The equation given in the image is:

$y = -x^2 - 8x - 15$

This is a quadratic equation representing a parabola. Let's break down the information requested.

1. Vertex

The vertex of a parabola given by the form $y = ax^2 + bx + c$ can be found using the formula for the x-coordinate of the vertex:

$x = \frac{-b}{2a}$

Here, $a = -1$ and $b = -8$. Substituting these values:

$x = \frac{-(-8)}{2(-1)} = \frac{8}{-2} = -4$

Now, substitute $x = -4$ back into the equation to find the y-coordinate:

$y = -(-4)^2 - 8(-4) - 15 = -16 + 32 - 15 = 1$

Thus, the vertex is:

$(-4, 1)$

2. Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is $-4$, the equation for the axis of symmetry is:

$x = -4$

3. Maximum or Minimum Value

Since the coefficient of $x^2$ (i.e., $a = -1$) is negative, the parabola opens downwards. This means the vertex represents the maximum value of the parabola.

The maximum value is the y-coordinate of the vertex, which is:

$1$

4. Range of the Parabola

Since the parabola opens downwards, the range is all values of $y$ less than or equal to the maximum value. Therefore, the range is:

$(-\infty, 1]$

Summary:

• Vertex: $(-4, 1)$
• Axis of Symmetry: $x = -4$
• Maximum Value: 1
• Range: $(-\infty, 1]$

Let me know if you'd like more details or have any questions!

Here are 5 related questions to explore:

1. How would the vertex change if the equation was $y = x^2 - 8x - 15$?
2. What happens to the axis of symmetry if we add a constant to the entire equation?
3. How do we determine whether a parabola opens upwards or downwards just by looking at the equation?
4. Can the vertex represent a minimum in any case, and when would that happen?
5. How does completing the square help in identifying the vertex form of a quadratic equation?

Tip: Always check the sign of $a$ in a quadratic equation to quickly determine if the parabola opens upwards (positive $a$) or downwards (negative $a$).