Given the following function, (a) find the vertex; (b) determine whether there is a maximum or a minimum value, and find the value; (c) find the range; and (d) find the intervals on which the function is increasing and the intervals on which the function is decreasing. 

f(x)= -1/2x^2+7x-8

Let's solve the problem step by step for the function \( f(x) = -\frac{1}{2}x^2 + 7x - 8 \).

### (a) **Find the vertex**

The vertex of a quadratic function \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[
x_{\text{vertex}} = -\frac{b}{2a}
\]
In this case, the function is \( f(x) = -\frac{1}{2}x^2 + 7x - 8 \), so:
- \( a = -\frac{1}{2} \)
- \( b = 7 \)

Now, calculate \( x_{\text{vertex}} \):
\[
x_{\text{vertex}} = -\frac{7}{2(-\frac{1}{2})} = \frac{7}{1} = 7
\]

To find the y-coordinate of the vertex, substitute \( x = 7 \) into the function:
\[
f(7) = -\frac{1}{2}(7)^2 + 7(7) - 8
\]
\[
f(7) = -\frac{1}{2}(49) + 49 - 8 = -24.5 + 49 - 8 = 16.5
\]

Thus, the vertex is \( (7, 16.5) \).

### (b) **Determine whether there is a maximum or minimum value, and find the value**

Since the coefficient of \( x^2 \) (i.e., \( a = -\frac{1}{2} \)) is negative, the parabola opens downward. This means the vertex represents a **maximum** point.

The maximum value of the function is the y-coordinate of the vertex, which is:
\[
\text{Maximum value} = 16.5
\]

### (c) **Find the range**

Since the parabola opens downward and the maximum value is 16.5, the range of the function is all values of \( y \) less than or equal to 16.5:
\[
\text{Range}: (-\infty, 16.5]
\]

### (d) **Find the intervals on which the function is increasing and decreasing**

For a quadratic function:
- The function is **increasing** to the left of the vertex.
- The function is **decreasing** to the right of the vertex.

Since the vertex occurs at \( x = 7 \), the function is:
- **Increasing** on the interval \( (-\infty, 7] \)
- **Decreasing** on the interval \( [7, \infty) \)

### Summary:
- **(a)** Vertex: \( (7, 16.5) \)
- **(b)** Maximum value: \( 16.5 \)
- **(c)** Range: \( (-\infty, 16.5] \)
- **(d)** Increasing on \( (-\infty, 7] \) and decreasing on \( [7, \infty) \)

Would you like further details or clarifications on any part of this solution?

Here are five related questions to consider:
1. How do you find the vertex of a parabola in a general quadratic function?
2. Why does the sign of the \( a \)-coefficient determine whether the function has a maximum or minimum value?
3. What is the significance of the vertex in a quadratic function?
4. How can you graph a quadratic function from its vertex form?
5. How would the vertex and range change if the \( a \)-coefficient was positive?

**Tip:** The vertex form of a quadratic function \( f(x) = a(x - h)^2 + k \) directly reveals the vertex \( (h, k) \), which can make analyzing the function easier.

Given the following function, (a) find the vertex; (b) determine whether there is a maximum or a minimum value, and find the value; (c) find the range; and (d) find the intervals on which the function is increasing and the intervals on which the function is decreasing. f(x)= -1/2x^2+7x-8

Learn how to solve the quadratic function f(x) = -1/2x^2 + 7x - 8. This detailed solution covers finding the vertex, identifying the maximum value, determining the range, and analyzing the intervals where the function is increasing and decreasing. Suitable for Grades 9-11 students studying algebra and quadratic functions.

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