homework help: Find a possible formula for the function in the figure below.?

Graphs like this with both vertical and horizontal asymptotes are usually graphs of rational functions of the form h(x) = f(x) / g(x). Such functions have vertical asymptotes at x values that make the denominator equal to zero. Meanwhile, x values that make the numerator equal to zero are x-intercepts on the graph.

In this graph, there are vertical asymptotes at x = -5 and x = 10,

and x-intercepts are x = -10 and x = 15 .

Using the above considerations, a first guess at the function is

h(x) = [(x + 10)(x - 15)] / [(x + 5)(x - 10)] …………. equation 1

I graphed that using Desmos, a graphing utility. The vertical asymptotes and x-intercepts were correct, but the orientation was wrong. For example, the ⋂ shape in the middle of the given graph was showing on my graph as a U shape. I fixed that by flipping my graph over the x-axis, accomplished by multiplying the above equation by -1 :

h(x) = - [(x + 10)(x - 15)] / [(x + 5)(x - 10)] …………. equation 2

Now I needed to deal with the horizontal asymptote at y = -5. Horizontal asymptotes are values that the function approaches as x gets very large negative or very large positive.

What happens in my equation 2 as gets large positive or negative? Well, without multiplying it all out, there is a quadratic expression in both the numerator and denominator (highest power of x in both cases is 2). For large positive or negative x values, the function essentially behaves like (-x² / x²) = -1, which is its horizontal asymptote. We want the horizontal asymptote to be -5, so I just multiplied by 5, to get

h(x) = -5 [(x + 10)(x - 15)] / [(x + 5)(x - 10)] …………. equation 3

and it turned out that was a match. In that last function, the y-intercept happened to turn out to be at the point (0, -15) so no further work was needed.

You can multiply out the numerator and denominator of equation 3 if you wish, but I think it's perfectly fine as an answer the way it is.

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In the graph linked to below, you can use the hollow circles on the left side next to the functions to turn on and off the three graphs. Right now, my first guess is showing. Look down the left side for my equation 2 and turn that one on, and turn off the first function. Then do that again, turning off equation 2 and turning on equation 3.

https://www.desmos.com/calculator/mpi3tk4gzc