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Learning Objectives
- Analyze a function and its derivatives to draw its graph.
We have shown how to use the first and second derivatives of a function to describe the shape of a graph. Now we put everything together with other features to graph a function .
Guidelines for Drawing the Graph of a Function
We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.
Problem-Solving Strategy: Drawing the Graph of a Function
Given a function use the following steps to sketch a graph of
- Determine the domain of the function.
- Locate the – and -intercepts.
- Evaluate and to determine horizontal or oblique asymptote.
- Determine whether has any vertical asymptotes.
- Calculate Find all critical numbers and determine the intervals where is increasing and where is decreasing. Determine whether has any local extrema.
- Calculate Determine the intervals where is concave up and where is concave down. Use this information to determine whether has any inflection points. The second derivative can also be used as an alternate means to determine or verify that has a local extremum at a critical number.
Now let’s use this strategy to graph several different functions. We start by graphing a polynomial function.
Sketching a Graph of a Polynomial
Sketch a graph of
Solution
Step 1. Since is a polynomial, the domain is the set of all real numbers.
Step 2. When Therefore, the -intercept is To find the -intercepts, we need to solve the equation gives us the -intercepts and
Step 3. We need to evaluate the end behavior of As and Therefore, As and Therefore,
Step 4. Since is a polynomial function, it does not have any vertical asymptotes.
Step 5. The first derivative of is
Therefore, has two critical numbers: Divide the interval into the three smaller intervals: and Then, choose test points and from these intervals and evaluate the sign of at each of these test points, as shown in the following table.
Interval | Test Point | Sign of Derivative | Conclusion |
---|---|---|---|
is increasing. | |||
is decreasing. | |||
is increasing. |
From the table, we see that has a local maximum at and a local minimum at Evaluating at those two points, we find that the local maximum value is and the local minimum value is
Step 6. The second derivative of is
The second derivative is zero at Therefore, to determine the concavity of divide the interval into the smaller intervals and and choose test points and to determine the concavity of on each of these smaller intervals as shown in the following table.
Interval | Test Point | Sign of | Conclusion |
---|---|---|---|
is concave down. | |||
is concave up. |
We note that the information in the preceding table confirms the fact, found in step 5, that has a local maximum at and a local minimum at In addition, the information found in step 5—namely, has a local maximum at and a local minimum at and at those points—combined with the fact that changes sign only at confirms the results found in step 6 on the concavity of
Combining this information, we arrive at the graph of shown in the following graph.
Sketch a graph of
Solution
Sketching a Rational Function
Sketch the graph of
Solution
Step 1. The function is defined as long as the denominator is not zero. Therefore, the domain is the set of all real numbers except
Step 2. Find the intercepts. If then so 0 is an intercept. If then which implies Therefore, is the only intercept.
Step 3. Evaluate the limits at infinity. Since is a rational function, divide the numerator and denominator by the highest power in the denominator: We obtain
Therefore, has a horizontal asymptote of as and
Step 4. To determine whether has any vertical asymptotes, first check to see whether the denominator has any zeroes. We find the denominator is zero when To determine whether the lines or are vertical asymptotes of evaluate and By looking at each one-sided limit as we see that
In addition, by looking at each one-sided limit as we find that
Step 5. Calculate the first derivative:
Critical numbers occur at points where or is undefined. We see that when The derivative is not undefined at any point in the domain of However, are not in the domain of Therefore, to determine where is increasing and where is decreasing, divide the interval into four smaller intervals: and and choose a test point in each interval to determine the sign of in each of these intervals. The values and are good choices for test points as shown in the following table.
Interval | Test Point | Sign of | Conclusion |
---|---|---|---|
is decreasing. | |||
is decreasing. | |||
is increasing. | |||
is increasing. |
From this analysis, we conclude that has a local minimum at but no local maximum.
Step 6. Calculate the second derivative:
To determine the intervals where is concave up and where is concave down, we first need to find all points where or is undefined. Since the numerator for any is never zero. Furthermore, is not undefined for any in the domain of However, as discussed earlier, are not in the domain of Therefore, to determine the concavity of we divide the interval into the three smaller intervals and and choose a test point in each of these intervals to evaluate the sign of in each of these intervals. The values and are possible test points as shown in the following table.
Interval | Test Point | Sign of | Conclusion |
---|---|---|---|
is concave down. | |||
is concave up. | |||
is concave down. |
Combining all this information, we arrive at the graph of shown below. Note that, although changes concavity at and there are no inflection points at either of these places because is not continuous at or
Sketch a graph of
Hint
A line is a horizontal asymptote of if the limit as or the limit as of is A line is a vertical asymptote if at least one of the one-sided limits of as is or
Solution
Sketching a Rational Function with an Oblique Asymptote
Sketch the graph of
Solution
Step 1. The domain of is the set of all real numbers except
Step 2. Find the intercepts. We can see that when so is the only intercept.
Step 3. Evaluate the limits at infinity. Since the degree of the numerator is one more than the degree of the denominator, must have an oblique asymptote. To find the oblique asymptote, use long division of polynomials to write
Since as approaches the line as The line is an oblique asymptote for
Step 4. To check for vertical asymptotes, look at where the denominator is zero. Here the denominator is zero at Looking at both one-sided limits as we find
Therefore, is a vertical asymptote, and we have determined the behavior of as approaches 1 from the right and the left.
Step 5. Calculate the first derivative:
We have when Therefore, and are critical numbers. Since is undefined at we need to divide the interval into the smaller intervals and and choose a test point from each interval to evaluate the sign of in each of these smaller intervals. For example, let and be the test points as shown in the following table.
Interval | Test Point | Sign of | Conclusion |
---|---|---|---|
is increasing. | |||
is decreasing. | |||
is decreasing. | |||
is increasing. |
From this table, we see that has a local maximum at and a local minimum at The value of at the local maximum is and the value of at the local minimum is Therefore, and are important points on the graph.
Step 6. Calculate the second derivative:
We see that is never zero or undefined for in the domain of Since is undefined at to check concavity we just divide the interval into the two smaller intervals and and choose a test point from each interval to evaluate the sign of in each of these intervals. The values and are possible test points as shown in the following table.
Interval | Test Point | Sign of | Conclusion |
---|---|---|---|
is concave down. | |||
is concave up. |
From the information gathered, we arrive at the following graph for
Find the oblique asymptote for
Hint
Use long division of polynomials.
Solution
Sketching the Graph of a Function with a Cusp
Sketch a graph of
Solution
Step 1. Since the cube-root function is defined for all real numbers and the domain of is all real numbers.
Step 2: To find the -intercept, evaluate Since the -intercept is To find the -intercept, solve The solution of this equation is so the -intercept is
Step 3: Since the function continues to grow without bound as and
Step 4: The function has no vertical asymptotes.
Step 5: To determine where is increasing or decreasing, calculate We find
This function is not zero anywhere, but it is undefined when Therefore, the only critical number is Divide the interval into the smaller intervals and and choose test points in each of these intervals to determine the sign of in each of these smaller intervals. Let and be the test points as shown in the following table.
Interval | Test Point | Sign of | Conclusion |
---|---|---|---|
is decreasing. | |||
is increasing. |
We conclude that has a local minimum at Evaluating at we find that the value of at the local minimum is zero. Note that is undefined, so to determine the behavior of the function at this critical number, we need to examine Looking at the one-sided limits, we have
Therefore, has a cusp at
Step 6: To determine concavity, we calculate the second derivative of
We find that is defined for all but is undefined when Therefore, divide the interval into the smaller intervals and and choose test points to evaluate the sign of in each of these intervals. As we did earlier, let and be test points as shown in the following table.
Interval | Test Point | Sign of | Conclusion |
---|---|---|---|
is concave down. | |||
is concave down. |
From this table, we conclude that is concave down everywhere. Combining all of this information, we arrive at the following graph for
Consider the function Determine the point on the graph where a cusp is located. Determine the end behavior of
Hint
A function has a cusp at a point if exists, is undefined, one of the one-sided limits as of is and the other one-sided limit is
Solution
The function has a cusp at For end behavior,
For the following exercises, sketch the function by finding the following:
- Determine the domain of the function.
- Determine the – and -intercepts.
- Determine any horizontal or vertical asymptotes.
- Determine the intervals where the function is increasing and where the function is decreasing. Determine whether the function has any local extrema.
- Determine the intervals where the function is concave up and where the function is concave down.
- Determine all inflection points (if any).
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For the following exercises, sketch the graph of by finding the following:
- Determine the domain of the function.
- Determine the – and -intercepts.
- Determine any horizontal or vertical asymptotes.
- Determine the intervals where is increasing and where is decreasing. Determine whether has any local extrema.
- Determine the intervals where is concave up and where is concave down.
- Determine all inflection points (if any).
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