Graph Exponential Function Worksheet - Free Printable

Let’s go step by step to fill in the blanks for both exponential growth and decay functions.

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First, let’s look at Exponential Growth Function:

Given: \( f(x) = 2^x \)

#### Graph behavior:
- As x increases (goes right), y gets bigger → so it’s increasing from left to right.
- The graph never touches the x-axis — it gets closer and closer but never reaches it. That means the asymptote is at y = 0.
- Domain: You can plug in any real number for x → all real numbers, or \( (-\infty, \infty) \)
- Range: Since \( 2^x \) is always positive, y > 0 → (0, ∞)
- Y-intercept: When x = 0, \( f(0) = 2^0 = 1 \) → so y-intercept is 1

✔ So for Exponential Growth:
- increasing from left to right
- asymptote at y = 0
- domain: all real numbers
- range: y > 0
- y-intercept: 1

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Now, Exponential Decay Function:

Given: \( f(x) = \left(\frac{1}{2}\right)^x \)

Note: This is the same as \( 2^{-x} \), which decreases as x increases.

#### Graph behavior:
- As x increases (goes right), y gets smaller → so it’s decreasing from left to right.
- Again, the graph approaches the x-axis but never touches it → asymptote at y = 0
- Domain: Still any real number → all real numbers
- Range: Always positive → y > 0
- Y-intercept: When x = 0, \( f(0) = \left(\frac{1}{2}\right)^0 = 1 \) → y-intercept is 1

✔ So for Exponential Decay:
- decreasing from left to right
- asymptote at y = 0
- domain: all real numbers
- range: y > 0
- y-intercept: 1

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Now let’s also fill in the definitions at the top:

> asymptote – a line that a graphed function approaches but never touches

> 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
> example: \( f(x) = \left(\frac{1}{2}\right)^x \) (already given)

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Final Answer:

Top Section Definitions:

- asymptote – a line that a graphed function approaches but never touches
- exponential growth function – an exponential function where b is greater than 1
- exponential decay function – an exponential function where b is between 0 and 1

Exponential Growth Function Table:

- increasing from left to right
- asymptote at y = 0
- domain: all real numbers
- range: y > 0
- y-intercept: 1

Exponential Decay Function Table:

- decreasing from left to right
- asymptote at y = 0
- domain: all real numbers
- range: y > 0
- y-intercept: 1