To sketch the graph of the quadratic function \( y = -x^2 + 6x - 8 \), let's follow these steps:

### 1. Identify the form of the quadratic function:
The equation is in the standard form \( y = ax^2 + bx + c \), where:
- \( a = -1 \) (coefficient of \( x^2 \))
- \( b = 6 \) (coefficient of \( x \))
- \( c = -8 \) (constant term)

### 2. Determine the vertex:
The vertex of 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 \( b \):
\[
x = \frac{-6}{2(-1)} = \frac{-6}{-2} = 3
\]
To find the \( y \)-coordinate of the vertex, substitute \( x = 3 \) into the equation:
\[
y = -(3)^2 + 6(3) - 8 = -9 + 18 - 8 = 1
\]
So, the vertex is at \( (3, 1) \).

### 3. Determine the axis of symmetry:
The axis of symmetry is the vertical line that passes through the vertex:
\[
x = 3
\]

### 4. Identify the direction of the parabola:
Since the coefficient of \( x^2 \) (i.e., \( a = -1 \)) is negative, the parabola opens downward.

### 5. Find the y-intercept:
The y-intercept occurs where \( x = 0 \):
\[
y = -(0)^2 + 6(0) - 8 = -8
\]
So, the y-intercept is at \( (0, -8) \).

### 6. Find the x-intercepts (roots):
To find the x-intercepts, solve the quadratic equation \( -x^2 + 6x - 8 = 0 \):
\[
-x^2 + 6x - 8 = 0
\]
Multiply by -1 to simplify:
\[
x^2 - 6x + 8 = 0
\]
Factor the quadratic equation:
\[
(x - 2)(x - 4) = 0
\]
So, the x-intercepts are \( x = 2 \) and \( x = 4 \).

### 7. Sketch the graph:
- Plot the vertex at \( (3, 1) \).
- Draw the axis of symmetry at \( x = 3 \).
- Mark the y-intercept at \( (0, -8) \).
- Plot the x-intercepts at \( (2, 0) \) and \( (4, 0) \).
- Draw the parabola opening downward, passing through these points.

Would you like a visual sketch of the graph as well?

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Here are 5 relative questions that build on this:

1. How would the graph change if the coefficient of \( x^2 \) were positive?
2. What is the significance of the vertex in a quadratic function?
3. How would you solve a quadratic equation if it couldn't be factored?
4. Can you determine the maximum value of the function \( y = -x^2 + 6x - 8 \)?
5. How does the discriminant of the quadratic equation relate to the number of x-intercepts?

**Tip:** The vertex of a quadratic function is either the maximum or minimum point on its graph, depending on the sign of the coefficient of \( x^2 \).

This detailed guide walks through sketching the graph of the quadratic function Y = -x^2 + 6x - 8. Learn about identifying the vertex, axis of symmetry, and intercepts, and how the parabola opens downward. Ideal for students in Grades 8-10.

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