Math 234, Calculus for BusinessLecture 2, Textbook Sections 2.1, 2.2, 2.3Quadratic, Polynomial, and Rational Functions,Translations & ReflectionsUniversity of Illinois, Urbana-ChampaignJanuary 23rd, 2017Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 1 / 25AnnouncementsIn Math 234 site, go to the side link “Frequently Asked Questions”.Quiz 1 is tomorrow in section. The coverage is Group Work 1 (abbr.,GW 1), GW2, Lecture 1, textbook Sections 1.1, 1.2, 2.1, andproblems 1–5 in HW1.HW1 is due Thursday 1/26, 11:59pm; late HWs are not accepted.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 2 / 25Section 2.1, Properties of Functions ReviewDefinitionA function is a way to take an input value (independent variable) andproduce an output value (dependent variable).DefinitionThe domain of a function is the set of all possible input values.DefinitionThe range of a function is the set of all possible output values.RemarkThe vertical line test asserts that if a vertical line intersects a graph attwo or more points, then this graph does not correspond to any function.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 3 / 25Section 2.2, Quadratic Functions, Translation and Reflection Graphing Quadratic FunctionsDefinitionA quadratic function is a function of the formf (x) = ax2+ bx + cwhere a, b, c are real numbers and a 6= 0.DefinitionThe graph of a quadratic function is called a parabola.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 4 / 25Section 2.2, Quadratic Functions, Translation and Reflection Graphing Quadratic FunctionsRemark (Graphs of f (x) = ax2+ bx + c and “openings”)Parabolas can open up or down. Without xy-axes, parabolas look likea > 0a < 0RemarkTwo methods to graph f (x) = ax2+ bx + c with xy-axes are:(1) Completing the square into vertex form, OR(2) Find x-intercepts (if any) as well as the vertex of the parabola.First, focus on Method 2.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 5 / 25Section 2.2, Quadratic Functions, Translation and Reflection Method 2: Vertex and x -interceptsDefinitionThe vertex of a parabola is the highest point of a downward openingparabola, or the lowest point of an upward opening parabola.DefinitionThe root(s) of f (x) = ax2+ bx + c are the value(s) of x that solvef (x) = ax2+ bx + c = 0.RemarkThe graph of f (x) crosses x-axis at these roots. The crossings are calledthe x-intercepts.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 6 / 25Section 2.2, Quadratic Functions, Translation and Reflection Method 2: Vertex and x -interceptsTheoremThe coordinates of the vertex of the graph of f (x) = ax2+ bx + c is(x, y) =−b2a, f−b2aTheorem (Quadratic Formula)The root(s) of f (x) = ax2+ bx + c arex =−b ±√b2− 4ac2aRemarkSee Section 2.2, Example 4 for further discussion of these theorems.Other ways of finding roots are factoring, completing the square, etc.Groupwork 2 entails the derivation of the quadratic formula.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 7 / 25Section 2.2, Quadratic Functions, Translation and Reflection Method 2: Vertex and x -interceptsExampleGraph f (x) = 2x − x2+ 3.Solution (Step 1: Find vertex and x-intercepts)In terms of f (x) = ax2+ bx + c, note a = −1, b = 2, c = 3. The vertex is(x, y ) =−b2a, f−b2a=−22(−1), f−22(−1)=1, f (1)=1, 2(1) − (1)2+ 3= (1, 4)The roots (and the location of the x-intercepts) arex =−b ±√b2− 4ac2a=−2 ±p22− 4(−1)(3)2(−1)=−2 ±√16−2= −1 as well as 3Or, find roots by factoring f (x) = −(x2− 2x − 3) = −(x + 1)(x − 3).Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 8 / 25Section 2.2, Quadratic Functions, Translation and Reflection Method 2: Vertex and x -interceptsExampleGraph f (x) = 2x − x2+ 3.Solution (Step 2: Draw the parabola through the vertex and x-intercepts)The vertex is (1, 4) and the x-intercepts are x = −1 and x = 3, soQuadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 9 / 25Section 2.2, Quadratic Functions, Translation and Reflection Method 1 Primer: Baseline ExampleNow, focus on Method 1.Use a baseline example: f (x) = x2Let’s modify this function and its graph.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 10 / 25Section 2.2, Quadratic Functions, Translation and Reflection Method 1: Vertex FormFor this slide, fix f (x) = x2as the baseline example.Graph of f (x) + 1 = x2+ 1 is Graph of f (x) −1 = x2− 1 isMoves graph of f (x) up by 1 Moves graph of f (x) down by 1TheoremThe graph of f (x) + k moves (or translates) graph of f (x) upward by k.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 11 / 25Section 2.2, Quadratic Functions, Translation and Reflection Method 1: Vertex FormFor this slide, fix f (x) = x2as the baseline example.Graph of f (x − 2) = (x − 2)2is Graph of f (x + 1) = (x + 1)2isMoves graph of f (x) right by 2 Moves graph of f (x) left by 1TheoremThe graph of f (x − h) moves (or translates) graph of f (x) right by h.Quadratic, Polynomial, and Rational Functions, Translations & Reflections (University of Illinois, Urbana-Champaign)Math 234, Calculus for Business January 23rd, 2017 12 / 25Section 2.2, Quadratic Functions, Translation and Reflection Method 1: Vertex FormFor