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Lesson 8: Graphing Rational Functions Graphing Rational Functions
Lesson 8: Graphing Rational Functions
Learning Outcome(s)
Learning Outcome(s): At the end of the lesson, the learner is able to find the domain and range, intercepts, zeroes, asymptotes of rational functions, graph rational functions, and solve problems involving rational functions.
Lesson Outline:
Lesson Outline:
1. Domain and range of rational functions.
2. Intercepts and zeroes of rational functions.
3. Vertical and horizontal asymptotes of rational functions.
4. Graphs of rational functions
Recall:
Recall:
(a) The domaindomain of a function is the set of all values that the variable x can take.
(b) The rangerange of the function is the set of all values that f(x) will take.
(c) The zeroeszeroes of a function are the values of x which make the function zero.
The real numbered zeroes are also x-interceptsx-intercepts of the graph of the function.
(d) The y-intercepty-intercept is the function value when x=0.
Example 1.
Example 1. Consider the function
(a) Find its domain, (b) intercepts, (c) sketch its graph and (d) determine its range.Solution.
(a) The domain of f(x) is
.Observe that the function is undefined at x = –2. This means that x = -2 is not part of the domain of f(x). In addition, other values of x will make the function undefined.
(b) The x-intercept of f(x) is 2 and its y-intercept is –1.
Recall that the x-intercepts of a rational function are the values of x that will make the function zero. A rational function will be zero if its numerator is zero.
Therefore the zeroes of a rational function are the zeroes of its numerator.
The numerator x – 2 will be zero at x=2. Therefore x=2 is a zero of f(x). Since it is a real zero, it is also an x-intercept.
The y-intercept of a function is equal to f(0). In this case,
.(c) In sketching the graph of f(x), let us look at what happens to the graph near the values of x which make the denominator undefined. Recall that in the previous lesson, we simply skipped connecting the points at integer values.
Let us see what happens when x takes on values that brings the denominator closer to zero.
The denominator is zero when x = –2. Let us look at the values of x close to –2 on its left side (i.e. x < –2, denoted –2--) and values of x close to –2 on its right side (i.e. x >
–2, denoted –2+).
D E P E D C O P Y
i. Table of values for x approaching –2 –
–3 –2.5 –2.1 –2.01 –2.001 –2.0001 As x approaches –2
5 9 41 401 4001 40001 f(x) increases without bound.Notation .. We use the notation “f(x) + as x –2 –” to indicate that f(x) increases without bound as x approaches –2 from the left.
ii. Table of values for x approaching –2+
–1 –1.5 –1.9 –1.99 –1.999 –1.9999 As x approaches –2+
–3 –7 –39 –399 –3999 –39999 f(x) decreases without bound.Notation .. We use the notation “f(x) – as x –2+” to indicate that f(x) decreases without bound as x approaches –2 from the right.
Plotting the points corresponding to these values on the Cartesian plane:
Figure 2.3 Note that the axes do not have the same scale.
Observe that as x approaches –2 from the left and from the right, f(x) gets closer and closer to the line x = –2, indicated in the figure with a dashed line.
We call this line a vertical asymptote, formally defined as follows:
Definition.
Definition. The vertical line x = a is a vertical asymptotevertical asymptote of a function f if the graph of f either increases or decreases without bound as the
-valuesapproach a from the right or left.
D E P E D C O P Y
Finding the Vertical Asymptotes of a Rational Function Finding the Vertical Asymptotes of a Rational Function
Find the values of a where the denominator is zero.
If this value of a does not make the numerator zero, then the line x = a is a vertical asymptote.
We will also look how the function behaves as x increases or decreases without bound.
We first construct a table of values for f(x) as x increases without bound, or in symbols, as x +.
iii. Table of values for f(x) as
5 10 100 1,000 10,000 As
0.43 0.67 0.96 0.9960 0.99960 f(x) approaches 1 – Next, construct a table of values for f(x) as x decreases without bound, or in symbols, as x –.iv. Table of values for f(x) as
–5 –10 –100 –1,000 –10,000 As
2.33 1.41 1.041 1.00401 1.0004001 f(x) approaches 1+ Plotting the points according to these on the Cartesian Plane:Figure 2.4: Note that the axes do not have the same scale.
Observe that as x increases or decreases without bound, f(x) gets closer and closer to 1. The line y= 1 is indicated in the figure with a dashed line.
We call this line a horizontal asymptote, formally defined as follows:
D E P E D C O P Y
Definition.
Definition. The horizontal line y = b is a horizontal asymptotehorizontal asymptote of the function f if f(x) gets closer to b as x increases or decreases without bound (x + or x
–.).
A rational function may or may not cross its horizontal asymptote. If the function does not cross the horizontal asymptote y = b, then b is not part of the range of the rational function.
Now that we know the behavior of the function as x approaches –2 (where the function is not defined), and also as x + or x –, we can complete the sketch of the graph by looking at the behavior of the graph at the zeroes.
Construct a table of signs to determine the sign of the function on the intervals determined by the zeroes and the intercepts. Refer to the lesson on rational inequalities for the steps in constructing a table of signs:
Interval
Test point
Test with the
rational function
The boundary between the intervals -2 < x < 2 and x >2 is a zero. Since the function is positive on the left of 2 and negative on the right, the function transitions from positive to negative at x = 2.
Plot the zeroes, y-intercept, and the asymptotes. From the table of signs and the previous graphs, we know that f(x)<1 as
. Draw a short segment across (2,0) to indicate that the function transitions from negative to positive at this point.We also know that f(x) increases without bound as
and f(x) decreases without bound as as
. Sketch some arrows near the asymptote to indicate this information.Figure 2.5 Zeroes and asymptotes of f(x).
D E P E D C O P Y
Trace the arrowheads along with the intercepts using smooth curves. Do not cross the vertical asymptote.
Figure 2.6 Tracing with smooth curves
Figure 2.7 The actual sketch of the graph of
for reference.(d) From the graph of the rational function, we observe that the function does not cross the horizontal asymptote. We also observe that the function increases and decreases without bound, and is asymptotic to the line y = 1. Therefore only the value 1 is not included in the range of f(x).
The rangerange of f(x) is
.Example 2.
Example 2. Find the horizontal asymptote of
.
Solution . We have seen from the previous example that the horizontal asymptotes can be determined by looking at the behavior of rational functions when |x| is very large (i.e., at extreme values of x).
D E P E D C O P Y
However, at extreme values of x, the value of a polynomial can be approximated using the value of the leading term.
For example, if x=1000, the value of
is 4,004,001. A good approximation is the value of
, which is 4,000,000.Similarly, for extreme values of
, the value of
can be approximated by
. Thus, for extreme values of
, then
can be approximated by
, andtherefore
approaches 4 for extreme values of
.This means that we have a horizontal asymptotehorizontal asymptote at y=4.
Example 3.
Example 3. Find the horizontal asymptote of
.Solution. Following the idea from the previous example, the value of
canbe approximated by
for extreme values of
.Thus, the horizontal asymptotehorizontal asymptote is
.
Example 4.
Example 4. Find the horizontal asymptote of
.Solution. Again, based on the idea from the previous example, the value of
can be approximated by
for extreme values of
.If we substitute extreme values of
in
, we obtain values very close to 0.Thus, the horizontal asymptotehorizontal asymptote is y=0.
Example 5.
Example 5. Show that
can be approximated by
.If we substitute extreme values of
in ,
we obtain extreme values as well.Thus, if
takes on extreme values, then y also takes on extreme values and does not approach a particular finite number. The function has no horizontal asymptoteno horizontal asymptote.D E P E D C O P Y
We summarize the results from the previous examples as follows:
Finding the Horizontal Asymptotes of a Rational Function Finding the Horizontal Asymptotes of a Rational Function
Let
be the degree of the numerator and
be the degree of the denominator. If
, the horizontal asymptotehorizontal asymptote is
. If
, the horizontal asymptotehorizontal asymptote is
, where
is the leading coefficient of the numerator and
is the leading coefficient of the denominator.
If
, there is no horizontal asymptotehorizontal asymptote.Properties of rational functions:
How
How to to find find the: the: Do Do the the following:following:
y-intercept Evaluate the function at x = 0.
x-intercept Find the values of x where the numerator will be zero.
Vertical asymptotes
Find the values of a where the denominator is zero. If this value of a does not make the numerator zero, then the line x = a is a vertical asymptote.
Horizontal asymptotes
Let n be the degree of the numerator and m the degree of the denominator
If n < m, the horizontal asymptote is y = 0.
If n = m, the horizontal asymptote is
,where aa is the leading coefficient of the numerator and bb is the leading coefficient of the denominator.
If n > m, there is no horizontal asymptote.
Example 6.
Example 6. Sketch the graph of
. Find its domain and range.Solution. The numerator and denominator of f(x) can be factored as follows:
D E P E D C O P Y
From the factorization, we can get the following properties of the function:
y-intercept:
zeroes:
vertical asymptotes:
and
horizontal asymptote: The polynomials in the numerator and denominator have equal degree. The horizontal asymptote is the ratio of the leading coefficients:
Plot the intercepts and asymptotes on the Cartesian plane:
Figure 2.8: Intercepts and asymptotes of f(x).
Construct a table of signs for the following intervals defined by the zeroes and the values where the denominator will be zero:
D E P E D C O P Y
Interval
Test point
3x + 1 – – + + +
x – 3 – – – – +
2x – 1 – – – + +
x + 4 – + + + +
abovex-axis++ belowx-axis – – abovex-axis++ belowx-axis – – Abovex-axis++Draw sections of the graph through the zeroes indicating the correct transition based on the table of signs.
Figure 2.9: Sketch the transitions across the zeroes based on the table of signs
D E P E D C O P Y
Draw sections of the graph near the asymptotes based on the transition indicated on the table of signs.
Figure 2.10: Sketch the graph near the asymptotes based on the table of signs.
Complete the sketch by connecting the arrowheads, making sure that the sketch passes through the y-intercept as well. The sketch should follow the horizontal asymptote as the x-values goes to the extreme left and right of the Cartesian plane.
Figure 2.11: Rough sketch of the graph following the information above.
D E P E D C O P Y
Figure 2.12: Actual sketch of the graph using a software grapher.
The domain of the function is all values of x not including those where the function is undefined. Therefore the domain of f(x) is
From the graph of the function, we observe that the function increases and decreases without bound. The graph also crosses the horizontal asymptote.
Therefore the range of the function is the set
of all real numbers.Solved Examples Solved Examples
1. Let
(a) Find its domain, (b) intercepts, (c) asymptotes. Next, (d) sketch its graph and (e) determine its range.Solution ..