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!

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.