The axis of symmetry of the graph of the function y = -x2 + mx +n is x = -2 and the y-intercept= -1 (a) Find the values of m and n (b) the maximum value

To solve this problem, we need to determine the values of \( m \) and \( n \) in the quadratic function \( y = -x^2 + mx + n \), given that the axis of symmetry is \( x = -2 \) and the y-intercept is \( -1 \).

### Step-by-Step Solution

#### (a) Finding \( m \) and \( n \)

1. **Axis of Symmetry:**

   The axis of symmetry for a quadratic function in the form \( y = ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \).

   For the given function \( y = -x^2 + mx + n \), \( a = -1 \) and \( b = m \).

   Since the axis of symmetry is \( x = -2 \):
   \[
   -2 = -\frac{m}{2 \cdot -1} \implies -2 = \frac{m}{2} \implies m = -4
   \]

2. **Y-Intercept:**

   The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) in the function gives the y-intercept:
   \[
   y = -0^2 + m \cdot 0 + n = n
   \]

   Given that the y-intercept is \( -1 \):
   \[
   n = -1
   \]

Therefore, the values of \( m \) and \( n \) are:
\[
m = -4 \quad \text{and} \quad n = -1
\]

#### (b) Finding the Maximum Value

The maximum or minimum value of a quadratic function \( y = ax^2 + bx + c \) is found at the vertex. For \( y = -x^2 - 4x - 1 \), since \( a = -1 \) (which is negative), the parabola opens downwards, indicating a maximum value.

The x-coordinate of the vertex is given by the axis of symmetry, \( x = -2 \).

To find the y-coordinate (the maximum value):
\[
y = -(-2)^2 - 4(-2) - 1 \implies y = -4 + 8 - 1 \implies y = 3
\]

### Summary

(a) The values of \( m \) and \( n \) are:
\[
m = -4 \quad \text{and} \quad n = -1
\]

(b) The maximum value of the function is:
\[
\text{Maximum value} = 3
\]

Would you like more details or have any questions about this solution? Here are five more questions you might be interested in:

1. How do you find the roots of a quadratic function?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you complete the square to rewrite a quadratic function?
4. Can you explain the vertex form of a quadratic function?
5. How do you solve a system of equations involving a quadratic and a linear equation?

**Tip:** When solving quadratic equations, always check if they can be factored easily before applying the quadratic formula for a quicker solution.

Learn how to find the values of m and n in the quadratic function y = -x^2 + mx + n with a detailed step-by-step solution. Discover the axis of symmetry and maximum value calculations. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation