Graphs of xy = ax + by + c


Consider the graphs of the above equation for various values of a, b, and c. We'll describe, discuss, and analyze families of curves. In each section, two of the variables will be fixed and the third varied.


Graphs of xy=ax+by+c with a and b fixed and c varied


Let a and b equal 1. Let c equal one for the first graph.

This appears to be a hyperbola with horizontal asymptote at x=1 and vertical asymptote at y=1. We'll graph x=1 and y=1 to see if this is true. The hyperbolas appear to open along an axis of y=x.

Yes, the asymptotes are x=1 and y=1. Now let's try c=2 to see what happens.

The curve of the hyperbola appears to be more blunted , but the asymptotes are the same. Let's try c=3,4, and 5.

As c increases the hyperbolas become more blunted, but the asymptotes remain the same. What will happen if c is a fraction? We'll look at c=1/2 and 1/4 on the same axes as c=1.

The c=1/2 is green and the c=1/4 is blue. Notice that the hyperbolas are more rounded and closer to each other. Again the asymptotes appear to be the same. What will happen if c is negative? Let's try c=-1 graphed together with c=1.

That's a surprise! The -1 graphed as x=1 and y=1. What about c=-2 or c=-3?

Now that's what I expected; the hyperbolas open along the line y=-x. In this graph c=-2 is blue and c=-3 is beige. All the above graphs are with a=b=1. Let's try a larger coefficient to see if there is any difference. Let's graph the equation with a=b=3 and c=1.

It appeared that the asymptotes were now x=3 and y=3, so those lines were added. Notice that this graph covers more area than the previous ones; that is to show more clearly how the graph of the equation approaches the asymptotes. Can we conclude that the coefficients of a and b determine the asymptotes for the hyperbolas when a=b?


xy=ax+by+c when a and c are fixed and b varies


Let's begin with a=c=1 and b=2.

Here we have asymptotes that are certainly x=0 and y=0. Let's try b=3, 4, and 5. Each one of these graphed the same line! What about negative numbers? Let's try b=-2,-3, and -4. The same thing happens. What about fractions? Say b=1/2 and b=1/4. The very same thing occurs. Variation in the y term seems to make no difference. Let's try different values for a and c. Let a=c=3. We'll graph this along with the one above.

Now the graph with a=c=3 has the same asymptotes, but is closer to the center and more square. Let's try a couple more say a=c=1/2 , 5, and -3.

 

The a=c=1/2 is beige and closer to the center ; the a=c=5 is the blue and is further from the center; and the a=c=-3 is the purple and is oriented the other direction. Notice that as long as a=c the asymptotes are x=0 and y=0.


xy=ax+by+c with b and c fixed, a varied


Let's begin with b=c=1 and a=2.

 

The asymptotes here appear to be y=2 and x=1. Let's put those equations in and enlarge the graph so we can see it more clearly.

Yes, that's right. Now what if a=3 , 4, or 5.

Now we have asymptotes at y=3,4,and 5 respectively. So it seems that the horizontal asymptotes correspond to the coefficients of the x term. Let's check a negative number and a fraction just to be sure. Try a=-5 , -10, and 1/2.

So now we have horizontal asymptotes at y=-5,-10, and 1/2 as expected. Surely the horizontal aymptotes are the value of a, the coefficient of the x term. Let's check to see what happens if b and c equal something other than 1. Let a=2, b=c=3.

The horizotal asymptote is still 2 just as expected; the vertical one appears to have changed to 3. Let's try a couple more variations on b and c to be sure. Let's try b=c=1/2, -5, and -10.

Notice that there are now vertical asymptotes at x=-5,-10, and 1/2. So the vertical aymptotes correspond to the coefficient of the y term and the value of the c term.


Extensions:

1. Try more variations of c with a=b, when a and b are negative integers or negative fractions.

2. With a constant, vary b and c at the same time. With b constant, vary a and c. With c constant vary both a and b.

3. Try making geometric designs of hyperbolas by using some of the above variations of this equation.


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