To write the quadratic equation in standard form, $y = ax^2 + bx + c$, we can start by using the vertex form of a quadratic equation, which is:

$y = a(x - h)^2 + k$

where $(h, k)$ is the vertex. In this case, the vertex is $(2, 3)$, so we substitute $h = 2$ and $k = 3$:

$y = a(x - 2)^2 + 3$

Next, we use the given point $(0, 2)$ to find the value of $a$. Substituting $x = 0$ and $y = 2$ into the equation:

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

$2 = a(4) + 3$

$2 - 3 = 4a$

$-1 = 4a$

$a = -\frac{1}{4}$

Now that we have $a = -\frac{1}{4}$, we substitute this back into the vertex form:

$y = -\frac{1}{4}(x - 2)^2 + 3$

To convert this to standard form, we expand the equation:

$y = -\frac{1}{4}(x^2 - 4x + 4) + 3$

$y = -\frac{1}{4}x^2 + \frac{1}{4}(4x) - \frac{1}{4}(4) + 3$

$y = -\frac{1}{4}x^2 + x - 1 + 3$

$y = -\frac{1}{4}x^2 + x + 2$

So, the standard form of the quadratic equation is:

$y = -\frac{1}{4}x^2 + x + 2$

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Here are 5 related questions to expand on this:

1. How do you convert any quadratic equation from vertex form to standard form?
2. What is the significance of the vertex in a quadratic equation?
3. How can you find the axis of symmetry from a quadratic equation in standard form?
4. What are the steps to determine the vertex given a quadratic equation in standard form?
5. How does the sign of $a$ in a quadratic equation affect the direction of the parabola?

Tip: The vertex form of a quadratic equation is particularly useful for identifying the maximum or minimum value of the function directly.