https://file.aiquickdraw.com/mathbot/file/1725583315756brsg67py.jpg

the new vertex is (-2,-3) and two other points are (-3,1) and (-1,1)

The image you've uploaded appears to be a graph of a quadratic function. Based on the information you provided:

- The vertex of the parabola is at \((-2, -3)\).
- Two other points on the parabola are \((-3, 1)\) and \((-1, 1)\).

Let's find the equation of the parabola in the standard form \(y = ax^2 + bx + c\).

### Step 1: Use the vertex form of a parabola

The vertex form of a parabola is given by:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola. Substituting the vertex \((-2, -3)\), we have:
\[
y = a(x + 2)^2 - 3
\]

### Step 2: Use one of the points to find \(a\)

Substitute one of the points \((-3, 1)\) into the equation to find the value of \(a\).

\[
1 = a(-3 + 2)^2 - 3
\]
\[
1 = a(-1)^2 - 3
\]
\[
1 = a(1) - 3
\]
\[
a = 4
\]

### Step 3: Write the equation of the parabola

Now that we have \(a = 4\), the equation of the parabola is:

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

### Step 4: Expand to the standard form

If you want the equation in the standard form \(y = ax^2 + bx + c\), expand the equation:

\[
y = 4(x^2 + 4x + 4) - 3
\]
\[
y = 4x^2 + 16x + 16 - 3
\]
\[
y = 4x^2 + 16x + 13
\]

So, the equation of the parabola is \(y = 4x^2 + 16x + 13\).

Would you like more details on how these steps work or have any other questions?

### Follow-Up Questions:
1. How would you verify if a point lies on this parabola?
2. What is the significance of the vertex in the context of a parabola?
3. How can the discriminant help in determining the nature of the roots of the quadratic equation?
4. How can you determine the axis of symmetry of the parabola from the equation?
5. Can you convert this parabola into its factored form?

### Tip:
When identifying the vertex of a parabola, remember that it provides crucial information about the graph's symmetry and direction (whether it opens upwards or downwards).

The new vertex is (-2,-3) and two other points are (-3,1) and (-1,1).

Learn how to find the equation of a parabola given the vertex at (-2, -3) and two points (-3, 1) and (-1, 1). This solution covers the use of vertex form and conversion to standard form, suitable for students in Grades 9-12.

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