Functions and Graphs

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Chapter 3

• Functions and Graphs

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Section 3.1 Introduction to Functions
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• 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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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.

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• 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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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.

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Examples

• Determine if the following relations are
  functions.
• 1) 2)

1 2 3
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Examples

• Determine if the following relations are
  functions.

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Functional Notation

• f(x) function of x x is the independent
  variable
• Note It 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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Examples
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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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Section 3.2 More about Functions
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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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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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Find the domain and range of each of the
functions.

• Function Domain
  Range

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Find the domain and range of each of the
functions.

• Function Domain
  Range

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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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Piecewise function
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Writing functions from Verbal Statements

• P. 88 in text 24, 32, 38

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Section 3.3 Rectangular Coordinate Plane
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Points in the Cartesian Plane
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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 lt 0
• All points for which xy gt 0
• All points for which xy lt 0

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Section 3.4 Graphing Functions
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Graphing Piecewise Functions

• Graph the function by hand

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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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Which are graphs of functions?

• 1)
• 2)
• 3)
• 4)
• 5)

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Section 3.5 Shifting, Reflecting, and Sketching
Graphs
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Seven Graphs of Common Functions
1
2
___________ Function ___________ Function Domain
_________ Domain _________ Range
_________ Range _________
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Seven Graphs of Common Functions (continued)
3
4

______________ Function _________________ Function
Domain _________ Domain _________
Range _________ Range _________

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Seven Graphs of Common Functions (continued)
5
6
_______________ Function ____________ Function
Domain _________ Domain _________
Range _________ Range _________
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Seven Graphs of Common Functions (continued)
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_______________ Function
Domain _________
Range _________
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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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Vertical and Horizontal Shifts
ExampleGraph 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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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
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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
____________________.
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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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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.
Reflection in x-axis
Parent Function
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• Examples
• Describe the transformations of each of the
  following graphs as compared to the graph of its
  parent function. Then sketch the graph of the
  transformed function.
• a) Parent
• Transformations

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b) Parent Transformations c)
Parent Transformations
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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.

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Example

• Solve

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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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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?