Let’s go step by step to fill in the blanks correctly.
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Top Section: Definitions
1.
asymptote – a _line_ that a graphed function approaches but never _touches or crosses_.
→ In exponential functions, this is usually the x-axis (y = 0).
2.
exponential growth function – an exponential function where b is _greater_ than _1_.
→ Example: f(x) = 3^x or f(x) = 2^x (like shown in the left graph)
3.
exponential decay function – an exponential function where b is _between 0 and 1_ (or “less than 1 and greater than 0”).
→ Example: f(x) = (1/2)^x or f(x) = 0.5^x (like shown in the right graph)
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Bottom Left: Exponential Growth Function — f(x) = 2^x
Looking at the graph:
- It goes
up from left to right → so: _increases_
- Asymptote: The graph gets closer and closer to y = 0 but never touches it → asymptote at _y = 0_
- Domain: All real numbers → (-∞, ∞)
- Range: Only positive y-values → (0, ∞)
- Y-intercept: When x = 0, f(0) = 2^0 = 1 → so y-intercept is _1_
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Bottom Right: Exponential Decay Function — f(x) = (1/2)^x
Looking at the graph:
- It goes
down from left to right → so: _decreases_
- Asymptote: Again, approaches y = 0 → asymptote at _y = 0_
- Domain: All real numbers → (-∞, ∞)
- Range: Only positive y-values → (0, ∞)
- Y-intercept: When x = 0, f(0) = (1/2)^0 = 1 → so y-intercept is _1_
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✔ Double-checking:
- Both have same domain, range, asymptote, and y-intercept — only difference is direction (growth vs decay).
- Base > 1 → growth; base between 0 and 1 → decay. Correct.
- Graphs match descriptions.
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Final Answer:
Top Section:
- asymptote – a _line_ that a graphed function approaches but never _touches_
- exponential growth function – where b is _greater_ than _1_
example: _f(x) = 3^x_ (or any base > 1)
- exponential decay function – where b is _between 0 and 1_
example: _f(x) = (1/3)^x_ (or any base between 0 and 1)
Exponential Growth Function (left):
- _increases_ from left to right
- asymptote at _y = 0_
- domain: _all real numbers_ or _(-∞, ∞)_
- range: _(0, ∞)_
- y-intercept: _1_
Exponential Decay Function (right):
- _decreases_ from left to right
- asymptote at _y = 0_
- domain: _all real numbers_ or _(-∞, ∞)_
- range: _(0, ∞)_
- y-intercept: _1_
Parent Tip: Review the logic above to help your child master the concept of algebra 1 exponential functions worksheet.
