# 3.3: Power Functions and Polynomial Functions - Mathematics

Learning Objectives

• Identify power functions.
• Identify end behavior of power functions.
• Identify polynomial functions.
• Identify the degree and leading coefficient of polynomial functions.

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in Table (PageIndex{1}).

 Year Bird Population 2009 2010 2011 2012 2013 800 897 992 1,083 1,169

The population can be estimated using the function (P(t)=−0.3t^3+97t+800), where (P(t)) represents the bird population on the island (t) years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes. ## Identifying Power Functions

In order to better understand the bird problem, we need to understand a specific type of function. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.)

As an example, consider functions for area or volume. The function for the area of a circle with radius (r) is

[A(r)={pi}r^2 onumber]

and the function for the volume of a sphere with radius (r) is

[V(r)=dfrac{4}{3}{pi}r^3 onumber]

Both of these are examples of power functions because they consist of a coefficient, ({pi}) or (dfrac{4}{3}{pi}), multiplied by a variable (r) raised to a power.

Definition: Power Function

A power function is a function that can be represented in the form

[f(x)=kx^p label{power}]

where (k) and (p) are real numbers, and (k) is known as the coefficient.

Q&A: Is (f(x)=2^x) a power function?

No. A power function contains a variable base raised to a fixed power (Equation ef{power}). This function has a constant base raised to a variable power. This is called an exponential function, not a power function. This function will be discussed later.

Example (PageIndex{1}): Identifying Power Functions

Which of the following functions are power functions?

[egin{align*} f(x)&=1 & ext{Constant function} f(x)&=x & ext{Identify function} f(x)&=x^2 & ext{Quadratic function} f(x)&=x^3 & ext{Cubic function} f(x)&=dfrac{1}{x} & ext{Reciprocal function} f(x)&=dfrac{1}{x^2} & ext{Reciprocal squared function} f(x)&=sqrt{x} & ext{Square root function} f(x)&=sqrt{x} & ext{Cube root function} end{align*}]

Solution

All of the listed functions are power functions.

The constant and identity functions are power functions because they can be written as (f(x)=x^0) and (f(x)=x^1) respectively.

The quadratic and cubic functions are power functions with whole number powers (f(x)=x^2) and (f(x)=x^3).

The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as (f(x)=x^{−1}) and (f(x)=x^{−2}).

The square and cube root functions are power functions with fractional powers because they can be written as (f(x)=x^{1/2}) or (f(x)=x^{1/3}).

Exercise (PageIndex{1})

Which functions are power functions?

• (f(x)=2x^2⋅4x^3)
• (g(x)=−x^5+5x^3−4x)
• (h(x)=frac{2x^5−1}{3x^2+4})

(f(x)) is a power function because it can be written as (f(x)=8x^5). The other functions are not power functions.

## Identifying End Behavior of Power Functions

Figure (PageIndex{2}) shows the graphs of (f(x)=x^2), (g(x)=x^4) and and (h(x)=x^6), which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin. To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol (infty) for positive infinity and (−infty) for negative infinity. When we say that “x approaches infinity,” which can be symbolically written as (x{ ightarrow}infty), we are describing a behavior; we are saying that (x) is increasing without bound.

With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as (x) approaches positive or negative infinity, the (f(x)) values increase without bound. In symbolic form, we could write

[ ext{as } x{ ightarrow}{pm}{infty}, ;f(x){ ightarrow}{infty} onumber]

Figure (PageIndex{3}) shows the graphs of (f(x)=x^3), (g(x)=x^5), and (h(x)=x^7), which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin. These examples illustrate that functions of the form (f(x)=x^n) reveal symmetry of one kind or another. First, in Figure (PageIndex{2}) we see that even functions of the form (f(x)=x^n), (n) even, are symmetric about the (y)-axis. In Figure (PageIndex{3}) we see that odd functions of the form (f(x)=x^n), (n) odd, are symmetric about the origin.

For these odd power functions, as (x) approaches negative infinity, (f(x)) decreases without bound. As (x) approaches positive infinity, (f(x)) increases without bound. In symbolic form we write

[egin{align*} & ext{as }x{ ightarrow}-{infty},;f(x){ ightarrow}-{infty} & ext{as }x{ ightarrow}{infty},;f(x){ ightarrow}{infty} end{align*}]

The behavior of the graph of a function as the input values get very small ((x{ ightarrow}−{infty})) and get very large (x{ ightarrow}{infty}) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.

Figure (PageIndex{4}) shows the end behavior of power functions in the form (f(x)=kx^n) where (n) is a non-negative integer depending on the power and the constant. How To: Given a power function (f(x)=kx^n) where (n) is a non-negative integer, identify the end behavior.

1. Determine whether the power is even or odd.
2. Determine whether the constant is positive or negative.
3. Use Figure (PageIndex{4}) to identify the end behavior.

Example (PageIndex{2}): Identifying the End Behavior of a Power Function

Describe the end behavior of the graph of (f(x)=x^8).

Solution

The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As (x) approaches infinity, the output (value of (f(x)) ) increases without bound. We write as (x→∞,) (f(x)→∞.) As (x) approaches negative infinity, the output increases without bound. In symbolic form, as (x→−∞,) (f(x)→∞.) We can graphically represent the function as shown in Figure (PageIndex{5}). Example (PageIndex{3}): Identifying the End Behavior of a Power Function.

Describe the end behavior of the graph of (f(x)=−x^9).

Solution

The exponent of the power function is 9 (an odd number). Because the coefficient is –1 (negative), the graph is the reflection about the (x)-axis of the graph of (f(x)=x^9). Figure (PageIndex{6}) shows that as (x) approaches infinity, the output decreases without bound. As (x) approaches negative infinity, the output increases without bound. In symbolic form, we would write

[egin{align*} ext{as }x{ ightarrow}-{infty},;f(x){ ightarrow}{infty} ext{as }x{ ightarrow}{infty},;f(x){ ightarrow}-{infty} end{align*}] Analysis

We can check our work by using the table feature on a graphing utility.

Table (PageIndex{2})
(x)(f(x))
-101,000,000,000
-51,953,125
00
5-1,953,125
10-1,000,000,000

We can see from Table (PageIndex{2}) that, when we substitute very small values for (x), the output is very large, and when we substitute very large values for (x), the output is very small (meaning that it is a very large negative value).

Exercise (PageIndex{2})

Describe in words and symbols the end behavior of (f(x)=−5x^4).

As (x) approaches positive or negative infinity, (f(x)) decreases without bound: as (x{ ightarrow}{pm}{infty}), (f(x){ ightarrow}−{infty}) because of the negative coefficient.

## Identifying Polynomial Functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius (r) of the spill depends on the number of weeks (w) that have passed. This relationship is linear.

[r(w)=24+8w onumber]

We can combine this with the formula for the area A of a circle.

[A(r)={pi}r^2 onumber]

Composing these functions gives a formula for the area in terms of weeks.

[ egin{align*} A(w)&=A(r(w)) &=A(24+8w) & ={pi}(24+8w)^2 end{align*}]

Multiplying gives the formula.

[A(w)=576{pi}+384{pi}w+64{pi}w^2 onumber]

This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

Definition: Polynomial Functions

Let (n) be a non-negative integer. A polynomial function is a function that can be written in the form

[f(x)=a_nx^n+...+a_2x^2+a_1x+a_0 label{poly}]

This is called the general form of a polynomial function. Each (a_i) is a coefficient and can be any real number. Each product (a_ix^i) is a term of a polynomial function.

Example (PageIndex{4}): Identifying Polynomial Functions

Which of the following are polynomial functions?

• (f(x)=2x^3⋅3x+4)
• (g(x)=−x(x^2−4))
• (h(x)=5sqrt{x}+2)

Solution

The first two functions are examples of polynomial functions because they can be written in the form of Equation ef{poly}, where the powers are non-negative integers and the coefficients are real numbers.

• (f(x)) can be written as (f(x)=6x^4+4).
• (g(x)) can be written as (g(x)=−x^3+4x).
• (h(x)) cannot be written in this form and is therefore not a polynomial function.

## Identifying the Degree and Leading Coefficient of a Polynomial Function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

Terminology of Polynomial Functions

We often rearrange polynomials so that the powers are descending. When a polynomial is written in this way, we say that it is in general form.

How To: Given a polynomial function, identify the degree and leading coefficient

1. Find the highest power of (x) to determine the degree function.
2. Identify the term containing the highest power of (x) to find the leading term.
3. Identify the coefficient of the leading term.

Example (PageIndex{5}): Identifying the Degree and Leading Coefficient of a Polynomial Function

Identify the degree, leading term, and leading coefficient of the following polynomial functions.

(f(x)=3+2x^2−4x^3)

(g(t)=5t^5−2t^3+7t)

(h(p)=6p−p^3−2)

Solution

For the function (f(x)), the highest power of (x) is 3, so the degree is 3. The leading term is the term containing that degree, (−4x^3). The leading coefficient is the coefficient of that term, −4.

For the function (g(t)), the highest power of (t) is 5, so the degree is 5. The leading term is the term containing that degree, (5t^5). The leading coefficient is the coefficient of that term, 5.

For the function (h(p)), the highest power of (p) is 3, so the degree is 3. The leading term is the term containing that degree, (−p^3); the leading coefficient is the coefficient of that term, −1.

Exercise (PageIndex{3})

Identify the degree, leading term, and leading coefficient of the polynomial (f(x)=4x^2−x^6+2x−6).

The degree is (6.) The leading term is (−x^6). The leading coefficient is (−1.)

## Identifying End Behavior of Polynomial Functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as (x) gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree (Table (PageIndex{3})).

Table (PageIndex{3})
Polynomial FunctionLeading TermGraph of Polynomial Function

(f(x)=5x4+2x3−x−4)

(5x^4) (f(x)=−2x^6−x^5+3x^4+x^3)(−2x^6) (f(x)=3x^5−4x^4+2x^2+1)(3x^5) (f(x)=−6x^3+7x^2+3x+1)

(−6x^3) Example (PageIndex{6}): Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in Figure (PageIndex{8}). Solution

As the input values (x) get very large, the output values (f(x)) increase without bound. As the input values (x) get very small, the output values (f(x)) decrease without bound. We can describe the end behavior symbolically by writing

[ ext{as } x{ ightarrow}{infty}, ; f(x){ ightarrow}{infty} onumber]

[ ext{as } x{ ightarrow}-{infty}, ; f(x){ ightarrow}-{infty} onumber]

In words, we could say that as (x) values approach infinity, the function values approach infinity, and as (x) values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

Exercise (PageIndex{1})

Describe the end behavior, and determine a possible degree of the polynomial function in Figure (PageIndex{9}). As (x{ ightarrow}{infty}), (f(x){ ightarrow}−{infty}); as (x{ ightarrow}−{infty}), (f(x){ ightarrow}−{infty}). It has the shape of an even degree power function with a negative coefficient.

Example (PageIndex{7}): Identifying End Behavior and Degree of a Polynomial Function

Given the function (f(x)=−3x^2(x−1)(x+4)), express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Solution

Obtain the general form by expanding the given expression for (f(x)).

[egin{align*} f(x)&=−3x^2(x−1)(x+4) &=−3x^2(x^2+3x−4) &=−3x^4−9x^3+12x^2 end{align*}]

The general form is (f(x)=−3x^4−9x^3+12x^2). The leading term is (−3x^4); therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (–3), so the end behavior is

[ ext{as }x{ ightarrow}−{infty}, ; f(x){ ightarrow}−{infty} onumber]

[ ext{as } x{ ightarrow}{infty}, ; f(x){ ightarrow}−{infty} onumber]

Exercise (PageIndex{7})

Given the function (f(x)=0.2(x−2)(x+1)(x−5)), express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

The leading term is (0.2x^3), so it is a degree 3 polynomial. As (x) approaches positive infinity, (f(x)) increases without bound; as (x) approaches negative infinity, (f(x)) decreases without bound.

## Identifying Local Behavior of Polynomial Functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the “middle” of the function. In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

We are also interested in the intercepts. As with all functions, the (y)-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one (y)-intercept ((0,a_0)). The (x)-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one (x)-intercept. See Figure (PageIndex{10}). Defintion: Intercepts and Turning Points of Polynomial Functions

A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The (y)-intercept is the point at which the function has an input value of zero. The (x)-intercepts are the points at which the output value is zero. Given a polynomial function, determine the intercepts.

1. Determine the (y)-intercept by setting (x=0) and finding the corresponding output value.
2. Determine the (x)-intercepts by solving for the input values that yield an output value of zero.

Example (PageIndex{8}): Determining the Intercepts of a Polynomial Function

Given the polynomial function (f(x)=(x−2)(x+1)(x−4)), written in factored form for your convenience, determine the (y)- and (x)-intercepts.

Solution

The (y)-intercept occurs when the input is zero, so substitute 0 for (x).

[ egin{align*}f(0)&=(0−2)(0+1)(0−4) &=(−2)(1)(−4) &=8 end{align*}]

The (y)-intercept is ((0,8)).

The (x)-intercepts occur when the output is zero.

[ 0=(x−2)(x+1)(x−4) onumber ]

[egin{align*} x−2&=0 & & ext{or} & x+1&=0 & & ext{or} & x−4&=0 x&=2 & & ext{or} & x&=−1 & & ext{or} & x&=4 end{align*}]

The (x)-intercepts are ((2,0)),((–1,0)), and ((4,0)).

We can see these intercepts on the graph of the function shown in Figure (PageIndex{11}). Example (PageIndex{9}): Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function (f(x)=x^4−4x^2−45), determine the (y)- and (x)-intercepts.

Solution

The (y)-intercept occurs when the input is zero.

[ egin{align*} f(0) &=(0)^4−4(0)^2−45 [4pt] &=−45 end{align*}]

The (y)-intercept is ((0,−45)).

The (x)-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.

[egin{align*} f(x)&=x^4−4x^2−45 &=(x^2−9)(x^2+5) &=(x−3)(x+3)(x^2+5)
end{align*}]

[0=(x−3)(x+3)(x^2+5) onumber]

[egin{align*} x−3&=0 & & ext{or} & x+3&=0 & & ext{or} & x^2+5&=0 x&=3 & & ext{or} & x&=−3 & & ext{or} & ext{(no real solution)} end{align*}]

The (x)-intercepts are ((3,0)) and ((–3,0)).

We can see these intercepts on the graph of the function shown in Figure (PageIndex{12}). We can see that the function is even because (f(x)=f(−x)). Exercise (PageIndex{5})

(PageIndex{5}): Given the polynomial function (f(x)=2x^3−6x^2−20x), determine the (y)- and (x)-intercepts.

Solution

(y)-intercept ((0,0)); (x)-intercepts ((0,0)),((–2,0)), and ((5,0))

## Comparing Smooth and Continuous Graphs

The degree of a polynomial function helps us to determine the number of (x)-intercepts and the number of turning points. A polynomial function of (n^ ext{th}) degree is the product of (n) factors, so it will have at most (n) roots or zeros, or (x)-intercepts. The graph of the polynomial function of degree (n) must have at most (n–1) turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

Intercepts and Turning Points of Polynomials

A polynomial of degree (n) will have, at most, (n) (x)-intercepts and (n−1) turning points.

Example (PageIndex{10}): Determining the Number of Intercepts and Turning Points of a Polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of (x)-intercepts and turning points for (f(x)=−3x^{10}+4x^7−x^4+2x^3).

Solution

The polynomial has a degree of 10, so there are at most (n) (x)-intercepts and at most (n−1) turning points.

Exercise (PageIndex{6})

Without graphing the function, determine the maximum number of (x)-intercepts and turning points for (f(x)=108−13x^9−8x^4+14x^{12}+2x^3)

There are at most 12 (x)-intercepts and at most 11 turning points.

Example (PageIndex{11}): Drawing Conclusions about a Polynomial Function from the Graph

What can we conclude about the polynomial represented by the graph shown in Figure (PageIndex{12}) based on its intercepts and turning points? Solution

The end behavior of the graph tells us this is the graph of an even-degree polynomial. See Figure (PageIndex{14}). The graph has 2 (x)-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.

Exercise (PageIndex{7})

What can we conclude about the polynomial represented by the graph shown in Figure (PageIndex{15}) based on its intercepts and turning points? Figure (PageIndex{15}).

Add texts here. Do not delete this text first.

Solution

The end behavior indicates an odd-degree polynomial function; there are 3 (x)-intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

Example (PageIndex{12}): Drawing Conclusions about a Polynomial Function from the Factors

Given the function (f(x)=−4x(x+3)(x−4)), determine the local behavior.

Solution

The (y)-intercept is found by evaluating (f(0)).

[egin{align*} f(0)&=−4(0)(0+3)(0−4) &=0 end{align*}]

The (y)-intercept is ((0,0)).

The (x)-intercepts are found by determining the zeros of the function.

[egin{align*} 0&=-4x(x+3)(x-4) x&=0 & & ext{or} & x+3&=0 & & ext{or} & x-4&=0 x&=0 & & ext{or} & x&=−3 & & ext{or} & x&=4 end{align*}]

The (x)-intercepts are ((0,0)),((–3,0)), and ((4,0)).

The degree is 3 so the graph has at most 2 turning points.

Exercise (PageIndex{8})

Given the function (f(x)=0.2(x−2)(x+1)(x−5)), determine the local behavior.

The (x)-intercepts are ((2,0)), ((−1,0)), and ((5,0)), the (y)-intercept is ((0,2)), and the graph has at most 2 turning points.

## Key Equations

• general form of a polynomial function: (f(x)=a_nx^n+a_{n-1}x^{n-1}...+a_2x^2+a_1x+a_0)

## Key Concepts

• A power function is a variable base raised to a number power.
• The behavior of a graph as the input decreases without bound and increases without bound is called the end behavior.
• The end behavior depends on whether the power is even or odd.
• A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.
• The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.
• The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.
• A polynomial of degree (n) will have at most (n) (x)-intercepts and at most (n−1) turning points.

## Glossary

coefficient

a nonzero real number that is multiplied by a variable raised to an exponent (only the number factor is the coefficient)

continuous function

a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph

degree

the highest power of the variable that occurs in a polynomial

end behavior

the behavior of the graph of a function as the input decreases without bound and increases without bound

the coefficient of the leading term

the term containing the highest power of the variable

polynomial function

a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

power function

a function that can be represented in the form (f(x)=kx^p) where (k) is a constant, the base is a variable, and the exponent, (p), is a constant

smooth curve

a graph with no sharp corners

term of a polynomial function

any (a_ix^i) of a polynomial function in the form (f(x)=a_nx^n+a_{n-1}x^{n-1}...+a_2x^2+a_1x+a_0)

turning point

the location at which the graph of a function changes direction

## 3.3: Power Functions and Polynomial Functions - Mathematics

Key 3.3 Polynomial and Other Functions

Key Concept: Know how to recognize the various shapes of n-th degree polynomials and be able to sketch a graph of their functions

1. Know the basic shapes polynomial of degrees 1, 2, 3, 4, 5 and 6

2. Know how the graph polynomial functions taking advantage of their symmetries

3. Know how to distinguish between even and odd functions 2-nd Degree Polynomial:   4-th Degree Polynomial:   6-th Degree Polynomial:   Even-degree Polynomial - General Shape Graphing Polynomial Functions

1. Find any symmetries of the graph (about y-axis or about the origin)

(a) symmetry about y-axis if f(x) = f(-x)

(b) symmetry about the origin if f(-x) = -f(x)

2. Find intercepts (y-intercept when x = 0 and x-interects when y = 0)

3. Determine where graph ius above and below the x-axis

4. Plot a few points is needed

5. Draw the graph as a continuous curve Step 1- Test for symmetry:

(a) y-axis: (b) Origin: - see (a) Step 2 - Intercepts, y-int is y = 0

And x-intercepts are x = -2, 0, 2

Step 3 - Test is above or below for each domain    Domain -5 -1 1 5 Test below above below above + / -  Step 1- Test for symmetry:

(a) y-axis: (b) Origin: - see (a) Step 2 - Intercepts, y-int is y = 1

And x-intercepts are x = -1, 1

Since and Step 3 - Test is above or below for each domain   Domain -5 0 5 Test above above above + / -  Step 1- Test for symmetry:

(a) y-axis: (b) Origin: - see (a) Step 2 - Intercepts, y-int is y = 1

And x-intercepts are x = 1

Step 4 - Plot a few points

 x -1 0 1 y 2 1 0 Odd and Even Functions

Odd Functions have graphs that are symmetrical about the origin

## Power Functions - Concept Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

A power function is a function where y = x ^n where n is any real constant number. Many of our parent functions such as linear functions and quadratic functions are in fact power functions. Other power functions include y = x^3, y = 1/x and y = square root of x. Power functions are some of the most important functions in Algebra. All power functions pass through the point (1,1) on the coordinate plane.

I want to introduce the idea of a power function. A power function is one of the form Y=X^N where N is any real number constant. A lot of our parent functions are actually power functions, for example, Y=X. One of our simplest functions is a power function where N is 1.
So here N=1. Y=X2, obviously a power function. N=2. Y=X3 . N=3, also a power function. Y=1/X is a power function. Here 1/X is the same as X-1 . So this is a power function with N=-1. And finally, Y= the square root of X. The square root of X is the same as X1/2 . So this is a power function with N=1/2.
Power functions are really important. And a lot of the basic functions we study are power functions. And so it's important to know the definition. One of the things that they all have in common is that they all pass through the point (1, 1). Anyway, power functions, functions of the form Y= X^N

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## Numerical Field Calculation for Charged Particle Optics

### 5.5.3 Improved Hermite Interpolation

The accuracy of the field interpolation is improved by the use of Hermite polynomials of fifth order. This makes sense only when the potentials at the nodes were calculated with a nine-point algorithm having a discretization error of sixth order. Otherwise the algorithms outlined here is still feasible but would bring only some smoothing of the results. Quite generally, it is better to determine first the partial derivatives at the nodes by sufficiently precise numerical formula and to store them before embarking on the task of interpolation because the differentiation of polynomials always causes some loss of accuracy.

We now assume equidistant grids in a u-v plane although it is certainly favorable to have equal spacings hu = hv in the two directions, this added assumption is not necessary here. For reasons of conciseness, we introduce the abbreviations

Because the interpolation polynomial and the accuracy of the FDM calculations are both of fifth order, a seven-point formula for numerical differentiation is here adequate we hence obtain

the error being of sixth order in all three cases. In the vicinity of boundaries or margins, where some of the necessary points are missing, the extrapolation rules based on symmetries, as outlined in Section 3.3.3 , must be evaluated appropriately. If this is impossible, then asymmetric formulas can be used at the price of some loss of accuracy.

We consider now the configuration shown in Fig. 5.28 . To carry out the Hermite interpolation at an arbitrary point Q inside the rectangle, we need to know the partial derivatives at the four corners. The four arrays P, U, V, and W are sufficient only for the bicubic interpolation, outlined in the preceding section. It is, of course, possible to extend the procedure in Eqs. (5.102) to higher orders, but this would require too much memory. An alternative way is shown in Fig. 5.29 : the rectangle of Fig. 5.28 now becomes the central one in a configuration of nine rectangles or of 16 points. The exceptional cases of marginal locations will be discussed below.

Figure 5.29 . Connection between the simplified inner labels referring to the node 0 with the global ones. The other three comers are to be treated analogously.

For reasons of symmetry and continuity, it is necessary to consider only the eight closest neighbors of each of the four inner nodes, because these remain in common with the corresponding neighboring cells. This is shown for the node (0) as an example. The nearest neighbors considered here are numbered sequentially from 1 to 8 the rigorous two-dimensional indexing must of course be used in a practical program.

The technique for determining derivatives of higher orders is quite simple. We assume that f(x) is a six times continuously differentiable function from which the function values and derivatives of first order may be given at three positions xh, x, and x + h. We can then write down the two Taylor series expansions for x ± h, the derivatives referring to the central position, and form the following linear combinations.

By elimination of f (4) and solving for f″(x) we obtain immediately

By repeated application of this formula, we can obtain the matrix elements in the following manner:

The elements for m ≤ 1 and n ≤ 1 are obtained from the stored arrays, and even the multiplications with hu and hv can be saved, if the corresponding multiplied arrays are stored instead of U, V, and W. The differentiation in the u-direction gives

and similarly for the v-direction:

The labels in parentheses are those for which the corresponding elements can be calculated but are not needed. The matrix scheme is completed by

These matrix elements are so easy to calculate that it is not necessary to store them permanently.

We now reconsider the configuration of Fig. 5.29 . With respect to the interpolation and differentiation at the position Q, it is favorable to introduce the normalized coordinates

The corresponding Hermite polynomials are then given by ( Eq. 3.93 ) with

with replacement of u by t and t = tu or t = tv, respectively. For the derivatives, we introduce the notation

the power of two arising from the factor 2 in Eqs. (5.107) . The interpolation polynomials can now be written in compact form as

This kind of interpolation furnishes very smooth results as even the normal component of the gradient at the mesh lines is still continuously differentiable, which is difficult to achieve by other techniques. With slight modifications, this kind of interpolation has been used by Killes  , who obtained very good results with it for ray tracing in electron guns.

There remains the task of interpolation in the vicinity of boundaries. The optic axis is a special case, which is dealt with in the next section. At other symmetry lines, these particular symmetries can be exploited to determine the missing matrix elements. For example if the potential has the property P(− u, v) = P(u, v) and u = 0, j = 0 is the lower boundary line, then we know that G0,1 (1,n) = 0 for all n and l and Eq. (5.106a) is then to be completed by

Near an outer boundary to field-free space or to at least a homogeneous field, the missing elements should all become zero, thereby satisfying the natural boundary conditions. In the vicinity of inner boundaries, material surfaces, a precise determination is hardly possible. It is then usually sufficient to assume constant derivatives within the respective mesh and use the values that can be calculated by Eqs. (5.106) .

## Increasing and Decreasing the Exponents of a Generating Function

When we have a generating function for a certain problem, we can manipulate it to solve other combinatorial problems.

Using this technique of "shift method" gives us a clean solution to the following problem:

## Introduction

Modeling of optimization problems frequently involves representing functions that are piecewise, discontinuous or nonsmooth. This includes inherently piecewise economical and physical characteristics [5, 24, 29], construction of surrogate models by sampling of simulators [15, 32, 58], and approximate or exact representation of nonconvex functions [2, 10, 37, 39, 47]. In this paper, we study the problem of efficiently representing and solving optimization problems containing piecewise polynomial (PWP) constraints. Piecewise polynomials are used in a wide range of disciplines, including efficiency curve modeling in electric-power unit commitment , rigid motion systems , image processing and data compression [44, 51], probability density estimation , flow networks [5, 15, 25] and in optimal control [4, 41].

We consider optimization problems where either or both of the objective function and a subset of the constraints are piecewise polynomial functions. Each polynomial may be nonconvex, and the piecewise polynomial function itself lower semi-continuous. There exist few targeted optimization methods for this class of optimization problems, while some approaches that exploit special structures of nonsmooth optimization problems are applicable, subject to certain modification methods: Womersley and Fletcher  developed a descent method for solving composite nonsmooth problems consisting of a finite number of smooth functions. Conn and Mongeau  constructed a method based on non-differentiable penalty functions for solving discontinuous piecewise linear optimization problems, sketching an extension to problems with PWP constraints. Scholtes  developed an active-set method for dealing with nonlinear programs (NLPs) with underlying combinatorial structure in the constraints. Li  used a conjugated gradient method for minimizing an unconstrained, strictly convex, quadratic spline. None of these methods are currently available in standard optimization software.

From a broader perspective, applicable solution approaches to PWP optimization problems include methods based on general nonsmooth optimization, smoothing techniques and mixed integer programming (MIP). Bundle-type and subgradient methods , originally developed for nonsmooth convex optimization, may be applied to optimization problems with general nonsmooth structures such as PWPs through Clark’s generalized gradients . These generalized methods for nonsmooth optimization are known to have poor convergence properties for nonconvex structures . Smoothing techniques for nonsmooth functions encompass a variety of techniques, seeking to ensure sufficient smoothness for gradient-based methods . Many of these methods are, however, designed for optimizing a nonsmooth function on a convex set, e.g [7, 38]. Meanwhile, smoothing techniques for discontinuities by means of step-function approximations (e.g. ) are known to be prone to numerical instabilities, particularly for increasing accuracies of the discontinuity [8, 60]. Exploitation of MIP for solving PWP optimization problems beyond complete approximative linearization  and direct solution as a nonconvex mixed integer nonlinear programming (MINLP) problem appears to be limited.

We adopt disjunctive representations of PWP constraints, drawing upon the extensive work on disjunctive programming (DP) formulations and representation of piecewise linear (PWL) functions [1, 39, 50, 54]. Modeling piecewise functions as disjunctions enables application of MIP techniques, or specialized branch-and-bound or branch-and-cut schemes with a set condition for representing the piecewise constraints [2, 28, 39]. While adopting MIP techniques and formulations for solving PWP-constrained optimization problems facilitates exploitation of advancements in global optimization solvers [35, 36, 57], careful constraint formulations are required to overcome the inherent problem complexity. To this end, polynomial spline formulations  such as the B-spline is an attractive approach. Polynomial splines are constructed from overlapping (piecewise) polynomials with local support, and embodies a versatile set of techniques for modeling PWPs with favorable smoothness and numerical properties. For decades, polynomial splines, which we simply refer to as splines in this paper, have played an important role in function approximation and geometric modeling. In particular, they have been popular as nonlinear basis functions in regression problems [12, 20], for example in kernel methods [22, 63], and in finite element methods . Yet, few references [5, 15] apply splines within mathematical programming beyond the optimization of spline design parameters [45, 59], trajectory optimization [18, 34, 43], and optimization of piecewise linear splines .

The availability of spline-compatible optimization algorithms and codes is limited. In a recent work,  develop a spatial branch-and-bound (sBB) algorithm for global optimization of spline-constrained problems. While the algorithm was shown to be highly efficient, it has only support for a limited set of algebraic functions, is only available as a specialized code and requires software for spline generation . To address the comparably high modeling and implementation effort required for using the specialized sBB algorithm of [14, 16] proposed an explicit constraint-formulation for continuous splines, yielding an ad-hoc mixed-integer quadratically constrained programming (MIQCP) model. In this paper, we build upon and significantly extend [14, 16] to construct a general-purpose framework for mathematical programming of piecewise polynomial constraints, subsuming spline constraints. The framework is based on an epigraph formulation and we show how it accommodates lower semi-continuous PWPs given in the monomial, Bernstein or B-spline basis. The extension to lower semi-continuous PWPs has not been explicitly covered in previous works. However, the epigraph formulation in  can be applied to lower semi-continuous PWPs written as B-splines.

The main advancement of our work from previous works is our representation of PWPs as a disjunction of polynomial pieces. This allows us to exploit the fact that all the polynomial pieces can be written as a linear combination of a single multivariate polynomial basis. This leads to formulations that are minimal in the number of nonlinear (non-convex) constraints. Furthermore, we exploit properties of the polynomial bases and the grid structure for bound tightening and derivation of Bernstein cuts. Exact reformulations of the DP models yield MINLP formulations, which we benchmark and compare with existing solution methods.

The remainder of the paper is organized as follows. In Sects. 2 and 3, we present background theory of Bernstein polynomials, piecewise polynomials and polynomial splines. In Sect. 4, we present DP formulations of PWPs which we in Sect. 5 reformulate to MINLP models. In Sect. 6, we present computational results of the proposed formulations, comparing the results with existing methods for optimizing PWP functions. Concluding remarks in Sect. 7 ends the paper.

## 3.3: Power Functions and Polynomial Functions - Mathematics

Polynomial functions are nothing more than a sum of power functions. As a result, certain properties of polynomials are very "power-like." When many different power functions are added together, however, polynomials begin to take on unique behaviors.

To understand polynomial behavior, it is important to separate the long term from the short term. Long term behavior refers to what a polynomial does far from the origin – with inputs of large absolute value. Short term behavior refers to what a polynomial does close to the origin – with inputs of small absolute value. The terms "long," "short," "large," and "small," of course, are all relative. They depend very much on the particular polynomial.

The long term behavior of a polynomial is very simple: It is indistinguishable from a single power function. A polynomial may be composed of many power functions, but one of these power functions always, eventually, dominates all of the others. The lesser power functions become insignificant by comparison, and the polynomial settles into the long term behavior of its dominant term.

It is the short term behavior of polynomials that makes them most interesting. Near the origin, polynomials may wiggle up and down – crossing the x-axis at many roots and hitting many highs and lows – before the dominant power function can take long term control. The area around the origin is the polynomial party shack. All of a polynomial's exuberance is expressed here.

It is this exuberant wiggling that ultimately distinguishes polynomials in modeling situations: