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To analyze the quadratic function \( y = -x^2 + 8x - 10 \), let’s identify the following:

1. **Axis of Symmetry**
2. **Vertex**
3. **y-intercept**
4. **Range**

### Step 1: Axis of Symmetry
For a quadratic function in the form \( y = ax^2 + bx + c \), the axis of symmetry is given by:
\[
x = -\frac{b}{2a}
\]
In this function, \( a = -1 \) and \( b = 8 \). Substituting these values:
\[
x = -\frac{8}{2(-1)} = \frac{8}{2} = 4
\]
So, the axis of symmetry is \( x = 4 \).

### Step 2: Vertex
The vertex lies on the axis of symmetry. To find the y-coordinate of the vertex, substitute \( x = 4 \) into the equation:
\[
y = -(4)^2 + 8 \cdot 4 - 10
\]
\[
y = -16 + 32 - 10
\]
\[
y = 6
\]
Thus, the vertex is at \( (4, 6) \).

### Step 3: y-intercept
The y-intercept is the value of \( y \) when \( x = 0 \):
\[
y = -(0)^2 + 8 \cdot 0 - 10 = -10
\]
So, the y-intercept is \( (0, -10) \).

### Step 4: Range
Since the coefficient of \( x^2 \) (i.e., \( a = -1 \)) is negative, the parabola opens downwards. The maximum value of \( y \) is at the vertex, \( y = 6 \). Therefore, the range of the function is:
\[
(-\infty, 6]
\]

### Summary
- **Axis of Symmetry**: \( x = 4 \)
- **Vertex**: \( (4, 6) \)
- **y-intercept**: \( (0, -10) \)
- **Range**: \( (-\infty, 6] \)

Would you like further details or have any questions?

#### Related Questions
1. How do you determine if a parabola opens upwards or downwards?
2. How can you find the x-intercepts of a quadratic function?
3. What is the domain of any quadratic function?
4. How do you convert a quadratic function to vertex form?
5. How can completing the square help find the vertex?

#### Tip:
Remember, for a quadratic function \( y = ax^2 + bx + c \), if \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.

Identify the axis of symmetry, vertex, y-intercept, and range for the quadratic function y = -x^2 + 8x - 10.

Explore the properties of the quadratic function y = -x^2 + 8x - 10, including finding the axis of symmetry, vertex, y-intercept, and range. Ideal for students in Grades 9-11 studying parabolas and quadratic functions.

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