To write the equation of the parabola in vertex form, we need two main pieces of information from the graph:

1. Vertex: This can be identified as the lowest (or highest) point on the parabola. From the graph, the vertex appears to be at $(-2, 0)$.

2. Another Point: A second point on the parabola is needed to determine the stretch/compression factor. From the graph, the point $(-1, 1)$ lies on the parabola.

Vertex Form of a Parabola:

The vertex form is given by: $y = a(x - h)^2 + k$ where $(h, k)$ is the vertex of the parabola, and $a$ is the stretch/compression factor.

From the graph:

• The vertex is $(-2, 0)$, so $h = -2$ and $k = 0$.

Now, the equation becomes: $y = a(x + 2)^2$

Find $a$ using the point $(-1, 1)$:

Substitute $(-1, 1)$ into the equation to solve for $a$: $1 = a(-1 + 2)^2$ $1 = a(1)^2$ $1 = a$

Thus, $a = 1$, and the equation of the parabola is: $y = (x + 2)^2$

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How would the equation change if the parabola was flipped upside down?
2. What would happen to the equation if the vertex shifted horizontally?
3. How does changing the value of $a$ affect the shape of the parabola?
4. Can you convert this vertex form equation into standard form?
5. How do you find the axis of symmetry for any given parabola?

Tip: The value of $a$ controls the direction and the "width" of the parabola. If $a > 0$, it opens upwards; if $a < 0$, it opens downwards.