QUICK TIPS FOR
GRAPHING RATIONAL EQUATIONS
(A topic for the 2020 School Closing Series)
(A topic for the 2020 School Closing Series)
The
‘artist’ in me loves graphing. I could watch my TI-84 (or any grapher) draw pictures
all day. I even like homework assignments that force me to graph by
hand. What I DON’T like are instructions that expect me to “calculate”
50 points before “sketching” a graph. When I just need to imagine what
the graph looks like, where it is positive or negative, what direction
the end points face, etc,
I’m in favor of estimating. I believe it’s a good practice even when a
more specific graph is eventually called for, because I can quickly
find silly errors.
Here’s a quick and dirty review on finding a few specific points when graphing rational expressions.
1. FIND VERTICAL ASYMPTOTES by setting the denominator to 0.
Here’s a quick and dirty review on finding a few specific points when graphing rational expressions.
1. FIND VERTICAL ASYMPTOTES by setting the denominator to 0.
2. FIND THE X-INTERCEPT by setting the numerator to 0.
Notice
that both of these tell you something about the X-axis and run left and
right in the original equation, like the X-axis does on the graph.
3. FIND THE HORIZONTAL ASYMPTOTE by reducing the leading variable.
4. FIND THE Y-INTERCEPT by reducing the constants.
5. FILL IN THE GRAPH.
Done. Sketched.
Other conditions tell even more about the graph.
-- Suppose in step 3, the exponent on the denominator’s variable is larger than the one on the numerator. Think of ‘end behavior,’ what happens way out there at infinity. If the leading coefficient of the demominator is LARGER than the one in the numerator, the fraction gets very, very small at infinity -- almost nothing. So the horizontal asymptote is 0.
-- Suppose reducing the leading coefficients leaves a variable in the numerator, like y = x or y = x/3. This indicates a SLANT (or SKEW) ASYMPTOTE, which is not considered “horizontal” even though it goes through the y-axis. Notice that it also goes through the x-axis.
-- Suppose there is no constant in the numerator. Fill in a place holder, 0. Then the fraction is zero divided by something and equals zero.
-- Suppose there is no constant in the denominator. Again, fill in a place holder, 0. The fraction becomes division by zero, which is undefined and the graph never crosses the y-axis.
-- Look at a bunch of rational graphs. Notice that asymptotes are like the poles of a magnet. They repel the graph. The segments separated by the asymptotes act in a similar way. In MOST (but not all) cases, if the graph is heading up on one side of the asymptote, the other side will NOT follow the same pattern. Simple rational expressions generally occupy only 2 quadrants formed by the vertical and horizontal asymptotes: I and III, or II and IV.
-- Suppose in step 3, the exponent on the denominator’s variable is larger than the one on the numerator. Think of ‘end behavior,’ what happens way out there at infinity. If the leading coefficient of the demominator is LARGER than the one in the numerator, the fraction gets very, very small at infinity -- almost nothing. So the horizontal asymptote is 0.
-- Suppose reducing the leading coefficients leaves a variable in the numerator, like y = x or y = x/3. This indicates a SLANT (or SKEW) ASYMPTOTE, which is not considered “horizontal” even though it goes through the y-axis. Notice that it also goes through the x-axis.
-- Suppose there is no constant in the numerator. Fill in a place holder, 0. Then the fraction is zero divided by something and equals zero.
-- Suppose there is no constant in the denominator. Again, fill in a place holder, 0. The fraction becomes division by zero, which is undefined and the graph never crosses the y-axis.
-- Look at a bunch of rational graphs. Notice that asymptotes are like the poles of a magnet. They repel the graph. The segments separated by the asymptotes act in a similar way. In MOST (but not all) cases, if the graph is heading up on one side of the asymptote, the other side will NOT follow the same pattern. Simple rational expressions generally occupy only 2 quadrants formed by the vertical and horizontal asymptotes: I and III, or II and IV.
FINDING THE BLACK HOLES OF A RATIONAL EQUATION
As problems become more complicated, factoring and reducing become necessary.
This reduced form gives a vertical asymptote at x = -1, but x can not
equal +3 either because it was part of the original equation. Although
there is no vertical barrier at x = 3, the graph will skip over that
point and create a hole.
WHAT DOES THE GRAPH LOOK LIKE WHEN THERE ARE 2 VERTICAL ASYMPTOTES?
The principle of repulsion....
And
some equations present wierdnesses like part of the graph crossing the horizontal
asymptote, looping around, and finally adopting the behavior we
expected.
WITNESS:
Notice the x and y intersections and the repulsion principle. For this one, a couple of calculated -x values could be useful in verifying that they fall BELOW the horizontal asymptote.
--- And another rule breaker....
How does it get that squiggle? No vertical asymptote because it's an
imaginary number, but a horizontal asymptote at 0. For X between -sqrt 2
and (roughly) 2.4, the y value is increasing; for all other values it
is ever decreasing. While the numerator can turn negative, the
denominator will always be positive. --- Here's a cute one that defies the repelling principle.
Will this graph ever enter negative territory? Nope. The fraction will always be positive.
With
just a few morsels of data, the general shape of most rational equation
graphs can provide valuable information to check more intricate
drawings, as well as inform Calculus questions later in math study. So let's get graphing!
