Write an equation for a rational function whose graph has the following properties:

x-intercept of 3 y-intercept of -3
vertical asymptote of x=-2
horizontal asymptote of y=2

x-intercept of 3 ---> needs a multiplier of x-3

vertical asymtote of x = -2 ----> needs a divisor of x+2

so far y = (x-3)/(x+2)

horizontal asymptote of y = 2
lets slap a 2 infront

y = 2(x-3)/(x+2) , as x gets huge y ---> 2 , ( try x = 100,000 )

we want y-intercept of -3, (when x = 0)
We lucked out, since if x = 0 , y = -6/2 = -3

y = 2(x-3)/(x+2)

To find the equation of a rational function that satisfies the given properties, we need to consider the key features.

1. The x-intercept of 3 implies that the function passes through the point (3, 0).
2. The y-intercept of -3 implies that the function passes through the point (0, -3).
3. The vertical asymptote of x = -2 indicates that there is a factor of (x + 2) in the denominator.
4. The horizontal asymptote of y = 2 indicates that the degree of the numerator and denominator are the same.

With these properties in mind, we can write the equation of the rational function as:

f(x) = (x + 2) * (ax + b) / (cx + d)

To find the values of a, b, c, and d, we substitute the given points into the equation:

1. The point (3, 0):
0 = (3 + 2) * (3a + b) / (3c + d)

2. The point (0, -3):
-3 = (0 + 2) * (0a + b) / (0c + d)

Simplifying the equations:

1. 0 = 5(3a + b) / (3c + d)
2. -3 = 2b / d

Solving these equations simultaneously will give us the values of a, b, c, and d.

To find the equation of a rational function with these properties, we need to consider the characteristics of the given function.

First, let's start with the fact that the x-intercept is 3. This means that the function passes through the point (3, 0).

Second, the y-intercept is -3, which means the function passes through the point (0, -3).

Third, the function has a vertical asymptote of x = -2. This indicates that the function approaches infinity or negative infinity as x approaches -2.

Finally, there is a horizontal asymptote of y = 2. This means that as x approaches positive or negative infinity, the function approaches 2.

Based on these properties, we can write the equation of the rational function as:

f(x) = (x - 3) / ((x + 2) * (a * x + b))

The numerator (x - 3) represents the x-intercept of 3, and the denominator (x + 2) represents the vertical asymptote of x = -2.

To find the values for 'a' and 'b', we use the fact that the function has a y-intercept of -3. Substituting x = 0 into the equation, we get:

-3 = (0 - 3) / ((0 + 2) * (a * 0 + b))

Simplifying this equation, we have:

-3 = -3 / (2b)

To cancel out the negative signs, multiply both sides by -1:

3 = 3 / (2b)

Cross-multiplying, we get:

3 * 2b = 3

Simplifying, we find:

6b = 3

Dividing both sides by 6, we obtain:

b = 1/2

Now that we know the value of 'b', we can substitute it back into the equation to find 'a'. The equation becomes:

f(x) = (x - 3) / ((x + 2) * (a * x + 1/2))

Next, we can use the fact that the function has a horizontal asymptote of y = 2. As x approaches infinity, the denominator becomes significant compared to the numerator, and the only term that dominates is 'a * x'. So, to ensure a horizontal asymptote at y = 2, the degree (exponent) of 'a * x' must be equal to 1.

Therefore, 'a' must be 1.

Finally, the equation of the rational function with the given properties is:

f(x) = (x - 3) / ((x + 2) * (x + 1/2))