Let’s solve each problem one by one by analyzing the graphs and matching them to the correct exponential function.
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Problem 1
Graph: Decreasing curve (exponential decay). Points marked:
- At x = -2, y ≈ 9
- At x = -1, y = 3
- At x = 0, y = 1
We are looking for a function of the form `f(x) = a^x` that passes through these points.
Check `f(x) = (1/3)^x`:
- f(-2) = (1/3)^(-2) = 3² =
9 ✔
- f(-1) = (1/3)^(-1) = 3¹ =
3 ✔
- f(0) = (1/3)^0 =
1 ✔
Perfect match.
✔ Answer: `f(x) = (1/3)^x`
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Problem 2
Graph: Increasing curve (exponential growth). Points marked:
- At x = 0, y = 1
- At x = 1, y = 2
- At x = 2, y = 4
Check `f(x) = (2)^x`:
- f(0) = 2⁰ =
1 ✔
- f(1) = 2¹ =
2 ✔
- f(2) = 2² =
4 ✔
Perfect match.
Note: The options list `f(x) = (1)^x` twice — that’s likely a typo. But `1^x` is always 1, so it can’t be correct.
✔ Answer: `f(x) = (2)^x`
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Problem 3
Graph: Increasing curve. Points marked:
- At x = 0, y = 1
- At x = 1, y = 2
- At x = 2, y = 4
Same as Problem 2!
Check `f(x) = (2)^x`:
- f(0) = 1
✔
- f(1) = 2
✔
- f(2) = 4
✔
Again, perfect match.
✔ Answer: `f(x) = (2)^x`
*(Note: This graph is identical to Problem 2’s — perhaps a duplicate or intentional repetition.)*
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Problem 4
Graph: Decreasing curve (exponential decay). Points marked:
- At x = -2, y ≈ 25
- At x = -1, y = 5
- At x = 0, y = 1
Try `f(x) = (1/5)^x`:
- f(-2) = (1/5)^(-2) = 5² =
25 ✔
- f(-1) = (1/5)^(-1) = 5¹ =
5 ✔
- f(0) = (1/5)^0 =
1 ✔
Perfect match.
✔ Answer: `f(x) = (1/5)^x`
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##
✔ Final Answers:
1.
`f(x) = (1/3)^x`
2.
`f(x) = (2)^x`
3.
`f(x) = (2)^x`
4.
`f(x) = (1/5)^x`
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## 💡 Explanation Summary:
- Exponential functions have the form `f(x) = a^x`.
- If `a > 1`, the graph increases (growth).
- If `0 < a < 1`, the graph decreases (decay).
- To identify the base `a`, plug in known points from the graph.
- Remember: `a^(-n) = 1/a^n`, so negative exponents give reciprocals — useful for decay functions when x is negative.
All answers are verified by plugging in key points from the graphs.
Parent Tip: Review the logic above to help your child master the concept of exponential function worksheet pdf.
