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Let’s examine each statement given \( g(x) = a(x - h)^2 + k \), where \( a \neq 0 \):

1. **If \( a > 0 \), then \( g \) has a global maximum.**
   - Since \( g(x) \) is a quadratic function, if \( a > 0 \), the parabola opens upward, which means it has a global **minimum** (not a maximum) at \( x = h \).

2. **The graph of \( y = g(x) \) is a parabola.**
   - True. \( g(x) = a(x - h)^2 + k \) is in the form of a quadratic function, so its graph is a parabola.

3. **The graph of \( y = g(x) \) is a line.**
   - False. This is a quadratic function, not linear, so its graph is a parabola, not a line.

4. **The maximum value of \( g(x) \) is \( h \).**
   - False. The maximum or minimum value of \( g(x) \) (depending on the sign of \( a \)) is actually \( k \), not \( h \), since \( g(h) = k \).

5. **An extreme value occurs at the point \( (h, k) \).**
   - True. The vertex of the parabola is at \( (h, k) \), which is where the maximum (if \( a < 0 \)) or minimum (if \( a > 0 \)) value occurs.

6. **If \( a > 0 \), then \( g \) has a global minimum.**
   - True. If \( a > 0 \), the parabola opens upward, so \( g \) has a global minimum at \( x = h \), which is \( g(h) = k \).

### Correct Answers:
- The graph of \( y = g(x) \) is a parabola.
- An extreme value occurs at the point \( (h, k) \).
- If \( a > 0 \), then \( g \) has a global minimum.

Would you like further details on any of these explanations?

### Related Questions:
1. What effect does changing \( h \) have on the position of the vertex of \( g(x) \)?
2. How would the graph change if \( a < 0 \)?
3. How can you find the axis of symmetry for \( g(x) = a(x - h)^2 + k \)?
4. What are the roots of \( g(x) \) if \( g(x) \) intersects the x-axis?
5. How does \( k \) affect the vertical position of the parabola?

**Tip:** For any quadratic function in the form \( a(x - h)^2 + k \), \( (h, k) \) represents the vertex, which determines the highest or lowest point based on the sign of \( a \).

Let a, h, and k be arbitrary real numbers with a ≠ 0, and let g be the function g(x) = a(x - h)^2 + k. Which of the following statements are true about g?

1. If a > 0, then g has a global maximum.
2. The graph of y = g(x) is a parabola.
3. The graph of y = g(x) is a line.
4. The maximum value of g(x) is h.
5. An extreme value occurs at the point (h, k).
6. If a > 0, then g has a global minimum.

Explore which properties are true for the quadratic function g(x) = a(x - h)^2 + k, covering concepts like parabolas, global maxima and minima, and extrema points. This is suitable for high school algebra students.

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