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We are given the vertex \((4, 3)\) and a point on the parabola \((3, -4)\). To find the equation of the parabola in vertex form, we will use the following formula:

\[
f(x) = a(x - h)^2 + k
\]

Where \((h, k)\) is the vertex of the parabola. In this case, \((h, k) = (4, 3)\). So, we have:

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

Now, we need to find the value of \(a\). We can do this by substituting the given point \((3, -4)\) into the equation and solving for \(a\).

Substitute \(x = 3\) and \(f(x) = -4\):

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

Simplify:

\[
-4 = a(-1)^2 + 3
\]
\[
-4 = a(1) + 3
\]
\[
-4 = a + 3
\]
\[
a = -4 - 3 = -7
\]

Thus, the equation of the parabola is:

\[
f(x) = -7(x - 4)^2 + 3
\]

Let me know if you need further clarification or details.

Here are 5 related questions you might want to explore:
1. How do you convert a parabola's equation from vertex form to standard form?
2. What is the significance of the vertex in a parabola's equation?
3. How does the value of \(a\) affect the shape of the parabola?
4. How can we derive the vertex form equation from a graph of a parabola?
5. What other forms can a quadratic equation take, besides vertex form?

**Tip**: When solving for \(a\), always substitute the coordinates of a known point on the parabola into the vertex form equation to maintain accuracy.

Write the equation of the parabola in vertex form. Vertex (4,3), point (3,-4).

Learn how to find the equation of a parabola given a vertex (4,3) and a point (3,-4). This problem explains how to use the vertex form equation and solve for the unknown constant.

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