The given equation is:

\[
x^2 = -8y - 16
\]

To better understand this equation, let's rewrite it in a more familiar form.

### Step 1: Rewrite in Standard Form

We can rewrite the equation by isolating \( y \). Let's start by rearranging the terms:

\[
x^2 = -8y - 16 \implies x^2 + 16 = -8y
\]

Now, solve for \( y \):

\[
y = -\frac{x^2 + 16}{8}
\]

Simplifying further:

\[
y = -\frac{x^2}{8} - 2
\]

### Step 2: Identify the Conic Section

The equation \( x^2 = -8y - 16 \) represents a **parabola**. In its standard form \( y = ax^2 + bx + c \), we can see that the equation represents a parabola that opens downward (since the coefficient of \( x^2 \) is negative).

Alternatively, the form \( x^2 = -8(y + 2) \) also represents a parabola that opens downward along the y-axis with its vertex at the point \((0, -2)\).

### Vertex and Direction

- **Vertex**: \((0, -2)\)
- **Direction**: Downward along the y-axis

Would you like more details, or do you have any questions?

### Related Questions:

1. How can we find the focus and directrix of this parabola?
2. What is the axis of symmetry for this parabola?
3. How do we convert this equation to vertex form?
4. Can we derive the parametric equations for this parabola?
5. How can we graph this parabola on a coordinate plane?

#### Tip:

For any conic section, identifying its standard form helps to quickly determine its properties such as vertex, focus, and directrix.

Learn how to rewrite and solve the quadratic equation x² = -8y -16 step-by-step. This problem covers key algebraic concepts such as quadratic equations and standard forms of parabolas. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation