Download - Lesson 2: A Catalog of Essential Functions
Section 2.2A Catalogue of Essential Functions
V63.0121.021/041, Calculus I
New York University
September 8, 2010
Announcements
I First WebAssign-ments are due September 13I First written assignment is due September 15I Do the Get-to-Know-You survey for extra credit!
. . . . . .
. . . . . .
Announcements
I First WebAssign-ments aredue September 13
I First written assignment isdue September 15
I Do the Get-to-Know-Yousurvey for extra credit!
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 2 / 31
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Objectives: A Catalog of Essential Functions
I Identify different classes ofalgebraic functions,including polynomial(linear, quadratic, cubic,etc.), polynomial(especially linear,quadratic, and cubic),rational, power,trigonometric, andexponential functions.
I Understand the effect ofalgebraic transformationson the graph of a function.
I Understand and computethe composition of twofunctions.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 3 / 31
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What is a function?
DefinitionA function f is a relation which assigns to to every element x in a set Da single element f(x) in a set E.
I The set D is called the domain of f.I The set E is called the target of f.I The set { y | y = f(x) for some x } is called the range of f.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 4 / 31
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Classes of Functions
I linear functions, defined by slope an intercept, point and point, orpoint and slope.
I quadratic functions, cubic functions, power functions, polynomialsI rational functionsI trigonometric functionsI exponential/logarithmic functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 5 / 31
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Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 6 / 31
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Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx+ b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Writethe fare f(x) as a function of distance x traveled.
AnswerIf x is in miles and f(x) in dollars,
f(x) = 2.5+ 2x
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 7 / 31
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Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx+ b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Writethe fare f(x) as a function of distance x traveled.
AnswerIf x is in miles and f(x) in dollars,
f(x) = 2.5+ 2x
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 7 / 31
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Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx+ b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Writethe fare f(x) as a function of distance x traveled.
AnswerIf x is in miles and f(x) in dollars,
f(x) = 2.5+ 2x
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 7 / 31
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Example
Biologists have noticed that the chirping rate of crickets of a certainspecies is related to temperature, and the relationship appears to bevery nearly linear. A cricket produces 113 chirps per minute at 70 ◦Fand 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.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 8 / 31
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Solution
I The point-slope form of the equation for a line is appropriatehere: If a line passes through (x0, y0) with slope m, then the linehas equation
y− y0 = m(x− x0)
I The slope of our line is80− 70
173− 113=
1060
=16
I So an equation for T and N is
T− 70 =16(N− 113) =⇒ T =
16N− 113
6+ 70
I If N = 150, then T =376
+ 70 = 7616◦F
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
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Solution
I The point-slope form of the equation for a line is appropriatehere: If a line passes through (x0, y0) with slope m, then the linehas equation
y− y0 = m(x− x0)
I The slope of our line is80− 70
173− 113=
1060
=16
I So an equation for T and N is
T− 70 =16(N− 113) =⇒ T =
16N− 113
6+ 70
I If N = 150, then T =376
+ 70 = 7616◦F
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
. . . . . .
Solution
I The point-slope form of the equation for a line is appropriatehere: If a line passes through (x0, y0) with slope m, then the linehas equation
y− y0 = m(x− x0)
I The slope of our line is80− 70
173− 113=
1060
=16
I So an equation for T and N is
T− 70 =16(N− 113) =⇒ T =
16N− 113
6+ 70
I If N = 150, then T =376
+ 70 = 7616◦F
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
. . . . . .
Solution
I The point-slope form of the equation for a line is appropriatehere: If a line passes through (x0, y0) with slope m, then the linehas equation
y− y0 = m(x− x0)
I The slope of our line is80− 70
173− 113=
1060
=16
I So an equation for T and N is
T− 70 =16(N− 113) =⇒ T =
16N− 113
6+ 70
I If N = 150, then T =376
+ 70 = 7616◦F
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
. . . . . .
Solution
I The point-slope form of the equation for a line is appropriatehere: If a line passes through (x0, y0) with slope m, then the linehas equation
y− y0 = m(x− x0)
I The slope of our line is80− 70
173− 113=
1060
=16
I So an equation for T and N is
T− 70 =16(N− 113) =⇒ T =
16N− 113
6+ 70
I If N = 150, then T =376
+ 70 = 7616◦F
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 9 / 31
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Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 10 / 31
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I Quadratic functions take the form
f(x) = ax2 + bx+ c
The graph is a parabola which opens upward if a > 0, downward ifa < 0.
I Cubic functions take the form
f(x) = ax3 + bx2 + cx+ d
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 11 / 31
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I Quadratic functions take the form
f(x) = ax2 + bx+ c
The graph is a parabola which opens upward if a > 0, downward ifa < 0.
I Cubic functions take the form
f(x) = ax3 + bx2 + cx+ d
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 11 / 31
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Example
A parabola passes through (0,3), (3,0), and (2,−1). What is theequation of the parabola?
SolutionThe general equation is y = ax2 + bx+ c. Each point gives anequation 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
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 12 / 31
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Example
A parabola passes through (0,3), (3,0), and (2,−1). What is theequation of the parabola?
SolutionThe general equation is y = ax2 + bx+ c.
Each point gives anequation 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
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 12 / 31
. . . . . .
Example
A parabola passes through (0,3), (3,0), and (2,−1). What is theequation of the parabola?
SolutionThe general equation is y = ax2 + bx+ c. Each point gives anequation 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
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 12 / 31
. . . . . .
Example
A parabola passes through (0,3), (3,0), and (2,−1). What is theequation of the parabola?
SolutionThe general equation is y = ax2 + bx+ c. Each point gives anequation 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
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 12 / 31
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Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a+ 6b−6 = 18a+ 6b
Subtract these two and we have −6 = −6a =⇒ a = 1. Substitutea = 1 into the first equation and we have
−12 = 12+ 6b =⇒ b = −4
So our equation isy = x2 − 4x+ 3
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 13 / 31
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Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a+ 6b−6 = 18a+ 6b
Subtract these two and we have −6 = −6a =⇒ a = 1.
Substitutea = 1 into the first equation and we have
−12 = 12+ 6b =⇒ b = −4
So our equation isy = x2 − 4x+ 3
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 13 / 31
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Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a+ 6b−6 = 18a+ 6b
Subtract these two and we have −6 = −6a =⇒ a = 1. Substitutea = 1 into the first equation and we have
−12 = 12+ 6b =⇒ b = −4
So our equation isy = x2 − 4x+ 3
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 13 / 31
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Solution (Continued)
Multiplying the first equation by 3 and the second by 2 gives
−12 = 12a+ 6b−6 = 18a+ 6b
Subtract these two and we have −6 = −6a =⇒ a = 1. Substitutea = 1 into the first equation and we have
−12 = 12+ 6b =⇒ b = −4
So our equation isy = x2 − 4x+ 3
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 13 / 31
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Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 14 / 31
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I Whole number powers: f(x) = xn.
I negative powers are reciprocals: x−3 =1x3
.
I fractional powers are roots: x1/3 = 3√x.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 15 / 31
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Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 16 / 31
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DefinitionA rational function is a quotient of polynomials.
Example
The function f(x) =x3(x+ 3)
(x+ 2)(x− 1)is rational.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 17 / 31
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Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 18 / 31
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I Sine and cosineI Tangent and cotangentI Secant and cosecant
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 19 / 31
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Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 20 / 31
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I exponential functions (for example f(x) = 2x)I logarithmic functions are their inverses (for example f(x) = log2(x))
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 21 / 31
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Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 22 / 31
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Transformations of Functions
Take the squaring function and graph these transformations:
I y = (x+ 1)2
I y = (x− 1)2
I y = x2 + 1I y = x2 − 1
Observe that if the fiddling occurs within the function, a transformationis applied on the x-axis. After the function, to the y-axis.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 23 / 31
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Transformations of Functions
Take the squaring function and graph these transformations:
I y = (x+ 1)2
I y = (x− 1)2
I y = x2 + 1I y = x2 − 1
Observe that if the fiddling occurs within the function, a transformationis applied on the x-axis. After the function, to the y-axis.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 23 / 31
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Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units
upward
I y = f(x)− c, shift the graph of y = f(x) a distance c units
downward
I y = f(x−c), shift the graph of y = f(x) a distance c units
to the right
I y = f(x+ c), shift the graph of y = f(x) a distance c units
to the left
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units upwardI y = f(x)− c, shift the graph of y = f(x) a distance c units
downward
I y = f(x−c), shift the graph of y = f(x) a distance c units
to the right
I y = f(x+ c), shift the graph of y = f(x) a distance c units
to the left
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units upwardI y = f(x)− c, shift the graph of y = f(x) a distance c units downwardI y = f(x−c), shift the graph of y = f(x) a distance c units
to the right
I y = f(x+ c), shift the graph of y = f(x) a distance c units
to the left
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units upwardI y = f(x)− c, shift the graph of y = f(x) a distance c units downwardI y = f(x−c), shift the graph of y = f(x) a distance c units to the rightI y = f(x+ c), shift the graph of y = f(x) a distance c units
to the left
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph ofI y = f(x) + c, shift the graph of y = f(x) a distance c units upwardI y = f(x)− c, shift the graph of y = f(x) a distance c units downwardI y = f(x−c), shift the graph of y = f(x) a distance c units to the rightI y = f(x+ c), shift the graph of y = f(x) a distance c units to the left
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 24 / 31
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Now try these
I y = sin (2x)I y = 2 sin (x)I y = e−x
I y = −ex
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 25 / 31
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Scaling and flipping
To obtain the graph ofI y = f(c · x), scale the graph of f
horizontally
by cI y = c · f(x), scale the graph of f
vertically
by cI If |c| < 1, the scaling is a
compression
I If c < 0, the scaling includes a
flip
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Scaling and flipping
To obtain the graph ofI y = f(c · x), scale the graph of f horizontally by cI y = c · f(x), scale the graph of f
vertically
by cI If |c| < 1, the scaling is a
compression
I If c < 0, the scaling includes a
flip
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Scaling and flipping
To obtain the graph ofI y = f(c · x), scale the graph of f horizontally by cI y = c · f(x), scale the graph of f vertically by cI If |c| < 1, the scaling is a
compression
I If c < 0, the scaling includes a
flip
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Scaling and flipping
To obtain the graph ofI y = f(c · x), scale the graph of f horizontally by cI y = c · f(x), scale the graph of f vertically by cI If |c| < 1, the scaling is a compressionI If c < 0, the scaling includes a
flip
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Scaling and flipping
To obtain the graph ofI y = f(c · x), scale the graph of f horizontally by cI y = c · f(x), scale the graph of f vertically by cI If |c| < 1, the scaling is a compressionI If c < 0, the scaling includes a flip
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 26 / 31
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Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 27 / 31
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Composition is a compounding of functions in
succession
..f .g
.g ◦ f
.x .(g ◦ f)(x).f(x)
.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 28 / 31
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Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solutionf ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2). Note they are not the same.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 29 / 31
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Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solutionf ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2). Note they are not the same.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 29 / 31
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Decomposing
Example
Express√x2 − 4 as a composition of two functions. What is its
domain?
SolutionWe 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. Toinsure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 30 / 31
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Summary
I There are many classes of algebraic functionsI Algebraic rules can be used to sketch graphs
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 8, 2010 31 / 31