DIRECTIONS FOR GROUP PROJECT
Chapter 2 Review Extended Application: Power Functions
69. Maximizing Area What would be the maximum area that could be enclosed by the college’s 380 ft of fencing if it
In Exercises 70 and 71, draw a sketch of the arch or culvert on coordinate axes, with the horizontal and vertical axes through the vertex of the parabola. Use the given information to label points on the parabola. Then give the equation of the parabola and answer the question.
70.Parabolic Arch An arch is shaped like a parabola. It is 30 m wide at the base and 15 m high. How wide is the arch 10 m from the ground?
71.Parabolic Culvert A culvert is shaped like a parabola, 18 ft across the top and 12 ft deep. How wide is the culvert 8 ft from the top?
38022x
function of degree 0, the constant function.) A polynomial function of degree 2 is a qua- dratic function.
Accurate graphs of polynomial functions of degree 3 or higher require methods of cal- culus to be discussed later. Meanwhile, a graphing calculator is useful for obtaining such graphs, but care must be taken in choosing a viewing window that captures the significant behavior of the function.
The simplest polynomial functions of higher degree are those of the form
Such a function is known as a power function. Figure 33 below shows the graphs of and as well as tables of their values. These functions are simple enough that they can be drawn by hand by plotting a few points and connecting them with a smooth curve. An important property of all polynomials is that their graphs are smooth curves.
The graphs of and shown in Figure 34 along with tables of their values, can be sketched in a similar manner. These graphs have symmetry about the y-axis, as does the graph of for a nonzero real number a. As with the graph of
the value of ain affects the direction of the graph. When the graph has the same general appearance as the graph of However, if the graph is reflected vertically. Notice that and are odd functions, while f1x25x4and f1x2 5x6are even functions.f1x2 5x3 f1x2 5x5
a,0, f1x2 5xn. a.0, f1x2 5axn
f1x2 5ax2,
f1x2 5ax2
f1x25x6, f1x25x4
f1x2 5x5, f1x2 5x3
f1x25xn.
FIGURE 33
0 x
f(x)
f(x) = x3
f(x) = x5
–2 2
–1 1
–2
2 4 6 8
–4 –6 –8
FIGURE 34
x x
2 8 1.5 7.6
1 1 1 1
0 0 0 0
1 1 1 1
2 8 1.5 7.6
2 2
2
2 2 2 2
2
f1x2 f1x2
f1x2 5x5 f1x25x3
x x
2 16 1.5 11.4
1 1 1 1
0 0 0 0
1 1 1 1
2 16 1.5 11.4
2
2 2
2
f1x2 f1x2
f1x2 5x6 f1x25x4
Translations and Reflections Graph
SOLUTION Using the principles of translation and reflection from the previous section, we recognize that this is similar to the graph of but reflected vertically (because of the negative in front of and with its center moved 2 units to the right and 3 units
up. The result is shown in Figure 35. TRY YOUR TURN 1
1x2223),
y5x3, f1x25 21x222313.
x f(x)
f(x) = x4
f(x) = x6
2
–1 1
–2
5 10 15
EXAMPLE 1
YOUR TURN 1 Graph f1x2 5642x6.
0 x
y 14 12 10 8 6 4
1 5
–1 2 3 4
2
–4 –6
f(x) = –(x – 2)3 + 3
FIGURE 35
2.3 Polynomial and Rational Functions 69 A polynomial of degree 3, such as that in the previous example and in the next, is known as a cubic polynomial.A polynomial of degree 4, such as that in Example 3, is known as a quartic polynomial.
Graphing a Polynomial Graph
SOLUTION Figure 36 shows the function graphed on thex- andy-intervals and In this view, it appears similar to a parabola opening downward. Zooming out to by we see in Figure 37 that the graph goes upward asxgets large. There are also two turning points near and (In a later chapter, we will introduce another term for such turning points:relative extrema.) By zooming in with the graphing calculator, we can find these turning points to be at approximately and
21.088662.
10.90825, 10.09175, 1.088662
x51.
x50 328, 84,
321, 24
322, 24. 320.5, 0.64
f1x2 58x3212x212x11.
EXAMPLE 2
20.5 0.6
2
22
y 5 8x3 2 12x2 1 2x 1 1
21 2
8
28
y 5 8x3 2 12x2 1 2x 1 1
FIGURE 36 FIGURE 37
Zooming out still further, we see the function on by in Figure 38.
From this viewpoint, we don’t see the turning points at all, and the graph seems similar in shape to that of This is an important point: when xis large in magnitude, either pos- itive or negative, behaves a lot like because the other terms are small in comparison with the cubic term. So this viewpoint tells us something useful about the function, but it is less useful than the previous graph for determining the turning points.
After the previous example, you may wonder how to be sure you have the viewing win- dow that exhibits all the important properties of a function. We will find an answer to this question in later chapters using the techniques of calculus. Meanwhile, let us consider one more example to get a better idea of what polynomials look like.
Graphing a Polynomial Graph
SOLUTION Figure 39 shows a graphing calculator view on by If you have a graphing calculator, we recommend that you experiment with various viewpoints and verify for yourself that this viewpoint captures the important behavior of the function.
Notice that it has three turning points. Notice also that as gets large, the graph turns downward. This is because as becomes large, the -term dominates the other terms, which are small in comparison, and the -term has a negative coefficient.
As suggested by the graphs above, the domain of a polynomial function is the set of all real numbers. The range of a polynomial function of odd degree is also the set of all real numbers. Some typical graphs of polynomial functions of odd and even degree are shown in Figure 40 on the next page. The first two graphs suggest that for every polynomial function fof odd degree, there is at least one real value of xfor which Such a value of xis called a real zeroof f; these values are also the x-intercepts of the graph.
f1x250.
x4
x4
0x0 0x0
3250, 504.
323, 54 f1x2 5 23x4114x3254x13.
8x3, 8x3212x212x11
y5x3.
32300, 3004 3210, 104
210 10
300
2300
y 5 8x3 2 12x2 1 2x 1 1
FIGURE 38
EXAMPLE 3
23 5
50
250
y 5 23x4 1 14x3 2 54x 1 3
FIGURE 39
TECHNOLOGY
TECHNOLOGY
Identifying the Degree of a Polynomial
Identify the degree of the polynomial in each of the figures, and give the sign ( or ) for the leading coefficient.
(a) Figure 41(a)
SOLUTION Notice that the polynomial has a range . This must be a polynomial of even degree, because if the highest power of xis an odd power, the polynomial can take on all real numbers, positive and negative. Notice also that the polynomial becomes a large positive num- ber as xgets large in magnitude, either positive or negative, so the leading coefficient must be positive. Finally, notice that it has three turning points. Observe from the previous examples that a polynomial of degree nhas at most turning points. In a later chapter, we will use calculus to see why this is true. So the polynomial graphed in Figure 41(a) might be degree 4, although it could also be of degree 6, 8, etc. We can’t be sure from the graph alone.
(b) Figure 41(b)
SOLUTION Because the range is this must be a polynomial of odd degree.
Notice also that the polynomial becomes a large negative number as xbecomes a large pos- itive number, so the leading coefficient must be negative. Finally, notice that it has four turning points, so it might be degree 5, although it could also be of degree 7, 9, etc.
12`, `2, n21
3k, `2
2 1
x y
Degree 3;
three real zeros
x y
Degree 6;
four real zeros y
x
Degree 3;
one real zero
FIGURE 40 EXAMPLE 4
x y
k
(a)
x y
(b)
FIGURE 41
Properties of Polynomial Functions
1. A polynomial function of degree ncan have at most turning points. Conversely, if the graph of a polynomial function has nturning points, it must have degree at least 2. In the graph of a polynomial function of even degree, both ends go up or both ends go
down. For a polynomial function of odd degree, one end goes up and one end goes down.
3. If the graph goes up as xbecomes a large positive number, the leading coefficient must be positive. If the graph goes down as xbecomes a large positive number, the leading coefficient is negative.
n11.
n21
2.3 Polynomial and Rational Functions 71
Rational Functions Many situations require mathematical models that are quotients.
A common model for such situations is a rational function.
Rational Function
Arational functionis defined by
where p1x2and q1x2are polynomial functions and q1x2 20.
f1x2 5p1x2 q1x2,
Since any values of xsuch that are excluded from the domain, a rational function often has a graph with one or more breaks.
Graphing a Rational Function
Graph
SOLUTION This function is undefined for since 0 is not allowed as the denominator of a fraction. For this reason, the graph of this function will not intersect the vertical line which is they-axis. Since xcan take on any value except 0, the values ofx can approach 0 as closely as desired from either side of 0.
x50,
x50, y5 1
x.
q1x2 50
EXAMPLE 5
approaches
x b0.01 0.1 0.2 0.5
100 10 5 2
gets larger and larger.a
0y0
2100 210
25 22 y5 1
x
20.01 20.1
20.2 20.5
0.
x
Values of 1/xfor Small x
x 1 4 10 100
1 0.25 0.1 0.01 21
20.25 20.1
20.01 y51
x
21 24
210 2100
Values of 1/xfor Large |x|
The table above suggests that as xgets closer and closer to 0, gets larger and larger.
This is true in general: as the denominator gets smaller, the fraction gets larger. Thus, the graph of the function approaches the vertical line (the y-axis) without ever touching it.
As gets larger and larger, gets closer and closer to 0, as shown in the table below. This is also true in general: as the denominator gets larger, the fraction gets smaller.
y51/x
0x0 x50
0y0
y
x 4
2
–2 –4
2 4
–2 –4
0
y = –1 x
FIGURE 42
The graph of the function approaches the horizontal line (the x-axis). The informa- tion from both tables supports the graph in Figure 42.
In Example 5, the vertical line and the horizontal line are asymptotes, defined as follows.
y50 x50
y50
Asymptotes
If a function gets larger and larger in magnitude without bound as xapproaches the number k, then the line is a vertical asymptote.
If the values of yapproach a number kas gets larger and larger, the line is a horizontal asymptote.
y5k 0x0
x5k
There is an easy way to find any vertical asymptotes of a rational function. First, find the roots of the denominator. If a numberkmakes the denominator 0 but does not make the numerator 0, then the line is a vertical asymptote. If, however, a number kmakes both the denominator and the numerator 0, then further investigation will be necessary, as we will see in the next example. In the next chapter we will show another way to find asymptotes using the concept of alimit.
Graphing a Rational Function Graph the following rational functions:
(a)
SOLUTION The value x 1 makes the denominator 0, and so 1 is not in the domain of this function. Note that the value x 1 also makes the numerator 0. In fact, if we fac- tor the numerator and simplify the function, we get
The graph of this function, therefore, is the graph of yx2 with a hole at x 1, as shown in Figure 43.
(b)
SOLUTION The value makes the denominator 0, but not the numerator, so the line is a vertical asymptote. To find a horizontal asymptote, let xget larger and larger, so that because the 2 is very small compared with 3x. Similarly, for xvery
large, Therefore, This
means that the line is a horizontal asymptote. (A more precise way of approaching this idea will be seen in the next chapter when limits at infinity are discussed.)
The intercepts should also be noted. When the y-intercept is . To make a fraction 0, the numerator must be 0; so to make it is necessary that Solve this for xto get (the x-intercept). We can also use these val- ues to determine where the function is positive and where it is negative. Using the tech- niques described in Chapter R, verify that the function is negative on and positive on With this information, the two asymptotes to guide us, and the fact that there are only two intercepts, we suspect the graph is as shown in Figure 44. A graphing calculator can support this. TRY YOUR TURN 2
Rational functions occur often in practical applications. In many situations involving environmental pollution, much of the pollutant can be removed from the air or water at a fairly reasonable cost, but the last small part of the pollutant can be very expensive to remove. Cost as a function of the percentage of pollutant removed from the environment can be calculated for various percentages of removal, with a curve fitted through the result- ing data points. This curve then leads to a mathematical model of the situation. Rational functions are often a good choice for these cost-benefit modelsbecause they rise rapidly as they approach a vertical asymptote.
Cost-Benefit Analysis Suppose a cost-benefit model is given by
y5 18x 1062x,
12`, 222< 122/3, `2. 122, 22/32
x5 22/3
3x1250. x50, y50, y52/451/2
y53/2
y5 13x122/12x142 < 13x2/12x253/2.
2x14<2x.
3x12<3x
x5 22 x5 22
y5 3x12 2x14.
y5 x213x12
x11 5 1x122 1x112
1x112 5x12 for x221.
y5x213x12 x11 .
x5k
EXAMPLE 6
–4 –3 –2 –1 0 1 2 3 4
–4 –3 –2 –1 1 2 3 4 y
x x2 + 3x + 2
x + 1 y =
FIGURE 43
y
x
x = –2 –4 4
–4 0
y = –3 2 3x + 2
2x + 4 y =