Let's analyze the graph to find the equation in the form $y = a(x - h)^2 + k$.

Step 1: Identify the vertex (h, k)

From the graph:

• The vertex is at the point $(1, 2)$.

Thus, $h = 1$ and $k = 2$.

So far, the equation looks like: $y = a(x - 1)^2 + 2$

Step 2: Identify another point on the graph

A visible point on the parabola is $(0, 3)$. We can substitute $x = 0$ and $y = 3$ into the equation to solve for $a$.

$3 = a(0 - 1)^2 + 2$ $3 = a(1) + 2$ $3 - 2 = a$ $a = 1$

Step 3: Write the final equation

Now that we know $a = 1$, the equation of the parabola is: $y = (x - 1)^2 + 2$

Let me know if you need further details or explanations!

Here are five questions you might explore next:

1. What happens to the graph if the value of $a$ is negative?
2. How do you convert a quadratic equation from vertex form to standard form?
3. How can you determine the axis of symmetry of a parabola from its equation?
4. How does the value of $k$ affect the vertical shift of a parabola?
5. How do we find the focus and directrix of a parabola?

Tip: When $a = 1$, the parabola has the standard "width". If $|a| > 1$, the parabola becomes narrower; if $|a| < 1$, it becomes wider.