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Announcements
Topics:
- section 1.2 (some basic common functions)
- section 1.3 (transformations and combinations of
functions)
- section 1.4 (exponential functions)
Homework:
read sections 1.2, 1.3, and 1.4 in your textbook
work on exercises from the textbook in sections 1.2, 1.3,
and 1.4
work on Assignment 1 and Assignment 2
Functions can be described in 4 ways:
• Numerically (table of values)
• Geometrically (graph)
• Algebraically (explicit formula)
• Verbally (description in words)
Modeling Exercise:
verbal description of a function
Example #20:
You place a frozen pie in an oven and bake it for
an hour. Then you take it out and let it cool
before eating it. Sketch a rough graph of the
temperature of the pie as a function of time.
Catalogue of Important Functions
• Download file from website and review +
memorize these functions (names, shape of
graph, important properties)
Linear Functions
slope:
point-slope equation:
slope-y-intercept equation:
y 2 - y1
x 2 - x1
Linear Functions
slope:
point-slope equation:
y 2 - y1
y - y1 = m( x - x1)
slope-y-intercept equation:
y = mx + b
x 2 - x1
Linear Model for the
Population of Canada
Data:
Year
Time,
t
Population, P(t)
(in thousands)
1996
0
28 847
2001
5
30 007
2006
10
31 613
Linear Model for the
Population of Canada
Create a linear model for
the population of Canada
as a function of time using
the first two data points.
Linear Model for the
Population of Canada
Use this model to predict
Canada’s population in
2006:
P = 232t +28847
Actual observed
population in 2006:
Polynomial Functions
A polynomial is a function of the form
where n is a nonnegative integer (0, 1, 2, 3, 4, …)
and the numbers
are constants
called the coefficients of the polynomial.
Domain:
Degree:
Polynomials
Example:
Quadratic Function:
f (x) = -x 2 + 6x - 5
Complete the square to find
the vertex:
f (x) = -(x - 3)2 + 4
Domain:
Range:
Polynomials
Example:
Cubic Function:
degree: n=3
f (x) = (x +1)3
Expand to standard form:
f (x) = x 3 + 3x 2 + 3x +1
Domain:
Range:
Polynomials
Note 1:
Polynomials have nice properties (domain is all
real numbers, graphs are smooth and
continuous, + more…) and for this reason are
used in calculus whenever possible for simple
calculations
Note 2:
A linear function, f(x)=mx+b, is just a polynomial
of degree 1.
Power Functions
A power function is a function of the form
f (x) = x
a
where a is a constant.
Note:
Although a can be any real number, we usually
omit the case when a = 0.
Power Functions
Some special cases:
a=2: f (x) = x 2
Shape: parabola
Vertex: (0,0)
Domain:
Range:
** Graphs of
look similar
x 4, x 6 ,...
Power Functions
Some special cases:
a=3: f (x) = x 3
Shape: cubic parabola
Domain:
Range:
**Graphs of
look similar
x 5, x 7,...
Power Functions
Some special cases:
1
a=1/2: f (x) = x 2 =
square root function
x
Shape: half of a parabola
Domain:
Range:
Power Functions
Some special cases:
1
3
a=1/3: f (x) = x 3 =
Shape: cubic parabola
Domain:
Range:
cube root function
x
Power Functions
Some special cases:
1
a=-1: f (x) = x =
x
-1
Shape: hyperbola
Domain:
Asymptotes:
Range:
rational function
Power Functions
Some special cases:
1
a=-2: f (x) = x = 2
x
-2
Shape: hyperbola
Domain:
Asymptotes:
Range:
rational function
Rational Functions
A rational function f is a ratio of two polynomials:
P(x)
f (x) =
Q(x)
where P and Q are polynomials and Q(x) ¹ 0.
Examples:
1
f (x) =
x-3
x+2
f (x) = 2
x -4
Algebraic Functions
A combination of any of the the previous
functions using algebraic operations (+, -, ´, ¸, n )
is called an algebraic function.
Example: f (x) =
3x - 4
x +1
2
• on your own:
In the text, read pages 31, 32, and 33 to briefly
review Trigonometric Functions, Exponential
Functions, and Logarithmic Functions
Transformations:
Scaling, Reflecting, Shifting
- It can be easier to graph a function if we
recognize it as a series of transformations of a
basic function
- Summary of rules on page 37 in text
- Use online graphing calculator and/or wolfram
alpha website to check your answers
Transformations:
Scaling, Reflecting, Shifting
Example 1: Graph f (x) = -2(x - 3)3 +1
f (x) = 2x
3
Reflect in the x-axis (multiply each y-coordinate
by -1):
f (x) = -2x
3
Transformations:
Scaling, Reflecting, Shifting
Example 2: Graph f (x) = 4 - x
First, re-write it so we can easily identify
transformations:
f (x) = -2(x - 2)
Graph base function:
f (x) = x
Transformations:
Scaling, Reflecting, Shifting
Example 3: Graph f (x) = -cos2x
f (x) = sin x
Compress horizontally by a factor of 2
f (x) = sin2x
Reflect graph in the x-axis
f (x) = -sin2x
(multiply each y-coordinate by -1):
Combinations of Functions
Adding/Subtracting Functions
The sum f + g of the functions f and g is the
function defined by
( f + g)(x) = f (x) + g(x)
The difference f - g of the functions f and g
is the function defined by
( f - g)(x) = f (x) - g(x)
Combinations of Functions
Multiplying/Dividing Functions
The product f × g of the functions f and g is
the function defined by
( f × g)(x) = f (x)× g(x)
The quotient f g of the functions f and g is
the function defined by
æfö
f (x)
ç ÷(x) =
g(x)
è gø
Combinations of Functions
Composition of Functions
The composition
of the functions f and g
is the function defined by
Note: f is called the outer function
g is called the inner function
Combinations of Functions
Diagram:
Note: In general,
Combinations of Functions
Example:
Use the table of values below to find the values
of (a)
(b)
(c)
x
1
2
3
4
5
6
f(x)
3
1
4
2
2
5
g(x)
6
3
2
1
2
3
Exponential Functions
An exponential function is a function of the form
f (x) = a
x
where a is a positive real number called the
base and x is a variable called the exponent.
Domain: xÎ R
Range: y > 0
Graphs of Exponential Functions
f (x) = 3
x
Memorize!!!
When a>1, the function is increasing.
f (x) = (
)
1 x
2
When a<1, the function is decreasing.
y=0 is a horizontal asymptote
Transformation of an Exponential
Function
Graph
f (x) = -e -1.
2x
Recall:
e is a special irrational
number between 2 and 3
that is commonly used in
calculus
Approximation:
e » 2.718
Laws of Exponents
1. a x × a y = a x +y
ax
2. y = a x-y
a
x y
xy
3. (a ) = a
4. (ab) x = a x b x
Examples:
a1 = a
1
a = x
a
-x
x
y
y
a0 = 1
y
a = ax = ( a )x
Exponential Models
P. 54 #24.
Suppose you are offered a job that lasts one
month. Which of the following methods of
payment do you prefer?
I. One million dollars at the end of the month.
II. One cent on the first day of the month, two
cents on the second day, four cents on the
third day, and, in general, 2 n-1 cents on the
nth day.