Graph the equation y, equals, minus, x, squared, minus, 6, x, minus, 8y=−x
2
−6x−8 on the accompanying set of axes. You must plot 5 points including the roots and the vertex.

To graph the equation \( y = -x^2 - 6x - 8 \), we need to identify and plot five points, including the roots and the vertex of the parabola. Let's go through the steps to find these points:

1. **Find the vertex:**
   The vertex form of a quadratic equation \( y = ax^2 + bx + c \) gives the x-coordinate of the vertex as \( x = -\frac{b}{2a} \). For the given equation:
   \[
   a = -1, \quad b = -6
   \]
   \[
   x = -\frac{-6}{2(-1)} = \frac{6}{-2} = -3
   \]
   To find the y-coordinate of the vertex, substitute \( x = -3 \) back into the equation:
   \[
   y = -(-3)^2 - 6(-3) - 8 = -9 + 18 - 8 = 1
   \]
   So, the vertex is \((-3, 1)\).

2. **Find the roots (x-intercepts):**
   To find the roots, set \( y = 0 \):
   \[
   0 = -x^2 - 6x - 8
   \]
   \[
   x^2 + 6x + 8 = 0
   \]
   Factor the quadratic equation:
   \[
   (x + 2)(x + 4) = 0
   \]
   \[
   x = -2, \quad x = -4
   \]
   So, the roots are \((-2, 0)\) and \((-4, 0)\).

3. **Find additional points:**
   Let's find two more points by selecting values of \( x \) around the vertex:
   - For \( x = -1 \):
     \[
     y = -(-1)^2 - 6(-1) - 8 = -1 + 6 - 8 = -3
     \]
     So, the point is \((-1, -3)\).
   - For \( x = -5 \):
     \[
     y = -(-5)^2 - 6(-5) - 8 = -25 + 30 - 8 = -3
     \]
     So, the point is \((-5, -3)\).

Now, we have the following points:
- Vertex: \((-3, 1)\)
- Roots: \((-2, 0)\) and \((-4, 0)\)
- Additional points: \((-1, -3)\) and \((-5, -3)\)

### Graph

To graph these points on the set of axes:

1. Plot the vertex \((-3, 1)\).
2. Plot the roots \((-2, 0)\) and \((-4, 0)\).
3. Plot the additional points \((-1, -3)\) and \((-5, -3)\).
4. Draw a smooth parabolic curve through these points, opening downwards.

Here is the graph of the equation:

```plaintext
  5 +--------------------------+
    |                          |
  4 +             *            + (1, -3)
    |                          |
  3 +                          |
    |                          |
  2 +                          |
    |                          |
  1 +       *                  + (3, 1)
    |                          |
  0 +   *       *              + (-2, 0), (-4, 0)
    |                          |
 -1 +                          |
    |                          |
 -2 +                          |
    |                          |
 -3 +  *       *               + (-1, -3), (-5, -3)
    |                          |
 -4 +--------------------------+
    -5 -4 -3 -2 -1  0  1  2  3
```

Would you like more details or have any questions?

### Related Questions

1. How do you derive the quadratic formula for finding the roots?
2. What are the properties of a parabola with respect to its vertex and axis of symmetry?
3. How does the sign of \( a \) in \( y = ax^2 + bx + c \) affect the direction of the parabola?
4. How can you determine if a quadratic equation has real or complex roots?
5. What is the significance of the discriminant in a quadratic equation?
6. How do you complete the square to convert a quadratic equation to its vertex form?
7. What are the steps to graph a quadratic equation by finding its intercepts and vertex?
8. How can you use the quadratic function to model real-world problems?

### Tip

When graphing a quadratic equation, always start by finding the vertex and the roots, as these provide key points that help shape the graph accurately.

Learn how to graph the quadratic equation y = -x^2 - 6x - 8 by finding its vertex, roots, and additional points. This problem covers key algebraic concepts including quadratic equations, vertex formula, and roots of a quadratic equation. Suitable for high school students.

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