https://file.aiquickdraw.com/mathbot/file/1729393572230k2l9ffn9.jpg

The problem provided states:

**Given that the maximum value of the function \( y = -x^2 + 6x + k \) is 12, find the value of \( k \).**

### Solution:

1. The function \( y = -x^2 + 6x + k \) is a quadratic function, and since the coefficient of \( x^2 \) is negative, the parabola opens downward, meaning it has a maximum value.
  
2. The formula for the x-coordinate of the vertex of a quadratic function \( y = ax^2 + bx + c \) is given by:
   \[
   x = \frac{-b}{2a}
   \]
   In this case, \( a = -1 \) and \( b = 6 \), so:
   \[
   x = \frac{-6}{2(-1)} = 3
   \]

3. The y-coordinate of the vertex represents the maximum value of the function. Substituting \( x = 3 \) into the equation to find the maximum value of \( y \):
   \[
   y = -(3)^2 + 6(3) + k
   \]
   Simplifying:
   \[
   y = -9 + 18 + k = 9 + k
   \]
   
4. We're told that the maximum value of \( y \) is 12, so:
   \[
   9 + k = 12
   \]
   Solving for \( k \):
   \[
   k = 12 - 9 = 3
   \]

Thus, the value of \( k \) is **3**.

---

Would you like any additional details or clarifications?

### Related Questions:
1. How do you find the vertex of a quadratic function in general?
2. What happens to the graph of the quadratic function when \( k \) changes?
3. How can you determine if a quadratic function has a maximum or minimum value?
4. How would the solution change if the coefficient of \( x^2 \) were positive?
5. How do you find the x-intercepts of the function \( y = -x^2 + 6x + k \)?

### Tip:
For quadratic functions, the maximum or minimum value always occurs at the vertex, which can be easily found using the formula \( x = \frac{-b}{2a} \).

Given that the maximum value of the function y = -x^2 + 6x + k is 12, find the value of k.

In this math problem, we determine the value of k in the quadratic function y = -x^2 + 6x + k when the maximum value of the function is 12. Learn how to use the vertex formula and algebraic techniques to find the solution, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation