write the standard form of the quadratic equation vertex (2,3) and point (0,2)

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:

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

Would you like further details or clarification?

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.

Learn how to convert a quadratic equation from vertex form to standard form using the vertex (2, 3) and the point (0, 2). This step-by-step solution involves key algebraic concepts, including solving for the coefficient 'a' and expanding to the standard form, suitable for Grades 9-11.

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