept: $ (0, 1) $ when $ b > 0 $
- Horizontal asymptote at $ y = 0 $
- The base $ b $ determines growth vs. decay:
- $ b > 1 $ → growth
- $ 0 < b < 1 $ → decay

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