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Let's solve the problem step by step.

### Part (a): Find a quadratic equation with a minimum point at \((-2, 4)\).

The general form of a quadratic function is:
\[ y = a(x - h)^2 + k \]
Where \((h, k)\) is the vertex of the parabola.

Given that the minimum point is at \((-2, 4)\), this point represents the vertex. So we substitute \(h = -2\) and \(k = 4\):

\[
y = a(x + 2)^2 + 4
\]

Now, to fully define the equation, we need the value of \(a\). Since the problem doesn't give any other specific information, we can assume \(a = 1\) (a common assumption for a simple quadratic equation). Therefore, the quadratic equation becomes:

\[
y = (x + 2)^2 + 4
\]

Expanding this, we get:
\[
y = (x^2 + 4x + 4) + 4
\]
\[
y = x^2 + 4x + 8
\]

So, the quadratic equation is:
\[
y = x^2 + 4x + 8
\]

### Part (b): Find a quadratic equation with a maximum point at \((3, -5)\).

Similarly, the vertex form of the quadratic equation is:
\[
y = a(x - h)^2 + k
\]
For a maximum point at \((3, -5)\), we substitute \(h = 3\) and \(k = -5\):

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

Since this is a maximum point, the parabola must open downwards, meaning \(a\) is negative. For simplicity, we assume \(a = -1\). Therefore, the equation becomes:

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

Expanding this, we get:
\[
y = -(x^2 - 6x + 9) - 5
\]
\[
y = -x^2 + 6x - 9 - 5
\]
\[
y = -x^2 + 6x - 14
\]

So, the quadratic equation is:
\[
y = -x^2 + 6x - 14
\]

### Summary:
- (a) The quadratic equation with a minimum point at \((-2, 4)\) is \(y = x^2 + 4x + 8\).
- (b) The quadratic equation with a maximum point at \((3, -5)\) is \(y = -x^2 + 6x - 14\).

Let me know if you'd like any further explanation or details!

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Here are 5 related questions:
1. How do you determine the direction a parabola opens?
2. What happens to the quadratic equation if \(a\) is negative versus positive?
3. How can you find the axis of symmetry for a quadratic equation?
4. What other methods can you use to find a quadratic equation from given points?
5. How do you find the roots of a quadratic equation if the vertex is known?

**Tip:** The vertex form of a quadratic function \(y = a(x - h)^2 + k\) is especially useful when dealing with maximum or minimum points.

Part (a): Find a quadratic equation with a minimum point at (-2, 4). Part (b): Find a quadratic equation with a maximum point at (3, -5).

Learn how to find quadratic equations with a minimum point at (-2, 4) and a maximum point at (3, -5). This problem covers key algebra concepts including the vertex form of a quadratic function. Suitable for students in Grades 9-12.

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