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1 Chapter 3 Functions and Graphs

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2 Section 3.1 Introduction to Functions

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3 A _____________ is a correspondence between two quantities. e.g. Students’ heights and their corresponding weight. e.g. Authors’ names and books they have written. A function is a special type of relation.

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4 A function is a correspondence between two sets such that every member of the first set, called the domain, maps to exactly one member of the second set, called the range. Definition:

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5 Mathematically, functions are often represented as equations. Functions can also be represented by mapping diagrams, tables, sets of ordered pairs, graphs.

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6 Function Diagram You can visualize a function by the following diagram which shows a correspondence between two sets: D, the domain of the function, gives the diameter of pizzas, and R, the range of the function gives the cost of the pizza. 10 12 16 9.00 12.00 10.00 domain D range R

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7 Examples Determine if the following relations are functions. 1) 2) 1515 3 -2 8 123123 6

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8 Examples Determine if the following relations are functions.

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Examples Determine if the following relations are functions. Here are some things to look for (these equations do NOT represent functions): 9

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10 Functional Notation f(x) “function of x” x is the independent variable Note: This notation does NOT mean f multiplied by x! Given f(x) = -2x + 5, a) find f(2) b) find f(a + 3)

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11 Examples

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12 Examples The surface area of a sphere is given by Find the surface area of a sphere with radius 3.1 in.

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13 Section 3.2 More about Functions

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14 Domain & Range The _______________ of a function is the set of possible values of the independent variable (input). The ______________ of a function is the set of possible resulting output values of the dependent variable. Note: We will consider only real numbers for the domain values of the functions we will study in this class.

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15 Two situations to watch for: 1) Input values that lead to a ZERO IN THE DENOMINATOR ----Function has a variable in the denominator 2) Input values that lead to a NEGATIVE NUMBER UNDER THE SQUARE ROOT (or any even root.) -Function has a variable under the radical In these situations, we often have to restrict the domain, excluding the “bad values”. For most other functions that we will study this semester, the domain will be the set of ALL REAL NUMBERS.

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16 Find the domain and range of each of the functions. Function Domain Range

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17 Find the domain and range of each of the functions. Function Domain Range

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18 Practical Domain Write the area of the rectangle as a function of x. What is the practical domain of A(x)?

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19 Piecewise function

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20 Writing functions from Verbal Statements P. 88 in text #24, 32, 38

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21 Section 3.3 Rectangular Coordinate Plane

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22 Horizontal (x) axis Ordered pair (4, 3) Vertical (y) axis Origin (0, 0) IV III III Points in the Cartesian Plane

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23 “Name that Quadrant” Name the quadrant(s) where we would find: 1.All points whose x-coordinate is 5 2.All points for which 3.All points for which y < 0 4.All points for which xy > 0 5.All points for which xy < 0

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24 Section 3.4 Graphing Functions

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25 The Vertical Line Test A graph can show you whether or not a given relation is a function. The VERTICAL LINE TEST: If a vertical line can be drawn that intersects the graph in more than one point, then the graph does NOT represent a function.

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26 Which are graphs of functions? 1) 2) 3) 4) 5)

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27 Section 3.5 Shifting, Reflecting, and Sketching Graphs

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28 Seven Graphs of Common (Elementary) Functions ___________ Function Domain: _________Range: _________ # 1# 2

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29 ______________ Function_________________ Function Domain: _________ Range: _________ # 3# 4 Seven Graphs of Common (Elementary) Functions (cont)

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30 _______________ Function____________ Function Domain: _________ Range: _________ # 5#6 Seven Graphs of Common (Elementary) Functions (cont)

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31 _______________ Function Domain: _________ Range: _________ # 7 Seven Graphs of Common (Elementary) Functions (cont)

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32 Vertical and Horizontal Shifts Shifts, also called translations, are simple transformations of the graph of a function whereby each point of the graph is shifted a certain number of units vertically and/or horizontally. The shape of the graph remains the same. Vertical shifts are shifts upward or downward. Horizontal shifts are shifts to the right or to the left.

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33 Vertical and Horizontal Shifts Example Graph the function on your calculator and describe the transformation that the graph of must undergo to obtain the graph of h(x).

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34 Vertical and Horizontal Shifts Vertical and Horizontal Shifts of the Graph of y = f(x) For c, a positive real number. 1.Vertical shift c units UPWARD: 2.Vertical shift c units DOWNWARD: 3.Horizontal shift c units to the RIGHT: 4.Horizontal shift c units to the LEFT:

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35 Vertical and Horizontal Shifts (continued) Take a look* Name the transformations that f(x) = x 2 must undergo to obtain the graph of g(x) = (x + 3) 2 – 5 f(x) g(x) Solution: You can obtain the graph of g by shifting f 3 units ____________ and 5 units ____________________.

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36 Reflecting Graphs A reflection is a mirror image of the graph in a certain line. Reflection in the x-axis:y = −f(x)

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37 Solution: You can obtain the graph of g by reflecting f in the x-axis. Reflecting Graphs (continued) Take a look* Sketch the graph of and. g(x) f(x) Elementary Function Reflection in x-axis

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38 Examples Describe the transformations of each of the following graphs as compared to the graph of its elementary function. Then sketch the graph of the transformed function. a) Elementary Function: Transformations:

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39 b) Elementary Function: Transformations: c) Elementary Function: Transformations:

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40 Solving Equations Using the TI-83 or TI-84 We can use our graphing calculators to solve equations graphically. Collect all of the terms on one side of the equation to get one side equal to zero. Enter the expression into y= editor Graph the function. You may want to start with a standard window ( ). Adjust your window manually using the setting.| Find the points where y = 0. These are called the x-intercepts of the graph (the points where the graph crosses or touches the x-axis.) i. Hit to access the CALC menu ii. Select 2: Zero iii. Left bound? Move the cursor to the left of the x-intercept using your arrow keys. Hit iv. Right bound? Move the cursor to the right of the x-intercept using your . Hit . v. Guess? Move the cursor close to the x-intercept. Hit . The x-coordinate is called a zero or a root of the function and is the solution of the equation f(x) = 0. You need to repeat these steps to find additional x-intercepts.

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41 Example Solve

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42 Example P 101 # 48 (Give answers to 2 sig. digits.) a) Find the height of the rocket 3.8 s into flight. b) Find the maximum height that the rocket attains. c) Determine when the rocket is at ground level.

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43 Example P 101 # 52 Write a function for the volume in terms of x. Find x when V(x) = 90 in 3 to 2 sig digits. What value of x will yield the maximum volume?

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