Lesson 2: A Catalog of Essential Functions

Section 2.2
A Catalogue of Essential Functions

V63.0121.021/041, Calculus I

New York University

September 8, 2010

Announcements

First WebAssign-ments are due September 13
First written assignment is due September 15
Do the Get-to-Know-You survey for extra credit!

. . . . . .

Announcements

First WebAssign-ments are
due September 13
First written assignment is
due September 15
Do the Get-to-Know-You
survey for extra credit!

. . . . . .

V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 2 / 31

Objectives: A Catalog of Essential Functions
Identify different classes of
algebraic functions,
including polynomial
(linear, quadratic, cubic,
etc.), polynomial
(especially linear,
quadratic, and cubic),
rational, power,
trigonometric, and
exponential functions.
Understand the effect of
algebraic transformations
on the graph of a function.
Understand and compute
the composition of two
functions. . . . . . .


What is a function?

Definition
A function f is a relation which assigns to to every element x in a set D
a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { y | y = f(x) for some x } is called the range of f.

. . . . . .


Classes of Functions

linear functions, defined by slope an intercept, point and point, or
point and slope.
quadratic functions, cubic functions, power functions, polynomials
rational functions
trigonometric functions
exponential/logarithmic functions

. . . . . .


Outline

Linear functions

Other Polynomial functions

Other power functions

Rational functions

Trigonometric Functions

Exponential and Logarithmic functions

Transformations of Functions

Compositions of Functions

. . . . . .


Linear functions

Linear functions have a constant rate of growth and are of the form

f(x) = mx + b.

. . . . . .


Linear functions


f(x) = mx + b.

Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write
the fare f(x) as a function of distance x traveled.

. . . . . .


Linear functions


f(x) = mx + b.

Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write
the fare f(x) as a function of distance x traveled.

Answer
If x is in miles and f(x) in dollars,

f(x) = 2.5 + 2x

. . . . . .


Example
Biologists have noticed that the chirping rate of crickets of a certain
species is related to temperature, and the relationship appears to be
very nearly linear. A cricket produces 113 chirps per minute at 70 ◦ F
and 173 chirps per minute at 80 ◦ F.
(a) Write a linear equation that models the temperature T as a function
of the number of chirps per minute N.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate the
temperature.

. . . . . .


Solution

. . . . . .


Solution

The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )

. . . . . .


Solution

has equation
y − y0 = m(x − x0 )

80 − 70 10 1
The slope of our line is = =
173 − 113 60 6

. . . . . .


Solution

has equation
y − y0 = m(x − x0 )

80 − 70 10 1
173 − 113 60 6
So an equation for T and N is

1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6

. . . . . .


Solution

has equation
y − y0 = m(x − x0 )

80 − 70 10 1
173 − 113 60 6
So an equation for T and N is

1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6

37
If N = 150, then T = + 70 = 76 1 ◦ F
6
6
. . . . . .


Outline

Linear functions



Rational functions





. . . . . .


Quadratic functions take the form

f(x) = ax2 + bx + c

The graph is a parabola which opens upward if a > 0, downward if
a < 0.

. . . . . .


Quadratic functions take the form

f(x) = ax2 + bx + c

The graph is a parabola which opens upward if a > 0, downward if
a < 0.
Cubic functions take the form

f(x) = ax3 + bx2 + cx + d

. . . . . .


Example
A parabola passes through (0, 3), (3, 0), and (2, −1). What is the
equation of the parabola?

. . . . . .


Example

Solution
The general equation is y = ax2 + bx + c.

. . . . . .


Example

Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:

3 = a · 02 + b · 0 + c
−1 = a · 22 + b · 2 + c
0 = a · 32 + b · 3 + c

. . . . . .


Example

Solution
The general equation is y = ax2 + bx + c. Each point gives an
equation relating a, b, and c:

3 = a · 02 + b · 0 + c
−1 = a · 22 + b · 2 + c
0 = a · 32 + b · 3 + c

Right away we see c = 3. The other two equations become

−4 = 4a + 2b
−3 = 9a + 3b

. . . . . .


Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives

−12 = 12a + 6b
−6 = 18a + 6b

. . . . . .



−12 = 12a + 6b
−6 = 18a + 6b

Subtract these two and we have −6 = −6a =⇒ a = 1.

. . . . . .



−12 = 12a + 6b
−6 = 18a + 6b

Subtract these two and we have −6 = −6a =⇒ a = 1. Substitute
a = 1 into the first equation and we have

−12 = 12 + 6b =⇒ b = −4

. . . . . .



−12 = 12a + 6b
−6 = 18a + 6b

Subtract these two and we have −6 = −6a =⇒ a = 1. Substitute
a = 1 into the first equation and we have

−12 = 12 + 6b =⇒ b = −4

So our equation is
y = x2 − 4x + 3

. . . . . .


Outline

Linear functions



Rational functions





. . . . . .


Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.

. . . . . .


Outline

Linear functions



Rational functions





. . . . . .


Definition
A rational function is a quotient of polynomials.

Example
x3 (x + 3)
The function f(x) = is rational.
(x + 2)(x − 1)

. . . . . .


Outline

Linear functions



Rational functions





. . . . . .


Sine and cosine
Tangent and cotangent
Secant and cosecant

. . . . . .


Outline

Linear functions



Rational functions





. . . . . .


exponential functions (for example f(x) = 2x )
logarithmic functions are their inverses (for example f(x) = log2 (x))

. . . . . .


Outline

Linear functions



Rational functions





. . . . . .



Take the squaring function and graph these transformations:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1

. . . . . .



Take the squaring function and graph these transformations:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
Observe that if the fiddling occurs within the function, a transformation
is applied on the x-axis. After the function, to the y-axis.

. . . . . .


Vertical and Horizontal Shifts

Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units

. . . . . .



y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units

. . . . . .



y = f(x) − c, shift the graph of y = f(x) a distance c units downward

. . . . . .



y = f(x − c), shift the graph of y = f(x) a distance c units to the right

. . . . . .



y = f(x − c), shift the graph of y = f(x) a distance c units to the right
y = f(x + c), shift the graph of y = f(x) a distance c units to the left

. . . . . .


Now try these

y = sin (2x)
y = 2 sin (x)
y = e−x
y = −ex

. . . . . .


Scaling and flipping

To obtain the graph of
y = f(c · x), scale the graph of f by c
y = c · f(x), scale the graph of f by c
If |c| < 1, the scaling is a
If c < 0, the scaling includes a

. . . . . .



y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f by c

. . . . . .



y = c · f(x), scale the graph of f vertically by c

. . . . . .



If |c| < 1, the scaling is a compression

. . . . . .



If |c| < 1, the scaling is a compression
If c < 0, the scaling includes a flip

. . . . . .


Outline

Linear functions



Rational functions





. . . . . .


Composition is a compounding of functions in
succession

g
. ◦f
.
x
. f
. . g
. . g ◦ f)(x)
(
f
.(x)

. . . . . .


Composing

Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.

. . . . . .


Composing

Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.

Solution
f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the same.

. . . . . .


Decomposing

Example
√
Express x2 − 4 as a composition of two functions. What is its
domain?

Solution
√
We can write the expression as f ◦ g, where f(u) = u and
g(x) = x2 − 4. The range of g needs to be within the domain of f. To
insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.

. . . . . .


Summary

There are many classes of algebraic functions
Algebraic rules can be used to sketch graphs

. . . . . .


Lesson 2: A Catalog of Essential Functions

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