Find a function f whose graph is a parabola that has the given vertex and that passes through the indicated point.
vertex (3, −2);    point (4, 3)

To find the equation of the parabola, we use the **vertex form** of a quadratic function:

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

Here, \((h, k)\) is the vertex, and \((x, y)\) is any point on the parabola.

Given:
- Vertex: \((3, -2)\), so \(h = 3\) and \(k = -2\).
- Point on the parabola: \((4, 3)\), so \(x = 4\) and \(y = 3\).

Substituting the vertex into the vertex form, we have:

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

Now, use the point \((4, 3)\) to solve for \(a\). Plug in \(x = 4\) and \(y = 3\):

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

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

\[
3 = a - 2
\]

Solving for \(a\):

\[
a = 5
\]

Thus, the equation of the parabola is:

\[
f(x) = 5(x - 3)^2 - 2
\]

Would you like more details or have any further questions?

Here are 5 related questions:
1. What is the general form of a quadratic equation?
2. How do you convert from vertex form to standard form?
3. How can you find the axis of symmetry for a parabola?
4. What role does the coefficient \(a\) play in the shape of the parabola?
5. How can you determine if the parabola opens upwards or downwards?

**Tip:** The value of \(a\) affects the "width" of the parabola. A larger absolute value of \(a\) makes the parabola narrower, while a smaller absolute value makes it wider.

Find a function f whose graph is a parabola that has the given vertex and that passes through the indicated point. Vertex (3, −2); point (4, 3).

Learn how to find the quadratic function of a parabola with a vertex at (3, -2) and passing through the point (4, 3). This example uses the vertex form of a quadratic function and explains how to calculate the value of 'a'. Suitable for students in Grades 9-11.

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