1. Write an equation of a rational function that satisfies all of these conditions (6 marks)

- Vertical asymptote at x = -8 and x = 5
- Horizontal asymptote at y = 0
- x-intercept at (-2, 0)
- f(0) = -2
- has a hole at x = 3

To write an equation for a rational function that satisfies these conditions, we need to consider the properties of a rational function. Remember that a rational function is formed by taking a ratio of two polynomial functions.

Let's break down the given information one by one:

1. Vertical asymptotes:
The vertical asymptotes occur when the denominator of the rational function is equal to zero. So, we have two vertical asymptotes at x = -8 and x = 5. This means that the denominator of our equation should contain factors of (x + 8) and (x - 5).

2. Horizontal asymptote:
The horizontal asymptote occurs when the degree of the numerator is less than or equal to the degree of the denominator. In this case, as the horizontal asymptote is y = 0, the numerator should have a degree less than or equal to the degree of the denominator.

3. x-intercept:
The x-intercept occurs when the function's value is zero. The x-intercept is given as (-2, 0), which means that our equation should have a factor of (x + 2) in the numerator.

4. f(0) = -2:
When x = 0, the function should have a value of -2. This means that our equation should have a constant term of -2 in the numerator.

5. Hole at x = 3:
A hole occurs at x = 3 when there is a common factor between the numerator and denominator. So, our equation should have a factor of (x - 3) in both the numerator and the denominator. This will cancel out, creating a hole at x = 3.

Combining these conditions, we can write the equation of the rational function as follows:
f(x) = (x + 2)(x - 3) / ((x + 8)(x - 5))

Now, this equation satisfies all the given conditions. However, it's important to note that there might be alternative equations that also satisfy these conditions.