Polynomial and Rational Functions along with Miscellaneous Equations. Chapter 3

Transcription

1 Polynomial and Rational Functions along with Miscellaneous Equations Chapter 3

2 Quadratic Functions and Inequalities 3.1

3 Quadratic Function A quadratic function is defined by a quadratic or second-degree polynomial. Standard Form f(x) = ax 2 + bx + c, where a 0. Vertex Form f(x) = a(x h) 2 + k, where a 0. The vertex form of f(x)=3x 2 +24x+46 is f(x)=3(x+4) 2-2

4 Vertex and Axis of Symmetry The point (h,k) is the vertex of the parabola. Finding the vertex: ( b., b.) 2a 2 f a The vertical line x = b 2a symmetry for the graph of f(x) = ax 2 + bx + c. is called the axis of

5 Opening and Maximum and Minimum If a > 0, the graph of a quadratic function opens upward; if a < 0, the graph opens downward. If a>0, k is the minimum value of the function If a<0, k is the maximum value of the function f(x)=3x 2 +24x+46 opens upward and has a minimum value, as a=3 Does f(x)=-x 2 +2x open upward or downward? Does it have a maximum or minimum value?

6 Intercepts The y-intercept is found by letting x=0 and solving for y. (0,y) The x-intercepts are found by letting y=0 and solving for x. Solve by factoring, square roots, or the quadratic formula.

7 Example Identify the vertex, axis of symmetry, maximum or minimum, y intercept, and x- intercepts of y=2x 2 +8x+6. Rewrite the equation in vertex form. Vertex: (-2,-2) Axis of Symmetry: x=-2 Minimum: -2 Y-intercept: (0,6) X-intercepts: (-3,0) and (-1,0) Vertex form: f(x)= 2(x+2) 2-2

8 Quadratic Inequalities Strategy: Solving a Quadratic Inequality by the Graphical Method 1. Get 0 on one side of the inequality and a quadratic polynomial on the other side. 2. Find all roots to the quadratic polynomial. 3. Graph the corresponding quadratic function. The roots found in step (2) determine the x- intercepts. 4. Read the solution set to the inequality from the graph of the parabola.

9 Example Solve the inequality and graph it. Write your answer in interval notation. x 2 +6x > -8 (-, -4)U(-2, ) Try 2x+15 < x 2 on your own

10 Zeros of Polynomial Functions 3.2

11 Division of Polynomials Long Division Can be used to divide any two polynomials. Synthetic Division Can only be used to divide two polynomials when dividing by x-k.

12 Remainder Theorem If R is the remainder when a polynomial P(x) is divided by x c, then R = P(c).

13 Long Division What is (x 2 +6x+8) (x+4)? x+2, remainder = 0 What is h(-4)? h(-4)=0 *notice: h(-4) = remainder

14 Synthetic Division What is (-3x 3 +5x 2-6x+1) (x+1) -3x 2 +8x-14, Remainder =15

15 Rational Zero Theorem If f(x) = a n x n + a n 1 x n 1 + a n 2 x n a 1 x + a 0 is a polynomial function with integral coefficients (a n 0 and a 0 0) and p/q (in lowest terms) is a rational zero of f(x), then p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n. To find the rational zeros, divided all the factors of the constant term by all the factors of the lead coefficient.

16 Examples List all the rational zeros and find all the real and imaginary zeros of h(x)=x 3 -x 2-7x+15 Rational zeros: ±(1,3,5,15) x=-3, 2+i, 2-i

17 Theory of Equations 3.3

18 Multiplicity and Conjugate Pairs Multiplicity: If the factor x c occurs k times in the complete factorization of the polynomial P(x), then c is called a root of P(x) = 0 with multiplicity k. Multiplicity is the number of times a zero occurs. Conjugate Pairs Theorem: If P(x) = 0 is a polynomial equation with real coefficients and the complex number a + bi (b 0) is a root, then a bi is also a root

19 Examples State the degree, find all real and imaginary zeros, and state their multiplicities. f(x)=x 5-6x 4 +9x 3 Degree 5 x=0, multiplicity 3 x=3, multiplicity 2

20 Find a polynomial with the given roots: 5, 4-3i x 3-13x 2 +65x-125

21 Graphs of Polynomial Functions 3.5

22 Symmetry Symmetric about the y-axis: f(x) is an even function if f(-x)= f(x) Symmetric about the origin: f(x) is an odd function if f(-x)= -f(x) A quadratic function is symmetric about the axis of symmetry if x= b 2a

23 Examples Is f(x)= x 6 -x 4 +x 2-8 an even or odd function? f(-x) = (-x) 6 -(-x) 4 +(-x) 2-8 = x 6 -x 4 +x 2-8 = f(x) f(-x)=f(x) so f(x) is an even function Is f(x)=4x 3 -x an even or odd function? f(-x)= 4(-x) 3 -(-x) = -4x 3 +x = -(4x 3 -x) = -f(x) f(-x)= -f(x) so f(x) is an odd function

24 Behavior of x-intercepts Even Multiplicity: the graph touches but does NOT cross the x-axis at the x- intercept Odd Multiplicity: the graph crosses the x- axis at the x-intercept

25 Examples Find the x-intercepts, discuss the behavior at each intercept, and graph. f(x)=x 3-3x 2 X-intercepts: (0,0) multiplicity 2-touches at x-axis (3,0) multiplicity 1-crosses at x-axis

26 Leading Coefficient Test

27 Polynomial Inequalities Very similar to solving Quadratic Inequalities. Strategy: Solving a Polynomial Inequality by the Graphical Method 1. Get 0 on one side of the inequality and a polynomial on the other side. 2. Find all roots to the polynomial. 3. Graph the corresponding function. The roots found in step (2) determine the x-intercepts.

28 Examples Solve x 3 +4x 2 -x-4>0 and write your answer in interval notation. (-4,-1)U(1,inf) Try x 3 +2x 2-2x-4<0 on your own

29 Rational Functions and Inequalities 3.6

30 Rational Functions If P(x) and Q(x) are polynomials, then a function of the form f x = P(x) Q(x) is called a rational function, provided that Q(x) is not the zero polynomial.

31 Vertical Asymptotes An invisible line that the function is always approaching but never reaching. Vertical asymptotes correspond to where Q(x)= 0.

32 Horizontal Asymptotes If the numerator has a lower degree than the denominator, the horizontal asymptote is the line y=0. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the line y=a/b where a is the lead coefficient of the numerator and b is the lead coefficient of the denominator.

33 Oblique (Slant) Asymptotes If the degree of the numerator is one degree higher than the degree of the denominator, the graph of the function has an oblique asymptote. Divide the numerator by the denominator, the quotient (without the remainder) is your oblique asymptote.

34 Example Find the horizontal and vertical asymptotes of f x = 2x x 2 +6x+9 To find the vertical asymptote solve x 2 +6x+9=0 X=-3 To find the horizontal asymptote notice that the degree of the numerator is less than the degree of the denominator. y=0

35 Example Find all asymptotes, x-intercepts, and y- intercepts of f x = 4x, and state the x 2 2x+1 domain. V.A. at x= 1 H.A. at y = 0 x & y intercept (0,0) Domain (-, 1)U(1, )

36 Rational Inequality Example: Solve x 1 x 3 > 0

37 Miscellaneous Equations 3.4

38 Miscellaneous Equations Equations involving absolute value can include more than one absolute value or contain higher degree polynomials where the definition for absolute value is used to determine the solutions. Equations involving square roots are solved by squaring both sides once a radical is isolated on one side of the equation. Equations with rational exponents are solved by raising both sides of the equation to a reciprocal power and considering positive and negative possibilities for even roots. Equations of quadratic type can be solved by substituting a single variable for a more complicated expression. Factoring is often the fastest method for solving an equation.

39 Copyright 2011 Pearson Education, Inc. Examples Absolute Value Examples: #1. v 2 3v = 5v #2. x + 5 = 2x + 1 #3. x 4 1 = 4x

40 Copyright 2011 Pearson Education, Inc. Examples Square Root Examples: #4. x + 1 = x 5 #5. 1 z = 3 4z+1 #6. x + 40 x = 4

41 Copyright 2011 Pearson Education, Inc. Examples Rational Exponent Examples: #7. x 2 3 = 2 #8. w 3 2 = 27 #9. (t 1) 1 2= 1 2

42 Copyright 2011 Pearson Education, Inc. Examples Quadratic Type Examples: #10. x 4 + 6x 2 7 = 0 #11. x 4 x 2 12 = 0 #12. x 7x = 0

43 Copyright 2011 Pearson Education, Inc. Examples Solving Higher Degree Polynomials with Factoring: # x 2x 2 = x 3 #14. 2x x 2 x 500 = 0 #15. x 4 81 = 0

44 Thank You for Coming!