405

5.7 SKETCHING GRAPHS OF RATIONAL FUNCTIONS 405

Suppose that the percentage of people using a new computer model t years after its introduction to the market is modelled by p(t) , where p is the percentage of people and is expressed as a percent. After how long does thegrowth rate in the percentage of people stop increasing and begin to decrease?Why might the answer to this question be useful to the company that markets thenew computer model?

To answer the first question, a specific point on the graph of the function mustbe determined.

A graphing calculator will help estimate this point, but using calculus willgive an exact answer. In this section, you will revisit the calculus techniques inChapter 4 used to analyze polynomial functions. You will apply these techniquesto analyze rational functions.

Sketching Graphs of Rational Functions

You have sketched the graphs of rational functions by first finding the domain,intercepts, and asymptotes. You have also learned how to find the extreme valuesof a function and the intervals where the function increases and decreases. Youcan also determine the concavity of a graph and how to find the points ofinflection.

Using all of this information, you can sketch the complete graph of a rationalfunction without technology.

Analyzing the Graph of a Rational Function

Without graphing technology, graph f (x) .

Solution

Use the equation of the function to find the domain, the intercepts, and theasymptotes.

DomainThere are no restrictions on the domain. The domain is the set of all realnumbers, {x x R}.Intercepts The zero of the function occurs when the numerator is 0, so the function has

one zero at x 0. The x-intercept is 0. f (0) 0, so the y-intercept is 0.

5xx2 1

100t2t2 500

5.7 Sketching Graphs of RationalFunctions

SETTING THE STAGE

EXAMINING THE CONCEPT

Example 1

1. Analyze f (x).

Asymptotes There are no vertical asymptotes since the denominator can never be equal

to 0.

limx

limx

1 0

0 0, so y 0 (the x-axis) is a

horizontal asymptote on the right, and

limx

limx

1 0

0 0, so y 0 is a horizontal

asymptote on the left. There are no oblique asymptotes, since the highest power of x in the

numerator is not one more than the highest power of x in the denominator.

SymmetryWill the graph of the function be symmetric? Because of the x2 in thedenominator, the denominator will always be positive. Therefore, the sign off (x) is determined by the sign of the numerator. In fact, f (x) f (x). So the graph will be symmetric about the origin. The shape of the graph on the left of the origin will be the same as that on the right of the origin. But one side will appear to be upside down.

The graph will approach the x-axis from above on the right and from belowon the left.

Critical NumbersUse f (x) to find any critical numbers.Apply the quotient rule to find f (x).

f (x)

f (x) is defined for all real x-values.f (x) 0 when the numerator is 0. x 1, so 1 and 1 are critical numbers.Intervals of Increase and Decrease

In this case, the sign of f (x) is completely determined by the numerator, theproduct (1 x)(1 x), since the denominator (x2 1)2 is always positive.

5(1 x)(1 x)(x2 1)2

5(1 x2)(x2 1)2

5(x2 1) (2x)(5x)(x2 1)2

5x

x2

x

x

22 x

12

5xx2 1

5x

x2

x

x

22 x

12

5xx2 1

406 CHAPTER 5 RATES OF CHANGE IN RATIONAL FUNCTION MODELS

At a critical number c, f (c) = 0 or f (c) does notexist.

The function isincreasing on the intervala < x < b if f (x) > 0 forall x in that interval.

The function isdecreasing on theinterval a < x < b if f (x) < 0 for all x in thatinterval.

Intervals

x < 1 1 < x < 1 x > 1

1 x + +

1 + x + +

y (+)() = (+)(+) = + ()(+) =

y decreasing increasing decreasing

2. Analyze f (x).

f is decreasing when x < 1, increasing on 1 < x < 1, and decreasing when x > 1.

Local Maximum and Minimum ValuesFrom the above table, you can see that f (x) changes from negative to positive atx 1, which means the graph has a local minimum at f (1) 2.5. f (1) 2.5The sign of f (x) changes from positive to negative at x 1, which means thegraph has a local maximum at f (1) 2.5.Concavity and Points of InflectionNow determine f . First expand f . f (x)

f (x) Use the quotient rule.

Simplify.

Factor.

Simplify.

In this case, f (x) is defined for all real values of x.f (x) 0 when the numerator equals 0. Let 10x 0 or x2 3 0. Thenx 0 or x 3. Points of inflection may occur at x 0 and x 3.

Create a table to analyze the second derivative on the four intervals defined bythese x-values.

The graph is concave up on 3 < x < 0 and on x > 3.

The graph is concave down on x < 3 and on 0 < x < 3.

10x(x2 3)(x2 1)3

10x(x2 3)(x2 1)(x2 1)4

10x(x4 2x2 3)(x2 1)4

10x(x4 2x2 2 2x4)[(x2 1)2]2

10x(x4 2x2 1) (20x)(1 x4)(x4 2x2 1)2

10x(x4 2x2 1) (5 5x2)(4x3 4x)(x4 2x2 1)2

5 5x2x4 2x2 1

5(1)(1)2 1


f (c) is a local minimumvalue if f (x) changesfrom negative to positiveat c.

f (c) is a local maximumvalue if f (x) changesfrom positive to negativeat c.

The graph of thefunction is concave upwhen f (x) is increasingi.e. when f (x) > 0.

The graph of thefunction is concavedown when f (x) isdecreasing i.e. whenf (x) < 0.

At a point of inflectionthe graph changes fromconcave up to concavedown or vice versa. f (c) 0 or f (c) doesnot exist.

Intervals

x < 3 3 < x < 0 0 < x < 3 x > 3

10x + +

x2 3 + +

(x2 + 1)3 + + + +

f (x) (

()+(+)

) =

(()+())

= + (+

(+)(

))

= (+

()+()+) = +

f (x) concave down concave up concave down concave up

3. Analyze f (x).

The graph changes from concave down to concave up at x 3. The graph changes from concave up to concave down at x 0. f 3 , or about 2.2, and f (0) 0, so the estimated points of inflection are (1.7, 2.2),(0, 0), and (1.7, 2.2). Putting all the information together, graph f (x) .

To sketch the graph of a rational function, follow these steps:

Apply these steps in the next example.

Curve Sketching

Without graphing technology, sketch the graph of f (x) .

Solution

In factored form, f (x) . The domain of the function is {x x 4, x R }. The x-intercepts are 0 and 3. The y-intercept is 0. (The graph passes through

the origin.)

x(x 3)

x 4

x2 3x

x 4

5xx2 1

534


2 4 6 82468

4

2

0

4

2

f(x)

x

maximum point(1, 2.5)

minimum point(1, 2.5)

inflection pointat (1.7, 2.2)

inflection pointat (1.7, 2.2)

x- and y-interceptand inflection point (0, 0)

horizontal asymptotegraph approachesx-axis from below

graph approachesx-axis from above

Sketching the Graph of a Rational Function

1. Use the function to determine the domain and any discontinuities determine the intercepts find any asymptotes

2. Use the first derivative to find the critical numbers determine where the function is increasing and where it is decreasing identify any local maxima or minima

3. Use the second derivative to determine where the graph is concave up and where it is concave down find any points of inflection

4. Calculate the values of y corresponding to critical points and points ofinflection. Use the information from steps 1 to 3 to sketch the graph.

Example 2

1. Analyze f (x).

4. Sketch the graph.

Let the denominator equal 0. The line x 4 is a vertical asymptote.f (3.999) 3995.001. So, as x 4, f (x) .f (4.001) 4005.01. So, as x 4, f (x) .As x , f (x) , and there is no horizontal asymptote.Since the highest power of x in the numerator is exactly one more than thehighest power of x in the denominator, the graph of the function will have an oblique asymptote. By long division, f (x) x 1

x

44. The line

y x 1 is an oblique asymptote.

f (x)

(in factored form) f (x) is defined for all values of x in the domain of the function. f (x) 0 when x 2 and when x 6, so 2 and 6 are critical numbers. For intervals of increase and decrease, note that (x 4)2 is always positive,

so consider only factors of the numerator. Also recall that f (4) does not exist.

The function is increasing when x < 2 and when x > 6. The function isdecreasing when 2 < x < 4 and when 4 < x < 6.

The function has a local maximum at f (2) 1 and a local minimum at f (6) 9.

f (x) , and x 4 f (x) is defined for all values of the domain of the original function. f (x) 0 for any value of x. The sign of the factor (x 4) in the denominator determines the sign of

f (x), since the numerator is always positive.

8(x 4)3

8(x 4)(x 4)4

(x 2)(x 6)(x 4)2

x2 8x 12(x 4)2

(2x 3)(x 4) (x2 3x)(1)(x 4)2


Intervals

x < 2 2 < x < 4 4 < x < 6 x > 6

x 2 + + +

x 6 +

f (x) = ()() = + (+)() = (+)() = (+)(+) = +

f (x) increasing decreasing decreasing increasing

maximum at x = 2 minimum at x = 6

(x 2)(x 6)

(x 4)2

3. Analyze f (x).

2. Analyze f (x).

Intervals x < 4 x > 4

x 4 +

f (x) = (x 8

4)3

((+))

= ((++

))

= +

f (x) concave down concave up

The graph is concave up on x < 4 and concave down on x > 4. There are no points of inflection. Since x 4 is not part of the domain of

f , the line x 4 is a vertical asymptote.

For applications, you must often consider one or more specific points on thegraph of a function.

Examining Growth Rate

Recall the problem in Setting the Stage:Suppose that the percentage of people using a new computer model t years

after its introduction to the market is modelled by p(t) , where p is the percentage of people and p is expressed as a percent. After how long doesthe growth rate in the percentage of people stop increasing and begin todecrease? Why might the answer to this question be useful to the company thatmarkets the new computer model?

Solution

The growth rate in the percentage of people is measured by p(t). The growth rate is increasing when p(t) is increasing and p(t), is positive.The growth rate is decreasing when p(t) is decreasing and p(t) is negative.Find the point of inflection at which p(t) changes from positive to negative.Check the domain and intercepts. Find any asymptotes. p(t) and p(t) will notbe defined at a vertical asymptote, and the curve may have different concavityon each side of such an asymptote. Since t represents time, the domain is the set of all real numbers greater than

or equal to 0. The x- and y-intercepts are both 0. (The graph passes through the origin.) There are no vertical or oblique asymptotes. The line y 100 is a horizontal

asymptote on the right. This asymptote makes sense since p(t) represents apercent.

100t2t2 500


4 10 16410

4

10

0

4

f(x)

x

local maximum at (2, 1)

local minimum at (6, 9)

vertical asymptote x = 4

oblique asymptotey = x + 1

Example 3

1. Analyze p(t).

4. Sketch the graph.

p(t) and is defined for all values of tin the domain of p.p(t) 0 when t 0. The growth rate in the percentage of people starts at 0.To determine p(t), first expand p(t).

p(t)

p(t)

p(t) is defined for all values of t in the domain of p(t).p(t) 0 when t 5030 or about 12.9. Ignore the negative root, since t 0.The factor (500 3t2) determines the sign of p(t), since the denominatorwill always be positive.

The inflection point occurs at t 5030. The graph changes from concave up to concave down. The growth rate in the percent of people stops increasingand starts decreasing after about 12 years and 11 months.

You could also trace, using a graphing calculator, to find the estimated pointat which the graph changes from concave up to concave down. However, usethe second derivative to find the exact value.

The company that makes the new computer model may need to adjustproduction levels as the market declines. The company should also developnew products to make and sell to keep their plants busy.

You can use the features of the graph of a function to graph its derivativefunctions.

100 000(500 3t2)(t2 500)3

100 000(500 3t2)(500 t2)(t2 500)4

100 000(250 000 1000t2 3t4)(t2 500)4

300 000t4 100 000 000t2 25 000 000 000[(t2 500)2]2

100 000(t4 1000t2 250 000) (4t3 2000t)100 000t[(t2 500)2]2

100 000t(t2 500)2

100 000t(t2 500)2

200t(t2 500) 100t2(2t)(t2 500)2


Intervals

0 < t < 5030 t > 5030500 3t 2 +

p(t ) = = + =

p(t ) concave up concave down

(+)()

(+)(+)(+)

(+)100 000(500 3t 2)

(t 2 + 500)3

3. Analyze p(t).

2. Analyze p(t).

Using the Graph of a Function to Graph Its First and Second Derivatives

The graph of f (x) is shown. Use this graph to sketch the graphs of f (x) and f (x).

Solution

Draw axes for the graphs of f (x) and f (x) beneath the graph of f (x).Examine the graph of f (x).Asymptotes: Since the y-axis is a vertical asymptote, f (x) and f (x) are undefined at

x 0. f (x) has a discontinuity at x 0, so the graphs of f (x) and f (x) willhave vertical asymptotes.

Critical Points: The graph of f (x) has a local minimum at point (1, 3) ( f (1) 3), so

f (1) 0 and f (x) will be positive. The graph of f (x) has a point of inflection at a point on the x-axis between 2

and 1, so f (x) 0 at this value. Since the graph of f (x) changes fromconcave up to concave down at this x-value, the sign of f (x) also changesfrom positive to negative here.

Also, since f (x) is 0 at this x-value, the graph of f (x) will have a localmaximum or minimum, and since the sign of f (x) changes from positive tonegative, it must be a local maximum point.

Slope and Intervals of Increase and Decrease: f (x) is decreasing for x < 0 and for 0 < x < 1, so f (x) will be negative for x on

these intervals. f (x) is increasing for all x > 1, so f (x) will be positive for x > 1.As x 0 and as x 0, the slope .

Concavity:The graph of f is concave up for all x > 0, so f is positive for all x > 0. Sketch the graphs of f and f using all of this information.

Sometimes you may have data about the derivative of a function rather than dataabout the function itself. What can you learn about a functions graph from thegraph of its derivative?

Sketching the Graph of a Function from the Graph ofIts Derivative Function

Here is the graph of f (x), the derivative of f (x), for 4 < x < 4.(a) For what x-values between 4 and 4 is f (x) increasing? decreasing?(b) At what x-values does the graph of f (x) have local maximum or minimum

points? Justify your answers.(c) For what values of x is the graph of f (x) concave up? concave down? Explain.(d) Sketch a possible graph for f (x) if f (0) 0.


Example 4

0

v

x

v = f(x)

44

8

8

0

y

xy = f'(x)

44

8

8

0

y

x

y = f ''(x)

44

8

8

Example 5

43214 3 2 1

4

3

2

1

0

y

x

Solution

(a) f (x) > 0 if f (x) is increasing. From the graph of f (x), f (x) > 0 on 1 < x < 1.f (x) < 0 if f (x) is decreasing. From the graph of f (x), f (x) < 0 on 4 < x < 1 and on 1 < x < 4.

(b) At a local maximum or minimum, f (x) 0. From the graph, the zeros forf (x) occur at x 1.At x 1, f (x) changes from negative to positive, so the graph of f (x) willhave a local minimum point at x 1.At x 1, f (x) changes from positive to negative, so the graph of f (x) willhave a local maximum point at x 1.

(c) f (x) is positive and f (x) is increasing where the graph of f (x) is concaveup. From the graph, f (x) is increasing between x 1.7 and x 0 andagain between x 1.7 and x 4. The graph of f will be concave up on1.7 < x < 0 and on 1.7 < x < 4.f (x) is negative and f (x) is decreasing where the graph of f (x) is concavedown. From the graph of f (x), the graph of f (x) will be concave down on4 < x < 1.7 and on 0 < x < 1.7 .

(d) Now sketch a possible graph for f (x).Since f (x) exists for all x-values between 4 and 4, f (x) must becontinuous in this interval.The value of the slope at different x-values can be read directly from thegraph of f (x). For example, the slope of f (x) is 4 when x 0. Also, as xapproaches 4 or 4, the slope is negative and close to 0.Given that f (0) 0, a possible graph for f (x) is shown on the left.

CHECK, CONSOLIDATE, COMMUNICATE

1. Explain why you might not use a graph created with graphing technologyto correctly answer an application problem.

2. What information can you get from f (x), f (x), and f (x) to help sketchthe graph of f (x)?

3. How can you sketch the graph of a function from the graph of itsderivative?

KEY IDEAS

The first and second derivatives of a rational function give informationabout the shape of the functions graph.

The number c is a critical number if f (c) exists, that is, c is in the domain of the function and f (c) 0 or f (c) does not exist.

The function is increasing for all x-values in an interval if f (x) > 0 on thatinterval.


43214 3 2 1

2

1

1

2

0

y

x

Questions 1 to 7 refer to the following functions:(a) f (x) (b) f (x) (c) f (x)

x

x

2 (d) f (x) x

35

x

35

1. Determine the domain and intercepts for each function.2. Find the equations of all asymptotes for each function.3. Find the first derivative. Determine the critical numbers for each function.4. Find the intervals on which the functions are

i. increasing ii. decreasing5. Find the second derivative. Determine the interval(s) on which the graph of

each function is i. concave up ii. concave down

6. Identify all local maximum and minimum points and any points ofinflection for the graph of each function.

7. Use your results from questions 1 to 6 to sketch the graph of each function. 8. Verify the sketches you created for question 7 using graphing technology.

Are any of the graphs symmetric about the y-axis or about the origin? How could you use the function equation to determine symmetry?

2xx2 1

x 1

x2

5.7 Exercises

A

The function is decreasing for all x-values in an interval if f (x) < 0 onthat interval.

Let c be a critical number for a continuous function. f (c) is a local maximum if the sign of f (x) changes from positive to

negative at c. f (c) is a local minimum if the sign of f (x) changes from negative to

positive at c. The graph of the function is concave up when f (x) is increasing and when

f (x) > 0. The graph of the function is concave down when f (x) is decreasing and

when f (x) < 0. A point of inflection is a point at which the graph changes from concave

up to concave down or vice versa. At a point of inflection, f (x) 0 or f (x) does not exist. Apply the steps for analyzing the features of any function to help sketch

its graph. These steps are introduced on page 408.


9. Knowledge and Understanding: For f (x) ,

(a) show that f (x) 1 and f (x) (b) without graphing technology, use the information provided by the

original function and its first and second derivatives to sketch the graphof the original function

10. For f (x) , show that f (x) and f (x) . Use the function and its derivatives to determine the domain, intercepts,asymptotes, intervals of increase and decrease, and concavity, and to locateany critical points and points of inflection. Then use this information tosketch the graph of the original function.

11. For f (x) , show that f (x) and f (x) . Follow the steps for analyzing the graph of a function to sketch f (x).

12. Determine the first and second derivatives for each function. Then analyzeeach function and sketch its graph.

(a) f (x) x 9x

(b) f (x) (c) f (x) (d) f (x) (e) f (x) x

x

12 (f) f (x) x

(g) f (x) (h) f (x) 4x2

x

3 (i) f (x) (j) f (x) 3

3

x

x (k) f (x) (l) f (x)

13. The concentration, c, of a certain drug in the bloodstream t hours after it is taken orally is modelled by c(t) . Analyze the function and sketch its graph. After how long does the concentration begin to decrease?How does the rate of change of the concentration vary over time?

14. The position, s, of a particle moving along a straight line at t seconds is described by s(t) , where 0 t 10. (a) Analyze the function and sketch its graph.(b) After what time does the particle stop speeding up and start to slow down?(c) When is the velocity greatest?(d) Sketch the graph of the velocity function.

15. Application: The population, p, of frogs in a newly created conservation area is modelled by p(t) 50 , where t is the time in years since the opening of the area. Analyze the model. According to this model,(a) how many frogs will populate this area in the long run?(b) when is the frog population increasing most rapidly?Sketch the graph of the frog population.

2500t225 t2

5t210 t2

5t2t2 5

x2 xx 1

xx2 2

x2 1x2 1

x2x 1

13x3

1x2 4x

x(x 2)2

x2 1

x3

2(3x2 1)(1 x2)3

2x(1 x2)2

11 x2

8(x 3)

x44(2 x)

x34(x 1)

x2

18x3

9x2

x2 3x 9

x

5.7 SKETCHING GRAPHS OF RATIONAL FUNCTIONS 4155.7 SKETCHING GRAPHS OF RATIONAL FUNCTIONS 415

B

16. Let R(x) be a revenue function. R(x) is the marginal revenue. Its value at any point is a measure of the estimated additional revenue from selling onemore item. Suppose the demand equation for a certain product is p(x) , where x is the number of items sold in thousands, and p is theprice in dollars. When does the marginal revenue reach its lowest value? What feature of the graph of the revenue function occurs at this point?

17. Given the following results of the analysis of a function, sketch a possiblegraph for the function:(a) f (0) 0, the horizontal asymptote is y 2, the vertical asymptote is

x 3, f (x) < 0 and f (x) < 0 for x < 3; f (x) < 0 and f (x) > 0 for x > 3.(b) f (0) 6, the horizontal asymptote is y 2, the vertical asymptote

is x 4; f (x) > 0 and f (x) > 0 for x < 4; f (x) > 0 and f (x) < 0 for x > 4.

18. Use the features of each functions graph to sketch the graphs of its first andsecond derivatives.(a) (b)

19. A functions derivative is shown in each graph. Use the graph to sketch apossible graph for the original function.(a) (b)

20. Communication: The graphs of a function and its derivatives, y f (x),y f (x), and y f (x), are shown in each graph. Which is which? Explainhow you can tell.(a) (b)

51 x2


33

2

0

2

y

xy = f (x)

0

y

x44

4

4

y = g(x)

55

2

0

2

y

x

A

B

C

55

4

0

4

y

x

EF

D

424 2

2

2

0

y

xy = f (x)

84

4

0

4

8

y

x

y = g(x)

21. Decide whether each statement is true or false. Explain and give examples tosupport your decisions.(a) Function f (x) has a local maximum or a local minimum when x 3 if

f (3) 0.(b) The graph of a rational function has a horizontal asymptote.(c) The graph of f (x) has a point of inflection when x 2 if f (2) 0.

22. Check Your Understanding: Without graphing technology, sketch the graph of f (x) using information about the function and its derivatives. Explain each step in the analysis.

23. The graph of the function f (x) ax2 bx has a horizontal tangent at point

(1, 3). Find a and b, and show that f (x) has a local minimum value at x 1.24. If a continuous function has a local maximum value and a local minimum

value, then the graph of the function must have a point of inflection betweenthese two extrema. Do you agree with this comment? Explain withexamples.

25. Create a rational function so that, in the graph of this function,(a) the y-axis and the line y 1 are asymptotes(b) the x-intercept is at x 1

26. Create a rational function for which i. the graph passes through (0, 4)

ii. limx

f (x) 0iii. lim

x a

f (x) , for any value of a in the domain

27. Thinking, Inquiry, Problem Solving

(a) Sketch the graph of a function that satisfies these conditions.i. f (x) < 0 and f (x) < 0 for x < 3 and for 3 < x < 6

ii. f (x) > 0 and f (x) < 0 for 3 < x < 3iii. f (x) < 0 and f (x) > 0 for x > 6iv. lim

x f (x) 0, lim

x 3f (x) , lim

x 3f (x) , and

limx

f (x) 0v. f (0) 0

(b) Label all the features of the graph.

x2 3xx 1


C

ADDITIONAL ACHIEVEMENT CHART QUESTIONS

Knowledge and Understanding: Let f (x) .(a) Determine the first and second derivatives; the domain, intercepts, and

asymptotes; intervals of increase and decrease and any critical points; andconcavity and points of inflection.

(b) Use the information from (a) to sketch f (x). Check your work with graphingtechnology.

Application: If a polio epidemic broke out in a small community with novaccine, the fraction of the population that would be infected after t months could be modelled by f (t) , 0 t 5. Graph the function, showing all essential features. When is the largest fraction of people infected? When is theepidemic spreading most rapidly? Show the results on your graph.

Thinking, Inquiry, Problem Solving: Graph a function f (x) for which lim

x 0f (x) ; for x < 0, f (x) 1 and f (x) 0; f (1) 0; for x > 0, the

only point at which f (x) 0 is (2, 2); f (3) 0; and limx

f (x) 1.

Communication: The growth in sales of a new product often follows a life-cyclecurve like the one shown. Describe the growth pattern. When will the salesexperience the most rapid growth?

64t2(4 t)4

3x2 4x 4

x2


1000

1500

2000

500

x100 20 305 15 25 35

y

The Chapter ProblemDesigning a Settling Pond

Apply what you learned in this section to answer these questions aboutThe Chapter Problem on page 342.CP16. For each concentration function, (a) determine any critical

numbers and (b) determine intervals of increase or decrease.CP17. Determine where the graph of each concentration function is

concave up or concave down. Are there any points of inflection?Explain.

CP18. Use the information from questions CP16 and CP17 to refine thegraphs you made for question CP8.

CP19. (a) Write a report that describes the buildup of arsenic in a pond if water contaminated with arsenic runs into it. Discuss the effect that the water runoff rate has on thebuildup of arsenic in the pond.

(b) In your report, also recommend the dimensions of a settlingpond that will minimize material costs. Illustrate your reportwith your graphs and calculations.

405

Documents