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11.1
POWER FUNCTIONS
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Proportionality and Power Functions
• Example 1 The area, A, of a circle is proportional to the square of its radius, r: A = π r2.
• Example 2 The weight, w, of an object is inversely proportional to the square of the object’s distance, d, from the earth’s center: w =k/d2 = kd−2.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
A quantity y is (directly) proportional to a power of x if y = k xn, k and n are constants.
A quantity y is inversely proportional to xn if y =k/xn , k and n are constants.
A power function is a function of the form f(x) = k xp, where k and p are constants.
Proportionality and Power Functions
• Example 3 Which of the following functions are power functions? For each power function, state the value of the constants k and p in the formula y = k xp.
(a) f(x) =
(b) g(x) = 2 (x + 5)3
(c) u(x) =
(d) v(x) = 6 ・ 3x
• Solution: The functions f (k=13, p=1/3) and u (k=5, p=-3/2) are power functions; the functions g and v are not power functions.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
313 x
3/25 x
The Effect of the Power p• Graphs of the Special Cases y = x0 and y = x1
• The power functions corresponding to p = 0 and p = 1 are both linear. The function y = x0 = 1, except at x = 0. Its graph is a horizontal line with a hole at (0,1). The graph of y = x1 = x is a line through the origin with slope +1. Both graphs contain the point (1,1).
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
1 1 2 3 4 5x
1
1
2
3
4
5
y
y = x1
y = x0(1,1)
●
The Effect of the Power pPositive Integer Powers
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
4 2 0 2 4x
2
4
6
8
10
12y
(1,1)
y = x4y = x2
(-1,1)● ●
2 1 1 2x
4
2
2
4
y y = x3 y = x5
(1,1)
(-1,-1)
●
●
Graphs of positive even powers of x are U-shaped
Graphs of positive odd powers of x are “chair”-shaped
3 2 1 0 1 2 3x
1
2
3
4
5
6y
3 2 1 1 2 3x
4
2
2
4
y
The Effect of the Power pNegative Integer Powers
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
(1,1)
(-1,-1)
y = x-2
(1,1)(-1,1)
Both graphs have a vertical asymptote of x = 0
Both graphs have a horizontal asymptote of y = 0
y = x-1
x 0.1 0.05 0.01 0.001 0.0001 0 1/x 10 20 100 1000 10000 undefined 1/x2 100 400 10,000 1,000,000 100,000,000 undefined
x 0 10 20 30 40 50 1/x undefined 0.1 0.05 0.033333 0.025 0.02 1/x2 undefined 0.01 0.0025 0.001111 0.000625 0.0004
4 2 2 4x
2
1
1
2y
4 2 2 4x
2
1
1
2y
The Effect of the Power pGraphs of Positive Fractional Powers
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
(1,1) y = x1/4
y = x1/2y = x1/3
y = x1/5(1,1)
(-1,-1)
Graphs of even roots of x are not defined for x < 0
Graphs of odd roots of x are defined for all values of x
11.2
POLYNOMIAL FUNCTIONS
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
A General Formula for the Family of Polynomial Functions
The general formula for the family of polynomial functions can be written as
p(x) = an xn + an−1 xn−1 + . . . + a1 x + a0, where n is a positive integer called the degree of p and where an ≠ 0.
• Each power function aixi in this sum is called a term.
• The constants an, an−1, . . . , a0 are called coefficients.
• The term a0 is called the constant term. The term with the
highest power, anxn, is called the leading term.
• To write a polynomial in standard form, we arrange its
terms from highest power to lowest power, going from left
to right.Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Like the power functions from which they are built, polynomials are defined for all values of x. Except for polynomials of degree zero (whose graphs are horizontal lines), the graphs of polynomials do not have horizontal or vertical asymptotes; they are smooth and unbroken. The shape of the graph depends on its degree; typical graphs are shown below.
Quadratic Cubic Quartic Quintic n = 2 n = 3 n = 4 n=5
The Long-Run Behavior of Polynomial Functions
When viewed on a large enough scale, the graph of the polynomial p(x) = anxn + an−1xn−1 + ・ ・ ・ + a1x + a0 looks like the graph of the power function y = anxn. This behavior is called the long-run behavior of the polynomial. Using limit notation, we write
• •
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
nn
xxxaxp
lim)(lim
nn
xxxaxp
lim)(lim
Zeros of PolynomialsExample 3
Given the polynomial q(x) = 3x6 − 2x5 + 4x2 − 1,where q(0) = −1, is there a reason to expect a solution to the equation q(x) = 0? Ifnot, explain why not. If so, how do you know?
Solution
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
1 .0 0 .5 0 .5 1 .0x
2
4
6
8
10
12
q
Plot of q(x)Since q(0) is negative and since the leading term, 3x6 , goes to positive infinity as x increases in either the positive or negative direction, we know that the graph must cross the x-axis for at least one positive and one negative value of x.
11.3
THE SHORT-RUN BEHAVIOR OF POLYNOMIALS
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Visualizing Short-Run and Long-Run Behaviors of a Polynomial
Example 1Compare the graphs of the polynomials f, g, and h given by
f(x) = x4 − 4x3 + 16x − 16, g(x) = x4 − 4x3 − 4x2 + 16x, h(x) = x4 + x3 − 8x2 − 12x.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
4 2 2 4x
30
20
10
10
20
30
y
10 5 5 10x
10 00
20 00
30 00
40 00y
A close-up look near zeros and turns A larger scale look resembling x4
Factored Form, Zeros, and the Short-Run Behavior of a Polynomial
Example 2Investigate the short-run behavior of the third degree polynomial u(x) = x3 − x2 − 6x.(a) Rewrite u(x) as a product of linear factors.(b) Find the zeros of u(x).
Solution(a) u(x) = x (x2 – x – 6) = x (x + 2) (x – 3)(b) u(x) = 0 if and only if one of its linear factors is zero, so
the zeros of u(x) occur at x = 0, x = -2, and x = 3.Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Example 2 (continued)Investigate the short-run behavior of the third degree polynomial u(x) = x3 − x2 − 6x.(c) Describe the graph of u(x). Where does it cross the x-
axis? the y-axis? Where is u(x) positive? Negative?
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
3 2 1 1 2 3 4x
20
10
10
20
y y = x3 y =u(x)
(-2,0)(3,0)
(0,0)
u negative → ↖ u negative
← u positive
u positive ↘
The graph of u(x) has zeros at x = -2, 0, and 3. The function changes sign at each of these zeros. Its long-run behavior resembles y = x3.
Factors and Zeros of Polynomials
Suppose p is a polynomial. • If the formula for p has a linear factor, that is,
a factor of the form (x − k), then p has a zero at x = k.
• Conversely, if p has a zero at x = k, then p has a linear factor of the form (x − k).
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
The Number of Factors, Zeros, and Bumps
The graph of an nth degree polynomial has at most n zeros and turns at most (n − 1) times.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
4 2 2 4x
30
20
10
10
20
30
y
Consider the three 4th degree polynomials in Example 1
The red graph has two zeros.
The blue graph has three zeros.
The green graph has four zeros.
Multiple Zeros
If p is a polynomial with a repeated linear factor, then p has a multiple zero. • If the factor (x − k) occurs an even number
of times, the graph of y = p(x) does not cross the x-axis at x = k, but “bounces” off the x-axis at x = k.
• If the factor (x − k) occurs an odd number of times, the graph of y = p(x) crosses the x-axis at x = k, but it looks flattened there.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Example 3Describe in words the zeros of the 4th-degree polynomials f(x), g(x), and h(x), in the graphs below
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
2 1 1 2 3x
3 0
2 0
1 0
1 0
y
2 1 1 2 3 4x
30
20
10
10
y
3 2 1 1 2 3x
30
20
10
10
y
The graph suggests that f has a single zero at x = −2. The flattened appearance near x = 2 suggests that f has a multiple zero there. Since the graph crosses the x-axis at x = 2 (instead of bouncing off it), this zero must occur an odd number of times. Since f is 4th degree, f has at most 4 factors, so there must be a triple zero at x = 2.
The graph of g has four single zeros.
The graph of h has two single zeros (at x = 0 and x = 3) and a double zero at x = −2. The multiplicity of the zero at x = −2 is not higher than two because h is of degree n = 4.
f(x) g(x) h(x)
Finding the Formula for a Polynomial from its Graph
Example 4Find a possible formula for the polynomial function f graphed to the right.SolutionBased on its long-run behavior, f is of odd degree greater than or equal to 3. The polynomial has zeros at x = −1 and x = 3. We see that x = 3 is a multiple zero of even power, because the graph bounces off the x-axis here instead of crossing it. Therefore, we try the formula f(x) = k (x + 1) (x – 3)2, where k represents a stretch factor. The shape of the graph shows that k must be negative. To find k, we use the fact that f(0) = kˑ1ˑ(-3)2= −3, so k= – 1/3 and f(x) = – 1/3 (x + 1) (x – 3)2 is a possible formula for this polynomial.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
2 2 4x
1 0
5
5
1 0
y
y = f(x)
11.4
RATIONAL FUNCTIONS
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
The Average Cost of Producing a Therapeutic Drug
A pharmaceutical company wants to begin production of a new drug. The total cost C, in dollars, of making q grams of the drug is given by the linear function
C(q) = 2,500,000 + 2000q.The fact that C(0) = 2,500,000 tells us that the company spends $2,500,000 before it starts making the drug. This quantity is known as the fixed cost because it does not depend on how much of the drug is made. It represents the cost for research, testing, and equipment. In addition, the slope of C tells us that each gram of the drug costs an extra $2000 to make. This quantity is known as the variable cost per unit. It represents the additional cost, in labor and materials, to make an additional gram of the drug.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
The Average Cost of Producing a Therapeutic Drug – continued
We define the average cost, a(q), as the cost per gram to produce q grams (q > 0) of the drug:
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
q
qqqC
qa2000000,500,2)(
)(grams ofNumber
cost Total
0 5 0 0 0 1 0 0 0 0 1 5 0 0 0 2 0 0 0 0q , n u m b e r o f g r a m s
1 0 0 0
2 0 0 0
3 0 0 0
4 0 0 0
5 0 0 0
6 0 0 0
y , a v e r a g e c o s t d o l l a r s p e r g r a m y = a(q)
y= 2000: horizontal asymptote
The graph of y = a(q), a rational function, has a horizontal asymptote at y = 2000 and a vertical asymptote at q = 0
What is a Rational Function?
If r can be written as the ratio of polynomial functions p(x) and q(x), that is, if
then r is called a rational function. We assume that q(x) is not the constant polynomial q(x) = 0.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
,)(
)()(
xq
xpxr
The Long-Run Behavior of Rational Functions
For x of large enough magnitude (either positive or negative), the graph of the rational function r looks like the graph of a power function. If r(x) = p(x)/q(x), then the long-run behavior ofy = r(x) is given by
Using limits, we write
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
q
py
of termLeading
of termLeading
q
p
xq
xpxx of termLeading
of termLeadinglim
)(
)(lim
The Long-Run Behavior of Rational Functions
Example 1For positive x, describe the positive long-run behavior of the rational function SolutionThe leading term in the numerator is x and the leading term in the denominator is x. Thus for large enough values of x,
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
.2
3)(
x
xxr
.11lim)(lim and 1)(xx
xrx
xxr
The Long-Run Behavior of Rational Functions
Example 1 - continuedFor positive x, describe the positive long-run behavior of the rational function Solution
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
.2
3)(
x
xxr
0 5 1 0 1 5 2 0 2 5x
0 .5
1 .0
1 .5
y
y = 1: horizontal asymptote
2
3
x
xy For large enough values of x, the
graph of y = r(x) looks like the line y = 1, its horizontal asymptote.However, for x > 0, the graph of r is above the line since the numerator is larger than the denominator.
The Long-Run Behavior of Rational Functions
Example 2For positive x, describe the positive long-run behavior of the rational function SolutionThe leading term in the numerator is 3x and the leading term in the denominator is x2. Thus for large enough values of x,
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
.2
13)(
2
xx
xxg
.03
lim)(lim and 33
)(xx2
xxg
xx
xxg
The Long-Run Behavior of Rational Functions
Example 2 - continuedFor positive x, describe the positive long-run behavior of the rational function SolutionPlotting both functions for values of x > 1:
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
.2
13)(
2
xx
xxg
2 4 6 8 10x
0 .5
1 .0
1 .5
2 .0
2 .5
y
y = 3/x
2
132
xx
xy
What Causes Asymptotes?
Examples 1 and 2 – Horizontal AsymptotesWe have seen that horizontal asymptotes, shown as red dashed lines, can occur as x approaches positive or negative infinity and these can be determined using the leading terms.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
5 5x
15
10
5
5
10
15y
4 2 2 4x
5
5
y
.2
3)(
x
xxr .
2
13)(
2
xx
xxg
x=-2 x=-2 x=1
y = 1y = 0
What Causes Asymptotes?
Examples 1 and 2 – Vertical AsymptotesVertical asymptotes occur whenever a denominator goes to zero, while the numerator does not. The functions given in Examples 1 and 2 both possess vertical asymptotes (blue vertical lines) as well as horizontal asymptotes (red dashed lines).
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
5 5x
15
10
5
5
10
15y
4 2 2 4x
5
5
y
.2
3)(
x
xxr .
2
13)(
2
xx
xxg
x=-2 x=-2 x=1
y = 0y = 1
11.5
THE SHORT-RUN BEHAVIOR OF RATIONAL FUNCTIONS
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
The Zeros and Vertical Asymptotes of a Rational Function
Example 2Find the zeros and vertical asymptotes of the rational function r(x). This was explored in the previous section.SolutionNoting that the numerator is zero when x = -3, we will expect the graph to cross the x-axis at x = -3. Noting that the denominator is zero when x = -2, we will expect a vertical asymptote when x = -2
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
.2
3)(
x
xxr
4 2 2 4x
4
2
2
4
6
8
y
2
3)(
x
xxr The horizontal asymptote, y = 1,
was determined in the previous section.
The Graph of a Rational Function
If r is a rational function given by
where p and q are polynomials with different zeros, then:• The long-run behavior and horizontal asymptote (if any)
of r are given by the ratio of the leading terms of p and q.• The zeros of r are the same as the zeros of the numerator,
p.• The graph of r has a vertical asymptote at each of the
zeros of the denominator, q.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
)(
)()(
xq
xpxr
Can a Graph Cross an Asymptote?The graph of a rational function never crosses a vertical asymptote. However, the graphs of some rational functions cross their horizontal asymptotes. The difference is that a vertical asymptote occurs where the function is undefined, so there can be no y-value there, whereas a horizontal asymptote represents the limiting value of the function as x →± ∞. Example:
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
2
2 32)(
x
xxxr
Horizontal asymptote at y = 1 and r(3/2) = 1
6 4 2 2 4 6x
3
2
1
1
y
y = 1y=r(x)
Rational Functions as Transformations of Power Functions
• The average cost function from Section 11.4 can be written as
• Thus, the graph of a is the graph of the power function y = 2,500,000q−1 shifted up 2000 units.
• Many rational functions can be viewed as translations of power functions.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
2000000,500,22000000,500,2
)( 1
qqa
6 4 2 2 4 6x
3
2
1
1
y
Finding a Formula for a Rational Function from its Graph
The graph of a rational function can give a good idea of its formula. Zeros of the function correspond to factors in the numerator and vertical asymptotes correspond to factors in the denominator.Exercise 34Find a possible formula for the function whose graph is given.
Solution
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
It is important to note that the horizontal asymptote at y =1 implies that the polynomials in the numerator and denominator have the same degree.
y = 1
2)2(
)1)(3()(
x
xxxr
(-3,0) (1,0)
(0,-3/4)
Possible rational function:
When Numerator and Denominator Have the Same Zeros: Holes
The rational function
is undefined at x = 1 because the denominator equalszero at x = 1. However, the graph of h does not have avertical asymptote at x = 1 because the numerator of halso equals zero at x = 1; h(1) = 0/0 which is undefined.
In fact, h(x) = x + 2 everywhere except at x = 1, so the graph of y = h(x) looks just like that for y = x + 2, except that there is a hole (not big enough to see) at the point (1,3) in the graph of y = h(x), since h(1) is not defined.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
1
2)(
2
x
xxxh
11.6
COMPARING POWER, EXPONENTIAL, AND LOG FUNCTIONS
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Comparing Power Functions
Example 1Let f(x) = 100x3 and g(x) = x4 for x > 0. Compare the long-term behavior of these two functions using graphs.Solution
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
When comparing power functions with positive coefficients, higher powers dominate.
0 2 4 6 8 1 0x
2 0 0 0 0
4 0 0 0 0
6 0 0 0 0
8 0 0 0 0
1 0 0 0 0 0y
Close-up view
0 100 200 300 400 500x
1 10 10
2 10 10
3 10 10
4 10 10
5 10 10y
Far-away view
0 20 40 60 80 10 0x
2 .0 10 7
4 .0 10 7
6 .0 10 7
8 .0 10 7
1 .0 10 8
1 .2 10 8
1 .4 10 8
y
Point of intersection: (100,1004)
f(x)
f(x)g(x)
g(x)
BlueWins
Comparing Exponential Functions and Power Functions
Example Compare the functions f(x) = x4 and g(x) = 2x for x > 0. Solution
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Any positive increasing exponential function eventually grows faster than any power function.
0 2 4 6 8 1 0x
2 0 0 0
4 0 0 0
6 0 0 0
8 0 0 0
1 0 0 0 0y
0 20 40 60 80 100x
2 107
4 107
6 107
8 107
1 108y
Close-up view
Far-away view
0 5 10 15 20x
50 000
100 000
150 000
200 000y
Point of intersection: (16, 65,536)
f(x)
f(x)
g(x)
g(x)
BlueWins
Decreasing Exponential Functions and Decreasing Power Functions
Example Compare the functions f(x) = x-2 and g(x) = 1.1-x for x > 0. Solution
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Any positive decreasing exponential function eventually approaches the horizontal axis faster than any positive decreasing power function.
0 20 40 60 80x
0 .002
0 .004
0 .006
0 .008
0 .010y
0 50 10 0 15 0 20 0 25 0x
0 .000 2
0 .000 4
0 .000 6
0 .000 8
0 .001 0yClose-up view
Far-away view
f(x)
f(x)
g(x)g(x)
BlueWins
Comparing Log and Power Functions
Example Compare the functions f(x) = x1/2 , g(x) = log x, and h(x) = ln x for x > 0. Solution
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Any positive increasing power function eventually grows more rapidly than y = log x and y = ln x.
0 5 10 15 20x
1
2
3
4
5y
f(x)
g(x)
h(x)
Red Wins
11.7
FITTING EXPONENTIALS AND POLYNOMIALS TO DATA
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
The Spread of AIDSVisualizing the DataThe data in the table gives the total number of deaths in the US from AIDS from 1981 to 1996. The graph suggests that a linear function may not give the best possible fit for these data.
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
t N1 1592 6223 21304 56355 126076 247177 411298 622489 90039
10 12157711 15819312 19928713 24392314 29258615 34095716 375904
0 2 4 6 8 10 12 14 16 180
50
100
150
200
250
300
350
400
US Deaths from AIDS (1981 - 1996)
t (years since 1980)
N (i
n th
ousa
nds)
The Spread of AIDSFitting Functions to the Data
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
US Deaths from AIDS (1981 - 1996)
t (years since 1980)
N (i
n th
ousa
nds)
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
US Deaths from AIDS (1981 - 1996)
t (years since 1980)
N (i
n th
ousa
nds)
Exponential Fit FunctionN = 630e0.47t
Power Fit FunctionN = 107t3.005
Linear Fit FunctionN = −97311 + 25946t
0 2 4 6 8 10 12 14 160
50
100
150
200
250
300
350
400
US Deaths from AIDS (1981 - 1996)
t (years since 1980)
N (i
n th
ousa
nds)
Which Function Best Fits the Data?
Despite the fact that all three functions fit the data reasonably well up to 1996, it is important to realize that they give wildly different predictions for the future. If we use each model to estimate the total number of AIDS deaths by the year 2010 (when t = 30), • the exponential model gives N = 630e(0.47)30 ≈
837,322,467, about triple the current US population;• the power model gives N = 107(30)3.005 ≈ 2,938,550, or
about 1% of the current population;• and the linear model gives N = −97311 + 25946ˑ30 =
681,069, or about 0.22% of the current population.Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
Analyzing the Results with Newer Data
Long-term Predictions Are Difficult
Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally
0 5 10 15 20 25 300
100
200
300
400
500
600
700
U S Deaths from AIDS1981–2007
t (years since 1980)
N (i
n th
ousa
nds)
When data from the entire period from 1981 to 2007 are plotted together, we see that the rate of increase of AIDS deaths reaches a peak sometime around 1995 and then begins to taper off. Since none of the three types of functions we have used to model AIDS deaths exhibit this type of behavior, some other type of function is needed to describe the number of AIDS deaths accurately over the entire 26-year period.