Math 234, Calculus for BusinessLecture 2, Textbook Sections 2.1, 2.2, 2.3Quadratic, Polynomial, and Rational Functions,Translations & ReflectionsJanuary 23rd, 2017Print out these sheets to fill out during or after the lecture. They are for your own learning benefit!Announcements(1) In Math 234 site, go to the side link “Frequently Asked Questions”. (2) 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, and problems1–5 in HW1. (3) HW1 is due Thursday 1/26, 11:59pm; late HWs are not accepted.Section 2.1, Properties of FunctionsDefinition. A function is a way to take and produce .Definition. The domain of a function isDefinition. The range of a function isThe vertical line test assertsSection 2.2, Quadratic Functions, Translation and ReflectionDefinition. A quadratic function is a function of the formwhere are and .Definition. The graph of a quadratic function is called a1If a > 0, then the graph looks like If a < 0, then the graph looks liksTwo methods to graph f(x) = ax2+ bx + c with xy-axes are:(1)(2)First, focus on Method 2.Method 2.Definition. The vertex of a isDefinition. The root(s) of f (x) = ax2+ bx + c areThe graph of f(x) crosses x-axis at these roots. The crossings are called the .Theorem 1. The coordinates of the vertex of the graph of f(x) = ax2+ bx + c isTheorem 2 (Quadratic Formula). The root(s) of f (x) = ax2+ bx + c areExample 3. Graph f(x) = 2x − x2+ 3.Solution. Step 1: Find andThe is/areThe is/are2Step 2: Draw the through the andMethod 1.Use a baseline example: f(x) = x2Graph of f(x) + 1 = x2+ 1 is Graph of f(x) − 1 = x2− 1 isTheorem 4. The graph of f(x) + k the graph of f(x) .Graph of f(x − 2) = (x − 2)2is Graph of f(x + 1) = (x + 1)2isTheorem 5. The graph of f(x − h) the graph of f(x) .Graph of 4f(x) = 4x2is Graph of −14f(x) = −14x2is3Theorem 6. The graph of af(x) the graph of f(x) .f(x − h)f(x + h)f(x) + kf(x) − k−f(x)f(−x)For further details and pictures, see the green box titled “Translations and Reflections of Functions” nearthe end of Section 2.2. Also, see tomorrow’s groupwork.Graph of (x − 2)2+ 1 is Graph of −3(x + 2)2− 1 isTheorem 7. Every quadratic function can be written in theThe vertex of this graph is .Question 1. To complete the square and go from theto thewhat is the expression of in terms of ?4Quadratic Function ExampleExample 8. The landlord of a 100-unit apartment complex wants to set rental rates. She finds that settingrent at $600 per month fills all units. Whenever she raises rent by $15 per month, 1 additional unit becomesvacant.For example, charging $ for rent means only units are filled.What is the rent that maximizes her revenue? What is the max revenue?Solution. Step 1: Find the independent variable. What does she control?The landlord controls . So, letx =Step 2: Determine the revenue function.Recall, revenue =Thus, the revenue function is R(x) = .Step 3: Maximize R(x).Multiplying out the two factors givesSince , the parabola opens : the vertex is the point!Coordinates of vertex =The max revenue is , with monthly rent .Section 2.3, Polynomial and Rational FunctionsDefinition. A polynomial function of degree n is of the formwhere are and . (Here, n ≥ 0 is an integer.)Definition. The zero polynomial isExample 9. The degree of 0x12+ x3− 3x4− 1 isDefinition. The roots (or zeros) of p(x) are5Definition. A power function isThe graphs of x2, x4, and x6areThe graphs of x3, x5, and x7areIncreasing power means the function (the graph is ).Theorem 10. A polynomial function of degree n• has at most roots• has at most“turning points” on its graph.Example 11. The graph of a mystery polynomial isWhat is the smallest degree of this polynomial?Answer: With roots and “turning points”, the smallest degree is n = 4.Definition. A rational function r(x) isand is not the .Example 12. The function r(x) =x + 1x − 3is a rational function. Observe thatAs x gets close to , the value of r(x) .6The vertical line is a of r(x).As x grows larger, the value of r(x) .The line is a of r(x).Review algebraic methods for finding horizontal asymptotes.The graphs of f(x) =1x7, f(x) =1x7+ 5, f(x) =1x2, and f(x)
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