MAT 105 SP09 Functions and Graphs Chapter 3 Functions and Graphs

Section 3.1 Introduction to Functions MAT 105 SP09 Section 3.1 Introduction to Functions

e.g. Students’ heights and their corresponding weight. 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.

Definition: 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.

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

MAT 105 SP09 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

Examples 1) 2) Determine if the following relations are functions. 1 5 1) 2) 1 5 3 -2 8 1 2 3 6

Examples Determine if the following relations are functions.

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

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)

Examples

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

Section 3.2 More about Functions

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.

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.

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

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

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

Piecewise function

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

Section 3.3 Rectangular Coordinate Plane

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

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

Section 3.4 Graphing Functions

The Vertical Line Test 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.

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

Section 3.5 Shifting, Reflecting, and Sketching Graphs MAT 105 SP09 Section 3.5 Shifting, Reflecting, and Sketching Graphs

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

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

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

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

Vertical and Horizontal Shifts MAT 105 SP09 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.

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).

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

Vertical and Horizontal Shifts (continued) Take a look* Name the transformations that f(x) = x2 must undergo to obtain the graph of g(x) = (x + 3)2 – 5 Solution: You can obtain the graph of g by shifting f 3 units ____________ and 5 units ____________________. f(x) g(x)

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

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

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:

MAT 105 SP09 b) Elementary Function: Transformations: c)

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.) Hit  to access the CALC menu Select 2: Zero Left bound? Move the cursor to the left of the x-intercept using your arrow keys. Hit  Right bound? Move the cursor to the right of the x-intercept using your . Hit . 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.

Example Solve

Example P 101 # 48 a) Find the height of the rocket 3.8 s into flight. (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.

Example P 101 # 52 Write a function for the volume in terms of x. Find x when V(x) = 90 in3 to 2 sig digits. What value of x will yield the maximum volume?