Lesson 2.1: Quadratic Functions

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1 Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard quadratic function form. *Use evaluation and solve for the missing value of a when given a vertex and a point. How does the vertex of a parabola relate to the Maximum or Minimum of a quadratic function? Is it possible to find the vertex of a parabola written in general form without completing the square? For each function find the (a)vertex (b) x & y intercepts (c) zeros (d) sketch it.

2 Write the following parabola in vertex form: Vertex (3, -2) passes through (5,7) Vertex (-1,2); passes through (4,4) #62, #64 True or False: Without using your calculator, do the graphs of and have the same axis of symmetry.

3 Lesson 2.2: Polynomial Functions of Higher Degree *Graphs of polynomial functions are continuous, meaning it has no breaks. *Graphs of polynomial functions only have smooth rounded turns as opposed to sharp pointed turns. Polynomial function: When n is odd: END BEHAVIOR: A graph with a positive leading coefficient falls to the left and rises to the right. Left end-behavior: Right end-behavior: A graph with a negative leading coefficient rises to the left and falls to the right. Left end-behavior: Right end-behavior: When n is even: A graph with a positive leading coefficient rises to the left and right. Left end-behavior: Right end-behavior: A graph with a negative leading coefficient falls to the left and right. Left end-behavior: Right end-behavior: The Intermediate Value theorem: Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a) f(b), then in the interval [a,b], f takes on every value between f(a) and f(b).

4 Polynomial function: The graph of f has at most n real zeros. The function f has at most (n 1) relative extrema. If f is a polynomial function and a is a real number: 1. x = a is a zero of the function f. 2. x = a is a solution of the polynomial equation f(x) = (x a) is a factor of the polynomial f(x). 4. (a, 0) is an x-intercept of the graph of f. In general, a factor of yields a repeated zero x = a of multiplicity k: If k is odd, the graph crosses the x-axis at x = a If k is even, the graph touches (but does not cross) the x-axis at x = a Find all the real zeros of the polynomial functions Find a polynomial function that has the given zeros. 2, -6 0,2,5 Can you solve these equations? From which polynomial (simplest) did we come from?

5 Analyze the given functions providing the x&y intercepts, zeros, action at the zeros, endbehavior. Then use your calculator to find the relative maximums & minimums, and give the increasing & decreasing intervals. #86, #88

6 Lesson 2.3: Real Zeros of Polynomial Functions Division Algorithm: Dividend = Divisor*Quotient + Remainder Long Division: Synthetic Division (when the divisor is in the form of x k) The Remainder Theorem: If a polynomial f(x) is divided by x k, the remainder is r = f(k) Use the remainder theorem to find the remainder. The Factor Theorem: A polynomial f(x) has a factor (x k) if and only if f(k) = 0 Is (x-1) a factor of the following function? f (x) = 2x 3 + 3x 2 8x + 3 The Rational Zero Test: see Pg. 118 List the possible rational zeros of f, use a calculator to narrow the list down, find the zeros. Given one real zero of a function find the others. ; x = ½

7 Lesson 2.4: Complex Numbers Complex number in standard form: a + bi, for real numbers a and b Conjugate: 4+6i -6 10i Complex plane: Simplify:

8 Lesson 2.5: The Fundamental Theorem of Algebra Linear Factorization Theorem: If f(x) is a polynomial of degree n, where n > 0, f has precisely n linear factors where are complex numbers and is the leading coefficient of f(x) If a + bi, where b 0, is a zero of a function, then the conjugate a bi is also a zero. Find all the zeros of the function and write the polynomial as a product of linear factors. Find the polynomial function with integer coefficients that has the given zeros. 2, 4 + i, 4 i 2,2,2,4i, -4i Use the given zero to find all the zeros of the function. ; zero: r = 2i ; zero:

9 Lesson 2.6: Rational Functions and Asymptotes Rational Function:, p(x) & q(x) are polynomials and q(x) isn t the zero polynomial. Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function y = f(x) if either ; where p(x) and q(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of q(x) 2. The graph of f has at most one horizontal asymptote, as follows a. If n < m, the x-axis (y = 0) is a horizontal asymptote b. If n = m, the line is a horizontal asymptote c. If n > m, the graph of f has no horizontal asymptote Analyzing graphs of rational functions 1.The y-int (if any) is the value of f(0) 2. The x-ints (if any) are the zeros of the numerator 3. The vert. asy.(if any) are the zeros of the denominator 4. The horizontal asy.: see a,b,and c above * It s all about the degree! Find the domain, and identify any horizontal and vertical asymptotes

10 f (x) = x 5 x Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. g(x) = x 2 4 x + 3 h(x) = x 2 +1 Write a rational function f having the specified characteristics. (There are many correct answers.) Vertical asymptotes: x = -2, x = 1 Vertical asymptotes: x = 0, x = 2.5 x = 5 Horizontal asymptotes: y = -3

11 Lesson 2.6: Rational Functions and Asymptotes Rational Function:, p(x) & q(x) are polynomials and q(x) isn t the zero polynomial. Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function y = f(x) if either ; where p(x) and q(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of q(x) 2. The graph of f has at most one horizontal asymptote, as follows a. If n < m, the x-axis (y = 0) is a horizontal asymptote b. If n = m, the line is a horizontal asymptote c. If n > m, the graph of f has no horizontal asymptote Analyzing graphs of rational functions 1.The y-int (if any) is the value of f(0) 2. The x-ints (if any) are the zeros of the numerator 3. The vert. asy.(if any) are the zeros of the denominator 4. The horizontal asy.: see a,b,and c above * It s all about the degree! Find the domain, and identify any horizontal and vertical asymptotes

12 f (x) = x 5 x Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. g(x) = x 2 4 x + 3 h(x) = x 2 +1 Write a rational function f having the specified characteristics. (There are many correct answers.) Vertical asymptotes: x = -2, x = 1 Vertical asymptotes: x = 0, x = 2.5 x = 5 Horizontal asymptotes: y = -3

13 Lesson 2.7: Graphs of rational functions If the degree of the numerator is exactly 1 more than the degree of the denominator, the graph of the rational function has a slant asymptote *use division to identify the slant asymptote. Find the intercepts and asymptotes to help you sketch the given rational functions.

Chapter 2 Formulas and Definitions:

Chapter 2 Formulas and Definitions: (from 2.1) Definition of Polynomial Function: Let n be a nonnegative integer and let a n,a n 1,...,a 2,a 1,a 0 be real numbers with a n 0. The function given by f (x)