Section 1.9: Proportionality, Power Functions, And
Polynomials
We have already studied:
Section 1.9: Proportionality, Power Functions, And
Polynomials
We have already studied:
I
Linear functions
Section 1.9: Proportionality, Power Functions, And
Polynomials
We have already studied:
I
Linear functions
I
Exponential functions
Section 1.9: Proportionality, Power Functions, And
Polynomials
We have already studied:
I
Linear functions
I
Exponential functions
I
Natural logarithm
Section 1.9: Proportionality, Power Functions, And
Polynomials
We have already studied:
I
Linear functions
I
Exponential functions
I
Natural logarithm
In this section, we will study a few more class of functions:
Section 1.9: Proportionality, Power Functions, And
Polynomials
We have already studied:
I
Linear functions
I
Exponential functions
I
Natural logarithm
In this section, we will study a few more class of functions:
I
Proportionality
Section 1.9: Proportionality, Power Functions, And
Polynomials
We have already studied:
I
Linear functions
I
Exponential functions
I
Natural logarithm
In this section, we will study a few more class of functions:
I
Proportionality
I
Power functions
Section 1.9: Proportionality, Power Functions, And
Polynomials
We have already studied:
I
Linear functions
I
Exponential functions
I
Natural logarithm
In this section, we will study a few more class of functions:
I
Proportionality
I
Power functions
I
Polynomial functions
Proportionality
Definition
We say y is (directly) proportional to x if there is a nonzero
constant k such that
y = kx
This k is called the constant of proportionality.
Proportionality
Definition
We say y is (directly) proportional to x if there is a nonzero
constant k such that
y = kx
This k is called the constant of proportionality.
. . . proportional to . . . ≈ . . . a (nonzero) constant multiple of . . .
Proportionality
Definition
We say y is (directly) proportional to x if there is a nonzero
constant k such that
y = kx
This k is called the constant of proportionality.
. . . proportional to . . . ≈ . . . a (nonzero) constant multiple of . . .
Example
“y is proportional to x” means exactly y = kx for some k , 0.
Example
The heart mass of a mammal is proportional to its body mass.
Write a formula for heart mass, H, as a function of body mass, B.
Example
The heart mass of a mammal is a constant multiple of its body
mass. Write a formula for heart mass, H, as a function of body
mass, B.
Example
The heart mass of a mammal is a constant multiple of its body
mass. Write a formula for heart mass, H, as a function of body
mass, B.
Solution:
H = kB
Example
The heart mass of a mammal is a constant multiple of its body
mass. Write a formula for heart mass, H, as a function of body
mass, B.
Solution:
H = kB
If a human with a body mass of 70kg has a heart mass of
0.42kg, find the constant of proportionality.
Example
The heart mass of a mammal is a constant multiple of its body
mass. Write a formula for heart mass, H, as a function of body
mass, B.
Solution:
H = kB
If a human with a body mass of 70kg has a heart mass of
0.42kg, find the constant of proportionality.
Solution:
0.42 = k · (70)
0.42
= 0.006
k=
70
Example
The area A of a circular disk is proportional to the square of its
radius r. Write a formula for A as a function of r.
Example
The area A of a circular disk is a constant multiple of the square
of its radius r. Write a formula for A as a function of r.
Example
The area A of a circular disk is a constant multiple of the square
of its radius r. Write a formula for A as a function of r.
Solution:
A = kr2
Example
The area A of a circular disk is a constant multiple of the square
of its radius r. Write a formula for A as a function of r.
Solution:
A = kr2
Indeed, we know that this constant of proportionality is the
famous constant π.
Inverse Proportionality
Definition
Similarly, we say y is inversely proportional to x if there is a
nonzero constant k such that
y=
k
x
Inverse Proportionality
Definition
Similarly, we say y is inversely proportional to x if there is a
nonzero constant k such that
y=
k
x
I.e.,
x·y=k
So we can also say two quantities are inversely proportional if
their product is a (nonzero) constant.
Example
The time, T, taken for a journey is inversely proportional to the
speed of travel v. Write a formula for T as a function of v.
Example
The time, T, taken for a journey is inversely proportional to the
speed of travel v. Write a formula for T as a function of v.
Solution:
k
T=
v
for some k , 0.
Example
The time, T, taken for a journey is inversely proportional to the
speed of travel v. Write a formula for T as a function of v.
Solution:
k
T=
v
for some k , 0.
Example
The amount G of gasoline one can purchase with $20 is
inversely proportional to the price P of each gallon of gasoline.
Write a formula for G as a function of P.
Example
The time, T, taken for a journey is inversely proportional to the
speed of travel v. Write a formula for T as a function of v.
Solution:
k
T=
v
for some k , 0.
Example
The amount G of gasoline one can purchase with $20 is
inversely proportional to the price P of each gallon of gasoline.
Write a formula for G as a function of P.
Solution:
k
G=
P
for some k , 0.
Power Function
Definition
We say that f (x) is a power function of x if f (x) is proportional
to a constant power of x. I.e.
f (x) = k · xp
for some constants k, p , 0.
Power Function
Definition
We say that f (x) is a power function of x if f (x) is proportional
to a constant power of x. I.e.
f (x) = k · xp
for some constants k, p , 0.
Example
5x3
2x100
are all power functions of x
1
3x 2
4x1.349
9xπ
Which of the following functions are power functions of x?
Which of the following functions are power functions of x?
f (x) = 2x4
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
Which of the following functions are power functions of x?
f (x) = 2x4
f (x) = x
power function: k = 2, p = 4
Which of the following functions are power functions of x?
f (x) = 2x4
f (x) = x = 1x
power function: k = 2, p = 4
1
power function: k = 1, p = 1
Which of the following functions are power functions of x?
f (x) = 2x4
f (x) = x = 1x
1
f (x) =
x
power function: k = 2, p = 4
1
power function: k = 1, p = 1
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
power function: k = 1, p = 1
power function: k = 1, p = −1
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3
x
power function: k = 1, p = 1
power function: k = 1, p = −1
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
4
f (x) = 2x + 1
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
4
f (x) = 2x + 1
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
not a power function
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
4
f (x) = 2x + 1
√
f (x) = 3 x
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
not a power function
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
4
f (x) = 2x + 1
√
f (x) = 3 x = 3x1/2
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
not a power function
power function: k = 3, p = 1/2
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
4
f (x) = 2x + 1
√
f (x) = 3 x = 3x1/2
√
f (x) = (3 x)3
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
not a power function
power function: k = 3, p = 1/2
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
4
f (x) = 2x + 1
√
f (x) = 3 x = 3x1/2
√
f (x) = (3 x)3 = 27x3/2
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
not a power function
power function: k = 3, p = 1/2
power function: k = 27, p = 3/2
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
4
f (x) = 2x + 1
√
f (x) = 3 x = 3x1/2
√
f (x) = (3 x)3 = 27x3/2
5
f (x) = 2
9x
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
not a power function
power function: k = 3, p = 1/2
power function: k = 27, p = 3/2
Which of the following functions are power functions of x?
f (x) = 2x4
power function: k = 2, p = 4
1
f (x) = x = 1x
1
f (x) = = 1x−1
x
5
f (x) = 3 = 5x−3
x
f (x) = 2x
4
f (x) = 2x + 1
√
f (x) = 3 x = 3x1/2
√
f (x) = (3 x)3 = 27x3/2
5
5
f (x) = 2 = x−2
9x
9
power function: k = 1, p = 1
power function: k = 1, p = −1
power function: k = 5, p = −3
not a power function
not a power function
power function: k = 3, p = 1/2
power function: k = 27, p = 3/2
5
power function: k = , p = −2
9
Graph of Power Function
Blackboard
Polynomials
Definition
Sums of power functions with non-negative integer exponents
are called polynomials.
Polynomials
Definition
Sums of power functions with non-negative integer exponents
are called polynomials. Functions given by polynomials are
called polynomial functions.
Polynomials
Definition
Sums of power functions with non-negative integer exponents
are called polynomials. Functions given by polynomials are
called polynomial functions.
Polynomial functions are of the form
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Polynomials
Definition
Sums of power functions with non-negative integer exponents
are called polynomials. Functions given by polynomials are
called polynomial functions.
Polynomial functions are of the form
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
in which n is a non-negative integer, called degree and an is a
non-zero number called leading coefficients.
Polynomials
Definition
Sums of power functions with non-negative integer exponents
are called polynomials. Functions given by polynomials are
called polynomial functions.
Polynomial functions are of the form
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
in which n is a non-negative integer, called degree and an is a
non-zero number called leading coefficients.
We call the term an xn the leading term.
Polynomials of low degrees have special names:
degree 1 linear
Polynomials of low degrees have special names:
degree 1 linear
degree 2 quadratic
Polynomials of low degrees have special names:
degree 1 linear
degree 2 quadratic
degree 3 cubic
Polynomials of low degrees have special names:
degree 1 linear
degree 2 quadratic
degree 3 cubic
degree 4 quartic
Polynomials of low degrees have special names:
degree 1 linear
degree 2 quadratic
degree 3 cubic
degree 4 quartic
degree 5 quintic
Polynomials of low degrees have special names:
degree 1 linear
degree 2 quadratic
degree 3 cubic
degree 4 quartic
degree 5 quintic
The study of polynomial function and equations is almost as
old as mathematics itself. The quadratic formula, for example,
was found in some Babylonian clay tablets (around 2000BC).
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
f (x) = x
polynomial function: deg = 2, LC = 2
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
not a polynomial function
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
not a polynomial function
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
not a polynomial function
polynomial function: deg = 4, LC = −3
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
f (x) = x4 − 3x2 − x4
not a polynomial function
polynomial function: deg = 4, LC = −3
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
not a polynomial function
f (x) = 2x − 3x + 1
polynomial function: deg = 4, LC = −3
f (x) = x4 − 3x2 − x4
polynomial function: deg = 2, LC = −3
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
not a polynomial function
polynomial function: deg = 4, LC = −3
f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3
√
f (x) = 2 x − 3x + 1
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
not a polynomial function
polynomial function: deg = 4, LC = −3
f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3
√
f (x) = 2 x − 3x + 1
not a polynomial function
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
not a polynomial function
polynomial function: deg = 4, LC = −3
f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3
√
f (x) = 2 x − 3x + 1
not a polynomial function
f (x) = 5x7
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
not a polynomial function
polynomial function: deg = 4, LC = −3
f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3
√
f (x) = 2 x − 3x + 1
not a polynomial function
f (x) = 5x7
polynomial function: deg = 7, LC = 5
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
not a polynomial function
polynomial function: deg = 4, LC = −3
f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3
√
f (x) = 2 x − 3x + 1
not a polynomial function
f (x) = 5x7
f (x) = 7
polynomial function: deg = 7, LC = 5
f (x) = an xn + an−1 xn−1 + · · · + a1 x1 + a0
Example
Which of the following functions are polynomial functions?
f (x) = 2x2 − 3x + 1
polynomial function: deg = 2, LC = 2
f (x) = x
polynomial function: deg = 1, LC = 1
f (x) = 2x−2 − 3x + 1
2
4
f (x) = 2x − 3x + 1
not a polynomial function
polynomial function: deg = 4, LC = −3
f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3
√
f (x) = 2 x − 3x + 1
not a polynomial function
f (x) = 5x7
polynomial function: deg = 7, LC = 5
f (x) = 7
polynomial function: deg = 0, LC = 7
End Behavior of Polynomial Functions
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