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:

plaintextCopy code
  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

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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.