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```Power Functions
вЂў A power function is a function of the form
f(x) пЂЅ kx
p
where k and p are constants.
вЂў Problem. Which of the following functions are power
functions?
( a) f(x) пЂЅ 13 пѓ— 3 x
( b) g(x) пЂЅ 2(x пЂ« 5)
( c) u(x) пЂЅ
25
x
3
( d) v(x) пЂЅ 6 пѓ— 3
x
3
Proportionality and Power Functions
вЂў A quantity y is (directly) proportional to a power of x if
y пЂЅ kx , k and n constants.
n
вЂў Example. The area, A of a circle is proportional to the square
A пЂЅПЂ r .
2
вЂў A quantity y is inversely proportional to xn if
yпЂЅ
k
x
n
, k and n constants, with n пЂѕ 0.
вЂў Example. The weight, w, of an object is inversely proportional
to the square of the objectвЂ™s distance, d, from the earthвЂ™s center:
w пЂЅkd
-2
пЂЅ
k
d
2
, k is constant
(but depends
on the
object).
Graphs of Positive Integer Powers
вЂў Graphs of functions y = xp, where p is even and positive satisfy:
вЂў Pass through (0,0) and (1,1) and (-1,1).
вЂў Decreasing for x < 0 and increasing for x > 0.
вЂў Are symmetric about the y-axis because the functions are even.
вЂў Are concave up on every interval (U-shaped).
вЂў Larger powers dominate for x large and positive or negative.
вЂў Smaller powers dominate for x near zero.
вЂў The graphs of functions y = xp, where p is odd and positive satisfy:
вЂў Pass through (0,0) and (1,1) and (-1,-1).
вЂў Increase on every interval.
вЂў Are symmetric about the origin since the functions are odd.
вЂў Are concave down, x < 0; concave up, x > 0 (вЂњchairвЂќ-shaped).
вЂў Larger powers dominate for x large and positive.
вЂў Smaller powers dominate for positive x near zero.
Graphs of Negative Integer Powers
вЂў The graphs of power functions with odd negative powers:
yпЂЅx ,x ,x
-1
-3
-5
,пЃЊ
all resemble the graph of y = x-1 = 1/x.
вЂў The graphs of power functions with even negative powers:
yпЂЅx
-2
,x
-4
,x
-6
,пЃЊ
all resemble the graph of y = x-2 = 1/(x2).
вЂў See the next three slides.
вЂў The graph of y = x-1:
вЂў Passes through (1,1) and (-1,-1) and does not have a y-intercept.
вЂў Is decreasing everywhere it is defined.
вЂў Is symmetric about the origin because the function is odd.
вЂў Is concave down for x < 0 and concave up for x > 0.
вЂў x-axis is horizontal asymptote; y-axis is vertical asymptote.
вЂў The graph of y = x-2:
вЂў Passes through (1,1) and (-1,1) and does not have a y-intercept.
вЂў Is increasing for x < 0 and decreasing for x > 0.
вЂў Is symmetric about the y-axis because the function is even.
вЂў Is concave up everywhere it is defined.
вЂў x-axis is horizontal asymptote; y-axis is vertical asymptote.
Comparison of graphs of y = x-1 and y = x-2 in first quadrant
вЂў As x gets large, y = x-2 approaches the x-axis more rapidly
than y = x-1.
вЂў The graph of y = x-2 climbs faster than the graph of y = x-1
as x approaches zero.
Graphs of Positive Fractional Powers
вЂў The graphs of power functions with power = 1/n, n even:
yпЂЅx
1/2
,x
1/4
,x
1/6
,пЃЊ
all resemble the graphs of y = x1/2 and y = x1/4.
вЂў The graphs of power functions with power = 1/n, n odd:
yпЂЅx
1/3
,x
1/5
,x
1/7
,пЃЊ
all resemble the graph of y = x1/3 and y = x1/5.
вЂў See the next two slides.
вЂў The graphs of y = x1/2 and y = x1/4:
вЂў Pass through (0,0) and (1,1).
вЂў Are increasing everywhere they are defined.
вЂў Are concave down everywhere they are defined.
вЂў The graphs of y = x1/3 and y = x1/5:
вЂў Pass through (-1,-1), (0,0) and (1,1).
вЂў Are increasing everywhere they are defined.
вЂў Are concave up for x < 0 and concave down for x > 0.
Finding the formula for a power function
вЂў Problem. Find a power function g which satisfies: g(3) = 1/3
and g(1/3) = 27.
Solution. We recall that a power function has the form kxp for
some constants k and p. Therefore, we assume that g(x) = kxp.
Taking the ratio of g(3) to g(1/3), we have
g(3)
g(3)
g(1/3)
пЂЅ
(1 / 3 )
27
, and therefore , 9
p
k(3)
пЂЅ
g(1/3)
пѓ¦ 3 пѓ¶
пѓ·пѓ·
пЂЅ пѓ§пѓ§
пѓЁ (1 / 3 ) пѓё
p
p
p
k(1/3)
пЂЅ
1
p
пЂЅ
1
81
81
We also have g(3) пЂЅ k( 3 )
Thus, k пЂЅ 3, and g(x) пЂЅ 3x
пЂЅ (9) , and
-2
-2
пЂЅ
k
9
.
пЂЅ
1
3
.
пЂЅ (9)
-2
, so p пЂЅ - 2.
Polynomial Functions
вЂў A polynomial function is a sum of power functions whose
exponents are nonnegative integers.
вЂў We have previously studied quadratic functions, which are
a special kind of polynomial function. We recall that a
quadratic function f(x) may be written in standard form as:
f(x) = ax2 + bx + c, where a, b, and c are constants, a п‚№ 0.
For a quadratic function, the highest power which appears
in the function is 2. However, many applications require
that we consider polynomials containing powers higher
than 2. The next example is one such application.
An example of a polynomial with a 3rd power term
вЂў Suppose a square piece of tin measures 12 inches on each
side. It is desired to make an open box from this material by
cutting equal sized squares from the corners and then
bending up the sides (see figure below). Find a formula for
the volume of the box as a function of the length x of the
side of the square cut out of each corner.
12
x
The volume V(x) = x(12 вЂ“ 2x)2 is found by multiplying
lengthпѓ—widthпѓ—height. The formula for V(x) may be expanded
to V(x) = 4x3вЂ“48x2+144x, which is a sum of power functions
with nonnegative exponents. For which values of x does this
formula represent the volume of the box?
A General Formula for the Family of Polynomial Functions
вЂў The general formula for the family of polynomial functions
can be written as:
p(x) пЂЅ a n x пЂ« a n -1 x
n
n -1
пЂ« пЃЊ пЂ« a1x пЂ« a 0 ,
where n is a positive integer called the degree of p and a n п‚№ 0 .
вЂў Each member of this sum, anxn, an-1xn-1, and so on, is
called a term.
вЂў The constants an, an-1, ... , a0 are called coefficients.
вЂў The term a0 is called the constant term. The highestpowered term, anxn, is called the leading term.
вЂў To write a polynomial in standard form, we arrange
its terms from highest power to lowest power, going
from left to right.
The Long-Run Behavior of Polynomial Functions
вЂў When viewed on a large enough scale, the graph of the
polynomial p(x) пЂЅ a n x n пЂ« a n -1 x n -1 пЂ« пЃЊ пЂ« a 1 x пЂ« a 0 looks like
the graph of the power function y = anxn. This behavior is
called the long-run behavior of the polynomial.
вЂў Example. Let p(x) = x2 + x. Then we have
x
2
пЂј x
2
пЂ« x пЂј 2x , for x пЂѕ 1.
2
It follows that the graph of p(x) is trapped between the
graph of x2 and the graph of 2x2 for x > 1. For this reason,
we say that the graph of p(x) looks like the graph of x2
when viewed on a large scale.
Using the long-run behavior of a polynomial to locate zeros
вЂў The zeros of a polynomial p are the values of x for which
p(x) = 0. These values are also called the x-intercepts,
because they tell us where the graph of p crosses the x-axis.
вЂў Problem. Given the polynomial
q(x) пЂЅ 3x
6
пЂ­ 2x
5
пЂ« 4x
2
пЂ­ 1,
where q(0) = вЂ“1, is there a reason to expect a solution to the
equation q(x) = 0?
Solution. Since the graph of the polynomial q(x) resembles
that of y = 3x6 for large (positive or negative) values of x, we
know that the graph of q(x) is greater than zero for these
large values of x. Since q(0) < 0 and the graph of q is
smooth and unbroken, it follows that this graph must cross
the x-axis at least twice.
Summary of Power and Polynomial Functions, Sections 9.1, 9.2
вЂў Power functions were defined and properties of graphs were
studied for various classes of power functions.
вЂў We discussed finding the formula for a power function when
two points on its graph are given.
вЂў Polynomials were defined to be a sum of power functions:
p(x) пЂЅ a n x пЂ« a n -1 x
n
n -1
пЂ« пЃЊ пЂ« a1x пЂ« a 0 .
вЂў The degree of a polynomial is the value n in the formula
above.
вЂў Long-run behavior of polynomials was discussed.
вЂў Zeros of polynomials were investigated and the long-run
behavior was used to show their existence in some cases.
The Short-Run Behavior of Polynomials
вЂў The shape of the graph of a polynomial depends on its degree;
typical shapes are shown below.
(degree = 2)
Cubic
(degree = 3)
Quartic
(degree = 4)
Quintic
(degree = 5)
вЂў The long-run behavior of a polynomial is determined by its
term may have very different short-run behaviors.
вЂў Example. Compare the graphs of f(x) = x3 вЂ“ x and g(x) = x3 + x.
Note that f has three zeros while g has only one.
The Factored Form of a Polynomial
вЂў To predict the long-run behavior of a polynomial, we write it in
standard form. However, to determine the zeros of a polynomial,
we write it in factored form, as a product of other polynomials.
Some, but not all, polynomials can be factored.
вЂў Problem. Rewrite the third-degree polynomial u(x) = x3 вЂ“ x2 вЂ“ 6x
as a product of linear factors.
Solution. By factoring out an x and then factoring the quadratic,
x2 вЂ“ x вЂ“ 6, we rewrite u(x) as
u(x) пЂЅ x
3
пЂ­x
2
пЂ­ 6x пЂЅ x(x
2
пЂ­ x пЂ­ 6) пЂЅ x(x пЂ­ 3)(x пЂ« 2).
вЂў The advantage of the factored form is that we can easily find the
zeros of the polynomial using the rule:
If a пѓ— b пЂЅ 0, then a or b (or both) must equal 0.
Linear Factors of a Polynomial
вЂў Suppose p is a polynomial. If the formula for p has a
linear factor, that is, a factor of the form (x вЂ“ k), then p
has a zero at x = k. Conversely, if p has a zero at x = k,
then p has a linear factor of the form (x вЂ“ k).
вЂў If a polynomial doesnвЂ™t have a zero, this doesnвЂ™t mean it
canвЂ™t be factored. For example, the polynomial
p(x) = x4 + 5x2 + 6 has no zeros. To see this, note that x4
and 5x2 are never negative. However, this polynomial can
be factored as p(x) = (x2 + 2)(x2 + 3). The point is that a
polynomial with a zero has a linear factor, and a
polynomial without a zero does not.
Polynomials, zeros, and linear factors
вЂў p(k) = 0
пѓі (x вЂ“ k) is a factor of polynomial p.
вЂў If p(x) = (xвЂ“3)(x+1)(xвЂ“2), name three zeros of p.
k1= ?, k2= ?, k3= ?.
вЂў If another polynomial p satisfies p(1) = 0, p(5) = 0, p(вЂ“3)=0,
name three linear factors of p.
(x вЂ“ ?), (x вЂ“ ?), (x вЂ“ ?).
The Number of Factors, Zeros, and Bumps
вЂў The number of linear factors is always less than or equal to
the degree of a polynomial. (Do you see why?) Since each
zero corresponds to a linear factor, the number of zeros is
less than or equal to the degree of the polynomial.
вЂў Between any two consecutive zeros, there is a bump in the
graph because it changes direction.
вЂў The above observations can be summarized by the following
statement: The graph of an nth degree polynomial has at
most n zeros and turns at most (nвЂ“1) times.
Multiple Zeros
вЂў If p is a polynomial with a repeated linear factor, then p has a
multiple zero.
вЂў If the factor (x вЂ“ k) is repeated an even number of
times, the graph of y = p(x) does not cross the x-axis
at x = k, but вЂњbouncesвЂќ off the x-axis there (see left below).
вЂў If the factor (x вЂ“ k) is repeated an odd number of times,
the graph of y = p(x) crosses the x-axis at x = k, but it
looks flattened there (see right below).
An Unknown Polynomial
вЂў If we know the polynomial f whose graph is shown below
has degree 4, what can we conclude about its zeros?
вЂў The graph suggests that f has a single zero at x = 1. The
flattened appearance near x = -1 suggests that f has a multiple
zero there. Since the graph crosses the x-axis at x = -1, we
conclude the zero there is repeated an odd number of times.
Since f has degree 4, there must be a triple zero at x = -1.
Finding a possible formula for a polynomial from its graph
вЂў Problem. Find a possible formula for the polynomial f which is
graphed below.
вЂў Solution. Based on its long-run behavior, f is of odd degree
greater than or equal to 3. The polynomial has zeros at x = -1 and
x = 3. We see that x = 3 is a multiple zero of even power, because
the graph вЂњbounces offвЂќ the x-axis here. Therefore, we try the
formula f(x) = k(x+1)(xвЂ“3)2, where k represents a stretch factor.
The shape of the graph shows that k is negative. We read f(0) = 3 from the graph, and we use the formula to get k = -1/3.
A General Formula for the Family of Rational Functions
вЂў If r is a function that can be written as the ratio of two
polynomials p and q so that
r(x) пЂЅ
p(x)
,
q(x)
then r is called a rational function. Here, it is assumed that
q(x) is not the constant polynomial with value zero.
вЂў Example. The function f(x) пЂЅ x пЂ« 3 пЂ­
2
is actually
x пЂ­2
a rational function. In order to see this, we must combine
the terms of f by finding a common denominator.
f(x) пЂЅ
x пЂ«3
1
пЂ­
2
x пЂ­2
пЂЅ
( x пЂ« 3 )( x пЂ­ 2) пЂ­ 2
x пЂ­2
пЂЅ
x
2
пЂ« x пЂ­8
x пЂ­2
.
The Long-Run Behavior of Rational Functions
вЂў For large enough values of x (either positive or negative),
the graph of the rational function r looks like the graph of a
power function. If
r(x) пЂЅ
p(x)
,
q(x)
then the long-run behavior of y = r(x) is given by
yпЂЅ
term of p(x)
term of q(x)
.
вЂў Example. Suppose that
r(x) пЂЅ
2x
2
пЂ«1
x пЂ­1
,
then, in the long run, r(x) behaves like 2x. The Maple plot
of y = r(x) is shown on the next slide.
>plot((2*x^2+1)/(x-1), x = -5..10, y = -50..50, discont =true);
Note that far from the origin, the graph resembles that of y = 2x.
Also note that the graph has a vertical asymptote at x = 1.
Compare and discuss the long-run behaviors of the following
functions:
вЂў f(x) =
вЂў g(x) =
вЂў h(x) =
x
2
пЂ«1
x
2
пЂ«5
x
3
пЂ«1
x
2
пЂ«5
x пЂ«1
x
2
пЂ«5
вЂў Note that none of the above functions has a vertical
asymptote. Can you explain why?
Summary of Polynomial and Rational Functions, Sections 9.3, 9.4
вЂў The number of linear factors, zeros, and bumps and their relation to
the degree was discussed.
вЂў Factored form and its linear factors were used to find the zeros and
analyze the short-run behavior of a polynomial.
вЂў If a multiple zero of a polynomial is repeated an even number of
times, the graph of the polynomial does not cross the x-axis at the
zero, but вЂњbouncesвЂќ off the x-axis there.
вЂў If a multiple zero of a polynomial is repeated an odd number of
times, the graph of the polynomial crosses the x-axis at the zero, but
it looks flattened there.
вЂў By analyzing long-run behavior and using the properties listed
above, we were able to find a possible formula for a polynomial if its
graph was given.
вЂў Rational functions r(x) are defined as the ratio of two polynomials,
p(x) in the numerator, and q(x) in the denominator.
вЂў Long run behavior of r(x): y = (lead term of p(x))/(lead term of q(x)).
The short-run behavior of rational functions
вЂў The first step in analyzing the short-run behavior of a
rational function
p(x)
r(x) пЂЅ
,
q(x)
where p and q are polynomials, is to factor p and q, if
possible.
вЂў Example. Let
3x пЂ« 2
r(x) пЂЅ
x
2
пЂ«xпЂ­2
.
As a first step in
analyzing the behavior of r(x), we factor the denominator
to obtain
3x пЂ« 2
r(x) пЂЅ
(x пЂ­ 1)(x пЂ« 2)
.
вЂў Note that the zeros of r(x) will be the same as the zeros of
the numerator, p(x), assuming that p and q have different
zeros. In the example, r(x) has a zero at x = вЂ“2/3.
The vertical asymptotes of a rational function
вЂў Just as we can find the zeros of a rational function by
looking at its numerator, we can find the vertical
asymptotes by looking at its denominator. A rational
function is large (positively or negatively) wherever its
denominator is small. This means that a rational function
r(x) = p(x)/q(x) has a vertical asymptote wherever q(x) is
zero (assuming that p(x) and q(x) have different zeros).
вЂў Example. Again let
r(x) пЂЅ
3x пЂ« 2
(x пЂ­ 1)(x пЂ« 2)
.
Since the zeros
of the denominator are 1 and вЂ“2, we conclude that the
graph of r has vertical asymptotes at 1 and вЂ“2. Note that
the zeros of the numerator and denominator are different.
The graph of a rational function
вЂў We summarize what we have learned about the graph of a
rational function r(x) = p(x)/q(x), where p(x) and q(x) are
polynomials with different zeros.
вЂў The long-run behavior of r is given by the ratio of
the leading terms of p and q.
вЂў The zeros of r are the same as the zeros of p.
вЂў The graph of r has a vertical asymptote at each of
the zeros of the denominator, q.
вЂў We can apply the above principles to help us draw the
graph of a rational function. Also of interest is the
yintercept of the graph, which is found by evaluating r(0),
assuming that r is defined at x = 0.
Example. The graph of
r(x) пЂЅ
3x пЂ« 2
(x пЂ­ 1)(x пЂ« 2)
.
Vertical Asymptote
at x = вЂ“2
Horizontal
Asymptote at y = 0
Vertical Asymptote
at x = 1
Note that the x-intercept is вЂ“2/3 and the y-intercept is вЂ“ 1.
Some rational functions are transformations of power functions
вЂў Problem. Show that the rational function r(x) = (x+3)/(x+2)
is a translation of the power function f(x) = 1/x.
Solution. The numerator of r may be rewritten as
r(x) пЂЅ
(x пЂ« 2) пЂ« 1
x пЂ« 2
.
Upon dividing, we have
r(x) пЂЅ
x пЂ« 2
x пЂ« 2
пЂ«
1
x пЂ« 2
пЂЅ1пЂ«
1
x пЂ« 2
.
Thus, the graph of y = r(x) is the graph of y = 1/x shifted two
units to the left and one unit up. The graph of y = r(x) is
shown below.
What is the vertical
asymptote? the horizontal
asymptote?
Finding the formula for a rational function from its graph
вЂў For a rational function, we can read the zeros and vertical
asymptotes from its graph. The zeros correspond to factors
in the numerator and the vertical asymptotes correspond to
factors in the denominator.
вЂў Problem. Given the graph, find a possible formula for r(x).
вЂў Solution.
r(x) пЂЅ k
(x пЂ« 1)
(x пЂ­ 1)(x пЂ« 2)
.
What is the value of k?
When p(x) and q(x) have the same zeros: holes
вЂў We have been studying rational functions r(x) = p(x)/q(x)
under the assumption that p(x) and q(x) have different zeros.
The following example indicates what can happen when this
assumption does not hold.
вЂў Example. Let r(x) =
x
2
пЂ« xпЂ­2
x пЂ­1
.
Here, p(x) = x2 + x вЂ“ 2
can be factored into (x вЂ“ 1) and (x + 2) so that p(x) and q(x)
have x = 1 as a common zero. If we cancel the common
factor, the formula for r becomes r(x) = x + 2, x п‚№ 1. Do you
see why x cannot equal 1? The graph of y = r(x) is shown
below:
The graph has a hole at
x = 1.
Comparing Power, Exponential, and Log Functions
вЂў Any positive increasing exponential eventually grows faster
than any power function.
вЂў Any positive decreasing exponential eventually approaches the
horizontal axis faster than any positive decreasing power
function.
вЂў Any positive increasing power function eventually grows more
rapidly than y = log x and y = ln x.
вЂў Futhermore, the three results stated above are not changed if
one of the functions is multiplied by a positive constant.
вЂў Problem. Which of the following functions will dominate for
large positive x: f(x) = 1,000,000x2 or g(x) = 0.000001ex ?
Solution. Using the first and fourth bullet points above, we see
that g will dominate f if x is chosen to be a sufficiently large
positive number.
Summary of Polynomial and Rational Functions, Sections 9.5, 9.6
вЂў Rational functions r(x) are defined as the ratio of two
polynomials, p(x) in the numerator, and q(x) in the denominator.
вЂў The long-run behavior of r(x) is given by:
yпЂЅ
term of p(x)
term of q(x)
.
вЂў Assuming that p and q do not have common zeros, the zeros of r
are the same as those of p, and the vertical asymptotes of r occur
at the zeros of q.
вЂў From the graph of a rational function we can determine its
formula.
вЂў If p(x) and q(x) have common zeros, the graph of y = r(x) will
have вЂњholesвЂќ at the common zeros.
вЂў We compared growth rates of powers, exponentials, and
logarithms.
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