Section 4 3A: Power Functions Limits A power function is a polynomial function with the x terms raised to powers that are positive integers. The terms are written in decreasing powers of x. Examples f (x) = 3x 2 + 2x + 4 f (x) = 2x 4 x 3 + 2x 2 5x 2 f (x) = 3x 6 5x 5 + 7x 3 x The graphs of power functions are smooth continuous curves with no breaks. The ends of the graphs have points whose y values increase to positive infinity or decrease to negative infinity. Between the ends the graph may cross the x axis or touch the x axis several times. We call each x value where this happens a zero or root of the polynomial. The graph may have several turns that form high or low points in the graph. The sketch graphs do not display the exact heights of the y values so the exact shape of the graph is not shown. The sketch graphs shown in these lectures display only the basic behaviors of roots, increasing and decreasing intervals of the graph and the the behavior of the graphs around gaps in the graph. The sketch graphs also display the behavior of the graphs as the values for x approach negative or positive infinity. Sample sketch graphs of power functions The sketch graphs do not display the exact heights of the y values so the exact shape of the graph is not shown. The graph below has 4 roots, 3 turns and! both ends increase without bound! y The graph below has 3 roots, 3 turns and both ends decrease without bound y 4 1 1 4 x 2 0 4 x! The graph below has 2 roots, 2 turns and! the left end increases without bound and! the right end decreases without bound! y The graph below has 3 roots, 4 turns and the left end decreases without bound and the right end increases without bound y 0 3 x 2 0 3 x! 4 3A Power Function Limits! Page 1 of 9! 2018 Eitel
A power function is a polynomial function with the x terms raised to powers that are positive integers. The terms are written in decreasing powers of x f (x) = a n x n + a n 1 x n 1 + a n 2 x n 2 + a n 3 x n 3 +... + a 2 x 2 + a 1 x + a 0 where n positive integers Z + and a n, a n 1, a n 2... R The first term a n x n has the highest power of x, x n, and a coefficient of a n. We call this the leading term of the Power Function. Examples of Power Functions f (x) = 3x 2 + 2x + 4 the leading term of 3x 2 has an even exponent and a positive coefficient!! f (x) = 6x 2 x 2 + 7 the leading term of 6x 2 has an even exponent and a negitive coefficient f (x) = x 3 5x 2 + 6x 1 the leading term of x 3 has an odd exponent and a positive coefficient!! f (x) = 4x 3 5x 2 + 6x 1 the leading term of 4x 3 has an odd exponent and a negitive coefficient The leading term of the power function has x raised either an odd or even power and the x variable has either a negative or positive coefficient. The combination of these 2 attributes determines what the right and left ends of the graph of the power function look like. We will investigate this farther as we develop the concepts of its in the following pages. The it question x + Limits of Power Functions f (x) =?? asks what the y values at the right end of the graph are getting close to as the values for x become larger and larger positive numbers. The it question x f (x) =?? asks what the y values at the left end of the graph are getting close to as the values for x become smaller and smaller negative numbers. 4 3A Power Function Limits! Page 2 of 9! 2018 Eitel
There are 4 cases that we will need to examine. They are represented by the 4 functions n even Z +! n even Z +! n odd Z +! n odd Z + The behavior of the Right and Left Ends of the Power Function Case 1 If n is a positive even integer and +x n has a positive coefficient The right sides of the graph has x values approaching larger and larger positive values x + and y values approaching larger and larger positive values ( y + )! The left side of the graph has x values approaching smaller and smaller negative values x + and y values approaching larger and larger positive values ( y + ) Theorem If n is a positive even integer and +x n has a positive coefficient left end of the graph! right end of the graph x + xn = + x + + xn = + 4 3A Power Function Limits! Page 3 of 9! 2018 Eitel
The behavior of the Right and Left Ends of the Power Function! Case 2 If n is a positive even integer and x n has a negative coefficient The right sides of the graphs has x values approaching larger and larger positive values x + and y values approaching smaller and smaller negative values ( y ) The left side of the graphs has x values approaching smaller and smaller negative values x and y values approaching smaller and smaller negative values ( y ) Theorem If n is a positive even integer and x n has a negative coefficient left end of the graph!! right end of the graph x xn = x + xn = 4 3A Power Function Limits! Page 4 of 9! 2018 Eitel
The behavior of the Right and Left Ends of the Power Function Case 3 If n is a positive ODD integer and +x n has a positive coefficient The right sides of the graphs has x values approaching larger and larger positive values x + and y values approaching larger and larger positive values ( y + ) The left side of the graph has x values approaching smaller and smaller negative values x + and y values approaching smaller and smaller negative values ( y ) Theorem If n is a positive ODD integer and +x n has a positive coefficient left end of the graph! right end of the graph x + xn = x + + xn = + 4 3A Power Function Limits! Page 5 of 9! 2018 Eitel
The behavior of the Right and Left Ends of the Power Function! Case 4 If n is a positive ODD integer and x n has a negative coefficient The right sides of the graphs has x values approaching larger and larger positive values x + and the y values approaching smaller and smaller negative values ( y ) The left side of the graphs has x values approaching smaller and smaller negative values x and the y values approaching larger and larger negative values ( y + ) Theorem If n is a positive ODD integer and x n has a negative coefficient left end of the graph! x xn = right end of the graph x + xn = 4 3A Power Function Limits! Page 6 of 9! 2018 Eitel
The it question The behavior of the Right and Left Ends of the Power Function x + Limits of Power Functions f (x) =?? asks what the y values at the right end of the graph are getting close to as the values for x become larger and larger positive numbers. The it question x f (x) =?? asks what the y values at the left end of the graph are getting close to as the values for x become smaller and smaller negative numbers. There are 4 cases that we will need to examine. They are represented by the 4 functions n even Z +! n even Z +! n odd Z +! n odd Z + 4 3A Power Function Limits! Page 7 of 9! 2018 Eitel
Expanding factored forms to find the leading terms It is easy to find the ratio of the leading terms of a power function if the polynomial is written with the terms listed in decreasing powers of x. The leading terms of f (x) = 3x 3 2x 2 + 4x + 3 is 3x 3 In many cases the rational expression will be in factored form like the function shown below. f (x) = (3x 4)3 (x + 3) 6x 2 (x 1) 2 If the polynomial is written factored form you will need to expand the factors to find the leading terms. Do I really need to FOIL the factors? That would be a larger job and take a lot to time. The good news is that all you really only need to find is the leading term of the polynomial. We can find the leading terms without FOILing the entire list of factors.! Example 1 Find the leading terms of the polynomial f (x) = (2x 1) 3 (x + 3 rewrite each factor with only the leading term for each factor = (2x) 3 (x) multiply those terms to find the leading term = 8x 3 x = 8x 4! Example 2 (3x 4) 3 (x + 3) 4 rewrite each factor with only the leading term for each factor = (3x) 3 (x) 4 multiply those terms to find the leading term = 27x 3 x 4 = 27x 7 4 3A Power Function Limits! Page 8 of 9! 2018 Eitel
! Example 3 Find the leading terms of the polynomial 4x 3 (x 1) 2 rewrite each factor with only the leading term for each factor = 4x 3 (x) 2 multiply those terms to find the leading term = 4x 3 x 2 = 4x 5! Example 4 Find the leading terms of the polynomial 8x(x + 5)(x 1) 2 rewrite each factor with only the leading term for each factor = 8x (x)(x) 2 multiply those terms to find the leading ter = 8x x x = 8x 4! 4 3A Power Function Limits! Page 9 of 9! 2018 Eitel