Sketch the graph of a function that is never negative and has the lines x = -1 and x = 1 as vertical asymptotes, the x-axis as a horizontal asymptote and the origin as an x-intercept.

Question

THEN find an equation for a possible corresponding rational function.

I have no idea where to even begin with this one.

Answer 1

For the function to:

(1) be never negative, you can square everything in the numerator and the denominator.

(2) have a vertical asymptote at x = -1, you can put x + 1 in the denominator.

(3) have a vertical asymptote at x = 1, you can multiply the denominator by x - 1.

(4) have the x-axis be a horizontal asymptote (at y = 0), you can put 1 in the numerator.

(5) have the origin as an x-intercept, you can multiply the numerator by x.

The resulting function is f(x) = x² / [ (x+1)² * (x-1)² ], the graph of which is at this link below:

desmos . com / calculator / 08rdgc7feh (delete the spaces).

Answer 2

never negative is an exponential function such as y = 2^x or a parabola y=x^2

vertical asymptotes mean a denominator with factors x+1 and x-1, multiplied together = (x^2-1)

x axis as a horizontal asymptote is just the same exponential function y=2^x

origin as the x intercept though means y =0 is the y intercept, which shifts the exponential function down by 1 unit, but then it has some negative values. so, just take the absolute value

y = |(2^x)/(x^2-1)|

y = the absolute value of 2^x/(x^2 -1)

or an infinite number of other equations are possible

plot a few points of the equation, and connect the dots