Question

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The rational function has a y-intercept of 7. What is the equation for this function?

A graph with horizontal axis x range from left to right labeled is negative 9 to 9; vertical axis y labeled from top to bottom is range from 12 to negative 4. The graph is of a hyperbola with horizontal asymptote x equals 5, and vertical asymptote y equals negative 2.

Answer 1

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).