Worksheet with questions about the Seven Wonders of the Ancient World.

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You've shared a list of questions titled "Worded Questions" related to Graphs and Functions, which appear to be mathematical problems involving functions, graphs, domain, range, transformations, and more.

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For now, I’ll proceed by solving each of the 10 worded questions as written, assuming standard interpretations. Let me know if you'd like me to adjust anything based on a visual component.

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Solutions to the Worded Questions

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#### 1. The equation of a quadratic function can be written in vertex form: y = a(x - h)² + k. What is the vertex?

- The vertex of a quadratic in vertex form $ y = a(x - h)^2 + k $ is at the point $ (h, k) $.
- ✔ Answer: The vertex is $ (h, k) $

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#### 2. Square a distance of 4 feet. Then add 5. What is the result?

- Square of 4 feet: $ 4^2 = 16 $
- Add 5: $ 16 + 5 = 21 $
- ✔ Answer: $ 21 $ (units: square feet, but since it's just a numerical result, we say 21)

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#### 3. In what direction will the parabola open if the coefficient of x² is negative?

- If the coefficient $ a < 0 $ in $ y = ax^2 + bx + c $, the parabola opens downward.
- ✔ Answer: Downward

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#### 4. If a function has a vertical asymptote at x = 2, what does this mean?

- A vertical asymptote at $ x = 2 $ means the function approaches infinity (or negative infinity) as $ x $ approaches 2.
- It indicates the function is undefined at $ x = 2 $, often due to division by zero.
- ✔ Answer: The function approaches infinity or negative infinity as $ x \to 2 $. The function is undefined at $ x = 2 $.

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#### 5. Graph the linear function f(x) = 2x + 3. What is its slope and y-intercept?

- This is in slope-intercept form: $ y = mx + b $
- Slope $ m = 2 $
- Y-intercept $ b = 3 $
- ✔ Answer: Slope = 2, y-intercept = (0, 3)

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#### 6. Describe the transformation of the graph of f(x) = x² when it is shifted 3 units to the right and 2 units up.

- Shift 3 units right: replace $ x $ with $ x - 3 $ → $ f(x) = (x - 3)^2 $
- Shift 2 units up: add 2 → $ f(x) = (x - 3)^2 + 2 $
- ✔ Answer: The new function is $ f(x) = (x - 3)^2 + 2 $. The graph is moved right by 3 and up by 2.

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#### 7. Find the domain and range of the function f(x) = √(x - 4).

- Domain: Expression under square root must be ≥ 0:
$$
x - 4 \geq 0 \Rightarrow x \geq 4
$$
So domain: $ [4, \infty) $

- Range: Square root outputs non-negative values, so $ f(x) \geq 0 $
As $ x \to \infty $, $ f(x) \to \infty $
So range: $ [0, \infty) $

- ✔ Answer:
- Domain: $ [4, \infty) $
- Range: $ [0, \infty) $

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#### 8. What is the inverse of the function f(x) = 3x + 1?

- To find the inverse:
1. Replace $ f(x) $ with $ y $: $ y = 3x + 1 $
2. Swap $ x $ and $ y $: $ x = 3y + 1 $
3. Solve for $ y $:
$$
x - 1 = 3y \Rightarrow y = \frac{x - 1}{3}
$$
- ✔ Answer: $ f^{-1}(x) = \frac{x - 1}{3} $

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#### 9. What is the amplitude of the function f(x) = 3 sin(2x)?

- General form: $ f(x) = A \sin(Bx) $
- Amplitude is $ |A| $
- Here, $ A = 3 $
- ✔ Answer: Amplitude = 3

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#### 10. Given the function f(x) = 2x² - 4x + 1, what is the axis of symmetry?

- For a quadratic $ f(x) = ax^2 + bx + c $, axis of symmetry is:
$$
x = -\frac{b}{2a}
$$
- Here, $ a = 2 $, $ b = -4 $
$$
x = -\frac{-4}{2 \cdot 2} = \frac{4}{4} = 1
$$
- ✔ Answer: Axis of symmetry is $ x = 1 $

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✔ Final Summary of Answers:

| Question | Answer |
|--------|--------|
| 1 | Vertex is $ (h, k) $ |
| 2 | 21 |
| 3 | Opens downward |
| 4 | Function approaches infinity as $ x \to 2 $; undefined at $ x = 2 $ |
| 5 | Slope = 2, y-intercept = (0, 3) |
| 6 | $ f(x) = (x - 3)^2 + 2 $ |
| 7 | Domain: $ [4, \infty) $, Range: $ [0, \infty) $ |
| 8 | $ f^{-1}(x) = \frac{x - 1}{3} $ |
| 9 | Amplitude = 3 |
| 10 | Axis of symmetry: $ x = 1 $ |

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