Functions & Graphs (1.2) What is a function? Review Domain & Range Boundedness Open & Closed...
-
Upload
patricia-dean -
Category
Documents
-
view
212 -
download
0
Transcript of Functions & Graphs (1.2) What is a function? Review Domain & Range Boundedness Open & Closed...
Functions & Graphs (1.2)What is a function?Review Domain & RangeBoundednessOpen & Closed IntervalsDistance from a point to a line
Even & Odd Functions...
Ex: Identify the domain, range, (use interval notation) and whether the function is odd or even or neither.
y = x2
y = √(1-x2)
y = √x
y = 1/x
y = 2x/(x-1)
Functions Defined in Pieces
While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domain.
These are called piecewise functions.
Examples:
-x ; x < 0 y = x2 ; 0 < x < 1 1 ; x > 1
-x ; 0 < x < 1y =
2x – 2 ; 1< x < 2
The Absolute Value Function
The absolute value function is defined piecewise:
Composite Functions
Suppose that some of the outputs of a function can be used as inputs of
a function . We can then link and to form a new function whose inputs
are inputs of and whose outputs are the numbers
g
f g f
x g ( )( )( )( ) ( )
.
We say that the function read of of is
. The usual standard notation for the composite is ,
which is read " of ."
f g x
f g x f g x
f g
f g
the composite
of and og f
Examples
f(x) = x2 + 1 g(x) = x- 7
Find:g(f(2))
f(g(2))
g(g(3))
f(f(x))
g(f(x))
g(g(x))
Trig ReviewComplete Packet (will be part of
HW #6) on your ownSeek help either during seminar
or at next week’s review session
1.3Exponential Functions
Slide 1- 11
Exponential GrowthExponential DecayApplicationsThe Number e
…and why
Exponential functions model many growth patterns.
What you’ll learn about…
Slide 1- 12
Exponential Function
Let be a positive real number other than 1. The function
( )
is the .
x
a
f x a
a
=
exponential function with base
The domain of f(x) = ax is (-∞, ∞) and the range is (0, ∞). Compound interest investment and population growth are examples of exponential growth.
Slide 1- 13
Exponential Growth
If 1 the graph of looks like the graph
of 2 in Figure 1.22ax
a f
y=
Slide 1- 14
Exponential Decay
If 0 1 the graph of looks like the graph
of 2 in Figure 1.22b.x
a f
y -=
Slide 1- 15
Exponential Growth and Exponential Decay
The function , 0, is a model for
if 1, and a model for if 0 1.
xy k a k
a a
exponential growth
exponential decay
= × >
> < <
Graphing Exponential Functions
Graph y = 2x
◦x-intercept:_______
◦y -intercept:_______
◦Domain: _______◦Range: _______◦Type:
_______Slide 1- 16
Graph y = 2-x
x-intercept:_______
y -intercept:_______
Domain: _______ Range: _______ Type:
_______
Slide 1- 17
Rules for Exponents
See page 21 to review these!
Half-life
Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation.
Use the Law of Exponents to expand or condense
1. ax ay
2. (ax)y
3. ax bx
4. (a/b)y
Slide 1- 18
Slide 1- 19
Example Exponential Functions
( )Use a grapher to find the zero's of 4 3.xf x = -
( ) 4 3xf x = -
[-5, 5], [-10,10]
Rewrite the exponential expression to have the indicated base
(9)2x , base 3(1/8) 2x , base 2
Slide 1- 20
Applications The Population of Knoxville is 500,000 and is increasing at the rate
of 3.75% annually. Approximately when will the population reach 1 million?
Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 g present initially. When will there only be 1 g of the substance left?
Interest ProblemsSimple Interest FormulaCompound Interest FormulaInterest compounded
continuously◦ How much would you get if P = $1, r =
100% and the principal were compounded continuously (every second of each day for 365 days) for one year?
Slide 1- 21
Slide 1- 22
The Number e
Many natural, physical and economic phenomena are best modeled
by an exponential function whose base is the famous number , which is
2.718281828 to nine decimal places.
We can define to be the numbe
e
e ( ) 1r that the function 1
approaches as approaches infinity.
x
f xx
x
æ ö÷ç= + ÷ç ÷çè ø
f(x) = (1 + 1/x)x
Slide 1- 23
The Number e
The exponential functions and are frequently used as models
of exponential growth or decay.
Interest compounded continuously uses the model , where is the
initial investment, is t
x x
r t
y e y e
y P e P
r
-= =
= ×
he interest rate as a decimal and is the time in years.t
Slide 1- 24
Example The Number e( ) 0.03
The approximate number of fruit flies in an experimental population after
hours is given by 20 , 0.
a. Find the initial number of fruit flies in the population.
b. How large is the populat
tt Q t e t= ³
ion of fruit flies after 72 hours?
c. Use a grapher to graph the function .Q
[0,100] by [0,120] in 10’s
Slide 1- 25
( )
( ) ( ) ( )
( ) ( )
0.03 0
0.03 2.72 16
0
a. To find the initial population, evaluate at 0.
20 20 20 1 20 flies.
b. After 72 hours, the population size is
20 2
0
0 173 flies.
c.
72
Q t t
Q e e
Q e e
=
= = = =
= = »
( ) 0.0320 , 0tQ t e t= ³
Slide 1- 26
Slide 1- 27
Quick Quiz Sections 1.1 – 1.3
( )
( )
( )
( )( )
You may use a graphing calculator to solve the following problems.
1. Which of the following gives an equation for the line through 3, 1
and parallel to the line: 2 1?
1 7A
2 21 5
B2 2
C 2 5
D 2
y x
y x
y x
y x
y x
-
=- +
= +
= -
=- +
=-
( )
7
E 2 1y x
-
=- +
Slide 1- 28
Quick Quiz Sections 1.1 – 1.3
( ) ( )( )( )
( )( )( )( )( )
22. If 1 and 2 1, which of the
following gives 2 ?
A 2
B 5
C 9
D 10
E 15
f x x g x x
f g
= + = -
o
Slide 1- 30
Warm-Up
( ) ( )( )
( )
In Exercises 1 3, write an equation for the line.
1. the line through the points 1, 8 and 4, 3
2. the horizontal line through the point 3, 4
3. the vertical line through the point 2, 3
In Exercises 4 - 6
-
-
-
2 2 2
2
, find the - and -intercepts of the graph of the relation.
4. 1 5. 19 16 16 9
6. 2 1
x y
x y x y
y x
+ = - =
= +
1.4Parametric Equations
Slide 1- 32
RelationsLines and Other Curves
What you’ll learn about…
…and why
Parametric equations can be used to obtain graphs of relations and functions.
Slide 1- 33
Relations
A relation is a set of ordered pairs (x, y) of real numbers.
The graph of a relation is the set of points in a
plane that correspond to the ordered pairs of the relation.
If x and y are functions of a third variable t, called a parameter, then the equations that define x and y are parametric equations.
Slide 1- 34
Parametric Curve, Parametric Equations
( ) ( )
( ) ( ) ( )( )
If and are given as functions
,
over an interval of -values, then the set of points , ,
defined by these equations is a . The equations are
of th
x y
x f t y g t
t x y f t g t
= =
=
parametric curve
parametric equations e curve.
Lines, line segments and many other curves can be defined parametrically.
General parametric equations involving angular measure:x = v0 cosθt and y = -16t2 + v0 sinθt + s
Ex. 1: Consider the path followed by an object that is propelled into the air as an angle of 45 degrees with an initial velocity of 48 ft/sec. The object will follow a parabolic path.
Write a Cartesian equation and a set of parametric equations to model this example.
Slide 1- 35
Graph each set of parametric equations, then find the Cartesian equation relating the variables (eliminate the parameter):
x = 2t + 1
y = 2 – t
Cartesian Equation:Slide 1- 36
t 0 1 2
x
y
Slide 1- 37
x = r2 – 3r + 1y = r + 1
Cartesian Equation:
r
x
y
x = sin ry = cos r
Cartesian Equation:
Slide 1- 38
r
x
y
x = t3
y = t2/2
Slide 1- 39
t
x
y