Rational Function: Graph
How to graph a rational function by finding its asymptotes: graph, 3 examples, and their solutions.
Graphy = a/x
This is the graph of y = a/x
(a > 0).
If the numerator is plus (a > 0),
the graph is
on the right top and the left bottom
of the axes.
This is the graph of y = a/x
(a < 0).
If the numerator is minus (a < 0),
the graph is
on the left top and the right bottom
of the axes.
Asymptotey = a/x
The graph of y = a/x
has two asymptotes:
a vertical asymptote
and a horizontal asymptote.
(An asymptote is a line that the graph follows.)
To find the asymptotes:
1) Set the denominator, x, 0.
x = 0.
2) Set the fraction part, a/x, 0.
Then y = 0.
ExampleGraph y = 4/(x - 1) + 2
To graph the rational function,
first find the asymptotes.
The denominator is (x - 1).
So set
x - 1 = 0.
Move -1 to the right side.
Then x = 1.
Next, the fraction part is 4/(x - 1).
Set this part 0.
Then y = 0 + 2.
Then y = 2.
So the asymptotes are
x = 1 and y = 2.
Then graph y = 4/(x - 1) + 2
on a coordinate plane.
First draw the asymptotes
x = 1 and y = 2.
Next, see the numerator of 4/(x - 1).
The numerator 4 is plus.
So draw the graph
on the right top and the left bottom
of the asymptotes.
This is the graph of
y = 4/(x - 1) + 2.
ExampleGraph y = 1/(3 - x) - 1
The coefficient of x should be 1.
So change 1/(3 - x)
to -1/(x - 3).
(Multiply -1
to both of the numerator and the denominator.)
To graph the rational function,
first find the asymptotes.
The denominator is (x - 3).
So set
x - 3 = 0.
Move -3 to the right side.
Then x = 3.
Next, the fraction part is -1/(x - 3).
Set this part 0.
Then y = 0 - 1.
Then y = -1.
So the asymptotes are
x = 3 and y = -1.
Then graph y = -1/(x - 3) - 1
on a coordinate plane.
First draw the asymptotes
x = 3 and y = -1.
Next, see the numerator of -1/(x - 3).
The numerator -1 is minus.
So draw the graph
on the left top and the right bottom
of the asymptotes.
This is the graph of
y = -1/(x - 3) - 1.
ExampleGraph y = (3x - 5)/(x - 2)
The numerator is 3x - 5.
To find the asymptotes,
the numerator should be a constant.
(= no variable)
So change the numerator to a constant.
First write the fraction bar
and the denominator (x - 2).
The x term of the numerator is 3x.
So write 3(x.
The denominator is (x - 2).
So write -2).
3(-2) is added.
To undo this change,
write +3⋅2.
Write -5.
So [3x - 5]/(x - 2)
= [3(x - 2) + 3⋅2 - 5]/(x - 2).
+3⋅2 = +6
+6 - 5 = +1
Split this into two parts:
3(x - 2) and +1.
3(x - 2)/(x - 2) = 3
Simplify a Rational Expression
So the given function is
y = 1/(x - 2) + 3.
Now the numerator of the fraction part
is a constant.
So you can find the asymptotes.
Find the asymptotes.
The denominator is (x - 2).
So set
x - 2 = 0.
Move 2 to the right side.
Then x = 2.
Next, the fraction part is 1/(x - 2).
Set this part 0.
Then y = 0 + 3.
So y = 3.
So the asymptotes are
x = 2 and y = 3.
Then graph y = 1/(x - 2) + 3
on a coordinate plane.
First draw the asymptotes
x = 2 and y = 3.
Next, see the numerator of 1/(x - 2).
The numerator 1 is plus.
So draw the graph
on the right top and the left bottom
of the asymptotes.
This is the graph of
y = 1/(x - 2) + 3.