Using the following equation y=-x^2+2x-5 , find the vertex and axis of symmetry.

To find the vertex and axis of symmetry for the quadratic equation \( y = -x^2 + 2x - 5 \), we can use the following steps:

### 1. **Identify the coefficients:**
The equation is in the form of \( y = ax^2 + bx + c \), where:
- \( a = -1 \)
- \( b = 2 \)
- \( c = -5 \)

### 2. **Find the axis of symmetry:**
The axis of symmetry for a parabola given by \( y = ax^2 + bx + c \) can be found using the formula:

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

Substituting the values of \( a \) and \( b \):

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

So, the axis of symmetry is \( x = 1 \).

### 3. **Find the vertex:**
The vertex lies on the axis of symmetry. To find the \( y \)-coordinate of the vertex, substitute \( x = 1 \) into the original equation:

\[
y = -(1)^2 + 2(1) - 5
\]
\[
y = -1 + 2 - 5 = -4
\]

So, the vertex of the parabola is \( (1, -4) \).

### **Summary:**
- **Vertex:** \( (1, -4) \)
- **Axis of Symmetry:** \( x = 1 \)

Would you like more details or have any questions?

### Related Questions:
1. How can you determine whether the vertex is a maximum or minimum point?
2. What does the sign of the coefficient \( a \) indicate about the direction of the parabola?
3. How do you find the roots of the quadratic equation?
4. What is the importance of the axis of symmetry in graphing a parabola?
5. How do you convert a quadratic equation to vertex form?
6. How do you determine the y-intercept of the parabola?
7. Can the vertex be used to find the range of the quadratic function?
8. How would the vertex and axis of symmetry change if the equation was \( y = x^2 - 2x + 5 \)?

**Tip:** The vertex form of a quadratic equation can often make it easier to identify the vertex directly.

Learn how to find the vertex and axis of symmetry of the quadratic equation y = -x^2 + 2x - 5. Understand the steps involved in identifying the vertex and axis, along with practical examples. Suitable for high school students and those studying quadratic equations.

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