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Solve this to vertex form and standard form

The given equation is:

\[
y = -3(x + 4)^2 - 7
\]

### Vertex Form:
The equation is already in vertex form, which is:

\[
y = a(x - h)^2 + k
\]

In this case:
- \( a = -3 \)
- \( h = -4 \) (because of the form \( x + 4 \), which can be written as \( x - (-4) \))
- \( k = -7 \)

So, the vertex is at \( (-4, -7) \), and the equation is already in vertex form.

### Standard Form:
To convert the equation into standard form, we need to expand the squared term and simplify.

\[
y = -3(x + 4)^2 - 7
\]

First, expand \( (x + 4)^2 \):

\[
(x + 4)^2 = x^2 + 8x + 16
\]

Now substitute this back into the equation:

\[
y = -3(x^2 + 8x + 16) - 7
\]

Distribute the \( -3 \):

\[
y = -3x^2 - 24x - 48 - 7
\]

Combine like terms:

\[
y = -3x^2 - 24x - 55
\]

Thus, the equation in standard form is:

\[
y = -3x^2 - 24x - 55
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you derive the vertex from the vertex form equation?
2. How does changing the value of \( a \) affect the graph of a parabola?
3. What is the difference between the vertex form and standard form of a quadratic equation?
4. How would you find the x-intercepts of this quadratic function?
5. Can you convert the standard form back to vertex form?

**Tip:** The vertex form of a quadratic equation is useful for identifying the vertex of a parabola quickly, which tells you the maximum or minimum point on the graph.

Solve this to vertex form and standard form.

This guide explains how to convert the quadratic equation y = -3(x + 4)^2 - 7 from vertex form to standard form. Explore the detailed steps and concepts behind quadratic equations, including how to expand and simplify expressions.

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