3.1 Power Functions & Polynomial Functions

Transcription

1 3.1 Power Functions & Polynomial Functions A power function is a function that can be represented in the form f() = p, where the base is a variable and the eponent, p, is a number. The Effect of the Power P y = 0 = 1 y = 1 = Positive integer power (Even) y = vs 4 (Odd) y = 3 vs 5 The same characteristic U-shaped. Symmetric about the y- ais. 4 is flatter near the origin and steeper away from the origin than. Negative odd integer powers The same characteristic chair -shaped. Symmetric about the origin. 5 is flatter near the origin and steeper away from the origin than 3. Negative even integer powers

2 Polynomial Functions and Their Graphs P and answer the following: a n : a 7 : a 6 : a 5 : a 4 : E 1: Consider the function Write P in standard form: The degree of this polynomial: The leading term: The leading coefficient: Constant: a 3 : a : a 1 : a 0 : E : Which of the following are polynomial functions: If they are polynomial, state the degree, a n and a 0 Polynomial: Degree a n P Q R S T 10 U V W 1 Graph of polynomial: Polynomial functions must be smooth and continuous functions. By smooth, the graph contains no sharp corners or cusps. By continuous, the graph has no gaps or holes. a 0

3 Turning Points Theorem: If f is a polynomial function of degree n, then f has at most (n-1) turning points. If the graph of a polynomial function f has (n-1) turning points, the degree of f is at least n. The Long-Run Behavior (End Behavior) of Polynomial The behavior of the graph of a function as the input takes on large negative values ( ) and large positive values ( ) as is referred to as the long run behavior of the function E. Describe the end behavior of the function f() = by completing the following statements: a) As, f()? b) As -, f()? Short Run Behavior Characteristics of the graph such as vertical (y-intercept) and horizontal intercepts (-intercepts) and the places the graph changes direction are part of the short run behavior of the polynomial. E. Find the vertical and horizontal intercepts of each function. f ( 5) k n 34 n (4n 3)

4 3. Quadratic Functions Forms of Quadratic Functions The standard form of a quadratic function is f ( ) a b c The transformation form/ verte form of a quadratic function is f ( ) a( h) k The verte of the quadratic function is located at (h, k), where h and k are the numbers in the transformation form of the function. Because the verte appears in the transformation form, it is often called the verte form. = a( h) + k E1. Use the given graph of f to find the following: Verte: Classify verte: lowest point or highest point? Graph has: minimum y-value or maimum y-value, which is Does the graph open upward or downward? Ais of symmetry: Solve for : 11 Ea Eb. 31 1

5 Find the verte, minimum or maimum, vertical and horizontal intercepts, ais of symmetry, rewrite the quadratic function into verte form, and graph f(). E3. Let 3 4 f. Determine if the graph of f opens upward or downward? The Verte is: The minimum or maimum value is: The Ais of symmetry: Vertical intercept (as a point): Horizontal intercept(s) (as point(s)): Verte form: E4. Let f 18. Determine if the graph of f opens upward or downward? The Verte is: The minimum or maimum value is: The Ais of symmetry: Vertical intercept (as a point): Horizontal intercept(s) (as point(s)): Verte form:

6 Write an equation for a quadratic with the given features 1. -intercepts (, 0) and (5, 0), and y intercept (0, 6) 5. Verte at (-3, ), and passing through (3, -) 7. A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by h t 4.9t 9t 34. a. From what height was the rocket launched? b. How high above sea level does the rocket reach its peak? c. Assuming the rocket will splash down in the ocean, at what time does splashdown occur?

7 3.3 Graphs of Polynomial Functions E1. Describe the end behavior of the function f() = by completing the following statements: a) Graph the End behavior: b)as, f()? c) As -, f()? E: P ( ) 1 Writing Equations using Intercepts Factored Form of Polynomials If a polynomial has horizontal intercepts at 1,,, n, then the polynomial can be written in the factored form p1 p p f ( ) a( ) ( ) ( n 1 n ) where the powers p i on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the horizontal intercept. E. Write an equation for a polynomial the given features. 31. Degree 3. Zeros at = -, = 1, and = 3. Vertical intercept at (0, -4)

8 Multiplicity (Repeated, or multiple zero of P) If c m 1 is a factor of a polynomial P and c m is not a factor of P, then c is called a zero of multiplicity m of P. Note: If m = even, the graph of P touches the -ais at c. If m = odd, the graph of P crosses the -ais at c. m = 1, straight cross m > 1, flattened appearance. 33. Degree 5. Roots of multiplicity at = 3 and = 1, and a root of multiplicity 1 at = -3. Vertical intercept at (0, 9) 35. Degree 5. Double zero at = 1, and triple zero at = 3. Passes through the point (, 15) Solve each inequality

9 E3. Answer the following of the polynomial f. f() = ( 4)(3 ) Degree of f: a n = a 0 = Graph the End behavior: Zeros: Multiplicity: Cross or Touch: E4. Answer the following of the polynomial f. f() = ( 1) 4 (3 ) Degree of f: a n = a 0 = Graph the End behavior: Zeros: Multiplicity: Cross or Touch:

10 E5. Answer the following of the polynomial f, and then graph the function. f ( ) 3 1 Degree of f: a n = Graph the End behavior: Zeros: Multiplicity: Cross or Touch: Test points: y-intercept (as a point): E. Write a formula for each polynomial function graphed

11 3.4 Rational Functions Dividing Polynomials Long Division of Polynomials Divide 84 by 15. Check: Is =? 84? Note: If the dividend is missing a term, we will replace that term with 0 to allow the terms to line up while we do the division process. Answer should be in the form of: Q() + R() D() Synthetic Division Synthetic Division can be used to divide (-k) into a polynomial. For eample, a + b + c k E E

12 Rational Functions r() = P() = a n n +a n 1 n 1 + +a 1 +a 0 Q() b m m +b m 1 m 1, where + +b 1 +b 0 P(), and Q() are polynomial functions and Q() 0. E.r() = E. r() = Domain: Domain: Vertical Asymptotes (Short run behavior) Locating Vertical Asymptotes A rational function r() = P() denominator Q. Note: The domain of r() is the set of all real numbers ecepts those for which the denominator Q is 0., in lowest terms, will have a vertical asymptotes = a if a is a real zero of the Q() Holes of Rational Functions A hole might occur in the graph of a rational function if an input causes both numerator and denominator to be zero. In this case, factor the numerator and denominator and simplify; if the simplified epression still has a zero in the denominator at the original input the original function has a vertical asymptote at the input, otherwise it has a hole. E. r() = ( )( + 1) + 1 Domain:

13 E.r() = V.A: Transformations of y = 1 Graph of y = 1 Graph of y = 1 Graph of y = Transformations of y = 1 Graph of y = 1 1 Graph of Graph of y = ( ) 1 y = ( + 3)

14 Finding Horizontal and Oblique (Slant) Asymptotes of a Rational Function No Horizontal and Oblique (Slant) Asymptotes(n m ): If n m, r has no Horizontal and Oblique (Slant) Asymptotes. In this case, for unbounded, ±, the graph of r will behave like the graph of the quotient. 3 E. r 1 3 r 1 Q 1 Oblique (Slant) Asymptotes (n m = 1): If n m = 1, the quotient obtained is of the form a + b, and the line y = a + b is the oblique asymptote. 4 E. r 1 4 r 1

15 Horizontal Asymptote E. r If n = m, then the graph will have the horizontal asymptote y = a n r b m E. r If n < m, then the graph will have the horizontal asymptote y = r 4 1 3

16 Note: The graph of a function will never intersect a vertical asymptote. The graph of a function may intersect a horizontal asymptote or an oblique asymptote. If n m, r has no Horizontal and Oblique (Slant) Asymptotes. If n m = 1, r has Oblique Asymptote, y = a + b. If n = m, r has Horizontal Asymptote y = a n. b m If n < m, r has Horizontal Asymptote y = 0. E. Of the rational function, find all the - and y- intercepts; the verticalasymptotes; the behavior near by the vertical asymptotes; the horizontal or oblique asymptotes; the intersection between the function and the horizontal or oblique asymptotes,if any; and obtain the graph of r. 4 r 6 5 Factor: -intercepts: y-intercepts: V.A: Behavior nearby the V.A: H.A or O.A: Find the intersection between the graph and H.A or O.A:

17 E: r 1 4 Factor: -intercepts: y-intercepts: V.A: Behavior nearby the V.A: H.A or O.A: Find the intersection between the graph and H.A or O.A:

18 Writing Rational Functions from Intercepts and Asymptotes If a rational function has horizontal intercepts at n,,, 1, and vertical asymptotes at m v v v,,, 1 then the function can be written in the form n n q m q q p n p p v v v a f ) ( ) ( ) ( ) ( ) ( ) ( ) ( where the powers p i or q i on each factor can be determined by the behavior of the graph at the corresponding intercept or asymptote, and the stretch factor a can be determined given a value of the function other than the horizontal intercept, or by the horizontal asymptote if it is nonzero. Write an equation for the function graphed

19 3.5 Inverses and Radical Functions In this section, we will eplore the inverses of polynomial and rational functions, and in particular the radical functions that arise in the process. When we try to find the inverse of polynomial functions, we have a slight difficulty: Because most polynomial functions are not one- to- one, they don t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that they become one- to- one. For Eample: Is the function f() = one- to-one? Is the function h() =, 0, one- to-one? Eercises: For each function, find a domain on which the function is one-to-one and non-decreasing, then find an inverse of the function on this domain.. f 3 6. f 4 Eercises: Find the inverse of each function. 8. f f

20 7 14. f 16. f A drainage canal has a cross-section in the shape of a parabola. Suppose that the canal is 10 feet deep and 0 feet wide at the top. If the water depth in the ditch is 5 feet, how wide is the surface of the water in the ditch? [UW]