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 r 2. 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/d 2 = kd −2. 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 x n, k and n are constants. A quantity y is (directly) proportional to a power of x if y = k x n, k and n are constants. A quantity y is inversely proportional to x n if y =k/x n, k and n are constants. A power function is a function of the form f(x) = k x p, where k and p are constants. A power function is a function of the form f(x) = k x p, 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 x p. (a) f(x) = (b) g(x) = 2 (x + 5) 3 (c) u(x) = (d) v(x) = 6 ・ 3 x 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

The Effect of the Power p Graphs of the Special Cases y = x 0 and y = x 1 The power functions corresponding to p = 0 and p = 1 are both linear. The function y = x 0 = 1, except at x = 0. Its graph is a horizontal line with a hole at (0,1). The graph of y = x 1 = 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 y = x 1 y = x 0 (1,1) ●

The Effect of the Power p Positive Integer Powers Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally (1,1) y = x 4 y = x 2 (-1,1) ●● y = x 3 y = x 5 (1,1) (-1,-1) ● ● Graphs of positive even powers of x are U-shaped Graphs of positive odd powers of x are “chair”-shaped

The Effect of the Power p Negative 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

The Effect of the Power p Graphs of Positive Fractional Powers Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally (1,1) y = x 1/4 y = x 1/2 y = x 1/3 y = x 1/5 (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) = a n x n + a n−1 x n− a 1 x + a 0, where n is a positive integer called the degree of p and where a n ≠ 0. Each power function a i x i in this sum is called a term. The constants a n, a n−1,..., a 0 are called coefficients. The term a 0 is called the constant term. The term with the highest power, a n x n, 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

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 QuarticQuintic n = 2 n = 3 n = 4n=5

The Long-Run Behavior of Polynomial Functions When viewed on a large enough scale, the graph of the polynomial p(x) = a n x n + a n−1 x n−1 + ・ ・ ・ + a 1 x + a 0 looks like the graph of the power function y = a n x n. This behavior is called the long- run behavior of the polynomial. Using limit notation, we write When viewed on a large enough scale, the graph of the polynomial p(x) = a n x n + a n−1 x n−1 + ・ ・ ・ + a 1 x + a 0 looks like the graph of the power function y = a n x n. 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

Zeros of Polynomials Example 3 Given the polynomial q(x) = 3x 6 − 2x 5 + 4x 2 − 1,where q(0) = −1, is there a reason to expect a solution to the equation q(x) = 0? If not, explain why not. If so, how do you know? Solution Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Plot of q(x) Since q(0) is negative and since the leading term, 3x 6, 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 1 Compare the graphs of the polynomials f, g, and h given by f(x) = x 4 − 4x x − 16, g(x) = x 4 − 4x 3 − 4x x, h(x) = x 4 + x 3 − 8x 2 − 12x. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally A close-up look near zeros and turnsA larger scale look resembling x 4

Factored Form, Zeros, and the Short- Run Behavior of a Polynomial Example 2 Investigate the short-run behavior of the third degree polynomial u(x) = x 3 − x 2 − 6x. (a) Rewrite u(x) as a product of linear factors. (b) Find the zeros of u(x). Solution (a)u(x) = x (x 2 – 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) = x 3 − x 2 − 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 y = x 3 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 = x 3.

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). 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 Consider the three 4 th 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 3 Describe 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 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 4 Find a possible formula for the polynomial function f graphed to the right. Solution Based 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 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, q. 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 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. 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

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 of y = r(x) is given by Using limits, we write 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 of y = r(x) is given by Using limits, we write Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

The Long-Run Behavior of Rational Functions Example 1 For positive x, describe the positive long-run behavior of the rational function Solution The 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

The Long-Run Behavior of Rational Functions Example 1 - continued For positive x, describe the positive long-run behavior of the rational function Solution Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally y = 1: horizontal asymptote 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 2 For positive x, describe the positive long-run behavior of the rational function Solution The leading term in the numerator is 3x and the leading term in the denominator is x 2. Thus for large enough values of x, Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

The Long-Run Behavior of Rational Functions Example 2 - continued For positive x, describe the positive long-run behavior of the rational function Solution Plotting both functions for values of x > 1: Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally y = 3/x

What Causes Asymptotes? Examples 1 and 2 – Horizontal Asymptotes We 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 x=-2 x=1 y = 1 y = 0

What Causes Asymptotes? Examples 1 and 2 – Vertical Asymptotes Vertical 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 x=-2 x=1 y = 0 y = 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 2 Find the zeros and vertical asymptotes of the rational function r(x). This was explored in the previous section. Solution Noting 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 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. 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

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 Horizontal asymptote at y = 1 and r(3/2) = 1 y = 1 y=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

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 34 Find 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 (-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 equals zero at x = 1. However, the graph of h does not have a vertical asymptote at x = 1 because the numerator of h also 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

11.6 COMPARING POWER, EXPONENTIAL, AND LOG FUNCTIONS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

Comparing Power Functions Example 1 Let f(x) = 100x 3 and g(x) = x 4 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. When comparing power functions with positive coefficients, higher powers dominate. Close-up view Far-away view Point of intersection: (100,100 4 ) f(x)f(x) f(x)f(x) g(x) Blue Wins Blue Wins

Comparing Exponential Functions and Power Functions Example Compare the functions f(x) = x 4 and g(x) = 2 x 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. Any positive increasing exponential function eventually grows faster than any power function. Close-up view Far-away view Point of intersection: (16, 65,536) f(x)f(x) f(x)f(x) g(x) Blue Wins Blue Wins

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. Close-up view Far-away view f(x)f(x) f(x)f(x) g(x) Blue Wins Blue Wins

Comparing Log and Power Functions Example Compare the functions f(x) = x 1/2, 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. f(x)f(x) g(x) h(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 AIDS Visualizing the Data The data in the table gives the total number of deaths in the US from AIDS from 1981 to 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

The Spread of AIDS Fitting Functions to the Data Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally Exponential Fit Function N = 630e 0.47t Power Fit Function N = 107t Linear Fit Function N = − t

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) ≈ 2,938,550, or about 1% of the current population; and the linear model gives N = − ˑ30 = 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 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.