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The given problem asks to write the equation of a parabola in vertex form, using the vertex \((4, 3)\) and another point \((3, -4)\).

The vertex form of a parabola is:

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

where \((h, k)\) is the vertex of the parabola. Here, the vertex is \((4, 3)\), so we can substitute \(h = 4\) and \(k = 3\):

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

Now, we need to find the value of \(a\). To do this, we can substitute the coordinates of the other given point \((3, -4)\) into the equation. This gives us:

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

Simplifying the equation:

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

Solving for \(a\):

\[
a = -4 - 3 = -7
\]

So the value of \(a\) is \(-7\). Now we can write the final equation of the parabola:

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

Would you like more details or have any questions?

Here are some related questions to consider:
1. How do you convert from vertex form to standard form?
2. How does the sign of \(a\) affect the direction of the parabola?
3. What are the steps to find the axis of symmetry from the vertex form?
4. How can you determine the focus and directrix of a parabola from its equation?
5. What role does the vertex play in determining the maximum or minimum point of the parabola?

**Tip:** When solving for \(a\), always substitute a known point on the parabola into the vertex form equation to find the precise value.

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

Solve for the equation of a parabola in vertex form using the given vertex (4,3) and point (3,-4). Learn the process of finding the value of 'a' and writing the final equation.

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