https://file.aiquickdraw.com/mathbot/file/17285964634930grgtp0q.jpg

Let's work through the problem based on the information from the image. The function given is:

\[
f(x) = -\frac{1}{2}x^2 + 7x - 1
\]

The task asks for:

1. The vertex of the parabola.
2. Whether the parabola has a maximum or minimum value and what that value is.
3. The range of the function.
4. The intervals where the function is increasing and decreasing.

### (a) Find the vertex:
For a quadratic function in the form \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is given by:

\[
x = \frac{-b}{2a}
\]

In the equation \( f(x) = -\frac{1}{2}x^2 + 7x - 1 \), we have \( a = -\frac{1}{2} \) and \( b = 7 \).

Thus, the x-coordinate of the vertex is:

\[
x = \frac{-7}{2(-\frac{1}{2})} = 7
\]

To find the y-coordinate of the vertex, substitute \( x = 7 \) into the function:

\[
f(7) = -\frac{1}{2}(7)^2 + 7(7) - 1
\]
\[
f(7) = -\frac{1}{2}(49) + 49 - 1 = -24.5 + 49 - 1 = 23.5
\]

So, the vertex is \( (7, 23.5) \).

### (b) Maximum or Minimum Value:
Since the coefficient of \( x^2 \) (i.e., \( a = -\frac{1}{2} \)) is negative, the parabola opens downward, meaning it has a **maximum** value. The maximum value occurs at the vertex, which is \( f(7) = 23.5 \).

### (c) Find the Range:
The function has a maximum value of \( 23.5 \) at the vertex. Since the parabola opens downward, the range is:

\[
(-\infty, 23.5]
\]

### (d) Intervals of Increase and Decrease:
The function increases before the vertex and decreases after the vertex. Therefore:

- The function is **increasing** on \( (-\infty, 7] \).
- The function is **decreasing** on \( [7, \infty) \).

### Answers for the problem:
1. **Vertex**: \( (7, 23.5) \)
2. **Maximum value**: \( 23.5 \) (since the parabola opens downward)
3. **Range**: \( (-\infty, 23.5] \)
4. **Intervals**:
   - Increasing on \( (-\infty, 7] \)
   - Decreasing on \( [7, \infty) \)

Would you like further clarification or details? Here are some related questions:

1. How do you determine the axis of symmetry for a parabola?
2. Can you explain why the parabola opens downward for negative \( a \) values?
3. What would change if the parabola opened upward?
4. How can you find the x-intercepts of a parabola?
5. What is the relationship between the discriminant and the number of x-intercepts?

**Tip**: Always check the sign of the coefficient of \( x^2 \) to determine whether a parabola opens upward or downward.

Given the quadratic function f(x) = -(1/2)x^2 + 7x - 1, find the vertex, determine whether the parabola has a maximum or minimum value, find the range, and identify the intervals where the function is increasing and decreasing.

This problem involves finding the vertex of the quadratic function f(x) = -(1/2)x^2 + 7x - 1, determining if it has a maximum value, and finding the range and intervals of increase and decrease. The solution uses the formula x = -b/(2a) to calculate the vertex and explains key concepts of quadratic functions.

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