To find the equation of this rational function with a y-intercept of 7, we can use the general form of a rational function:
y = (ax + b) / (cx + d)
Since the function has a y-intercept of 7, that means that when x is 0, y is 7:
7 = b / d
Since the graph has horizontal asymptote x = 5, this means that the denominator has a factor of x + 5.
Since the graph has a vertical asymptote at y = -2, this means that the numerator has a factor of y + 2.
Putting all this together, we can write the equation of the function as:
y = (ay + 7)(x + 5) / (x - 5)
Expanding this out gives:
y = (ayx + 5ay + 7x + 35) / (x - 5)
Now, we know that when x is 0, y is 7:
7 = 35 / (-5)
7 = -7
This tells us that a = -1, so the equation of the rational function is:
y = (-1)yx + 5(-1)y + 7x + 35 / (x - 5)
y = -yx - 5y + 7x + 35 / (x - 5)
So, the equation for this function is y = -yx - 5y + 7x + 35 / (x - 5).