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5.2 Power Functions and Polynomial Functions

3 min read•june 24, 2024

Power functions are the building blocks of polynomial math. They come in various forms, from simple squares to complex . Understanding their behavior is key to mastering more advanced algebraic concepts.

These functions have unique characteristics that affect their graphs and behaviors. By studying their , , and other properties, we gain insights into how polynomials work in general.

Power Functions

Characteristics of power functions

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  • Power functions have the form [f(x)](https://www.fiveableKeyTerm:f(x))=kxn[f(x)](https://www.fiveableKeyTerm:f(x)) = kx^n, where kk is a constant and nn is a real number
    • Positive integer nn creates polynomial functions (quadratic, cubic)
    • Negative integer or rational nn creates rational functions (1x,1x2\frac{1}{x}, \frac{1}{x^2})
    • Irrational nn creates irrational functions (x,x3\sqrt{x}, \sqrt[3]{x})
  • Graph passes through the point (1,k)(1, k)
  • is all real numbers, except when nn is a negative integer or rational with even denominator, which excludes x=0x = 0
  • depends on nn and sign of kk
    • Even nn and k>0k > 0 has range [0,∞)[0, \infty) (parabola opening upward)
    • Even nn and k<0k < 0 has range (−∞,0](-\infty, 0] (parabola opening downward)
    • Odd nn has range (−∞,∞)(-\infty, \infty) (cubic, square root)
  • Graph symmetry
    • Even integer nn is symmetric about y-axis (parabola)
    • Odd integer nn is symmetric about origin (cubic)

End behavior of functions

  • End behavior describes the graph as xx approaches positive or negative infinity
  • determines end behavior
    1. Even , positive approaches positive infinity as xx approaches ±∞\pm \infty (upward parabola)
    2. Even degree, negative approaches negative infinity as xx approaches ±∞\pm \infty (downward parabola)
    3. Odd degree, positive leading approaches positive infinity as x→∞x \to \infty and negative infinity as x→−∞x \to -\infty (increasing cubic)
    4. Odd degree, negative leading coefficient approaches negative infinity as x→∞x \to \infty and positive infinity as x→−∞x \to -\infty (decreasing cubic)

Polynomial Functions

Classification of polynomial functions

  • Polynomial functions have the form f(x)=[an](https://www.fiveableKeyTerm:an)xn+an−1xn−1+...+a1x+a0f(x) = [a_n](https://www.fiveableKeyTerm:a_n) x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where nn is a non-negative integer and a0,a1,...,ana_0, a_1, ..., a_n are constants with an≠0a_n \neq 0
  • Degree is the highest power of the variable
    • Degree 0: constant functions (f(x)=5f(x) = 5)
    • Degree 1: linear functions (f(x)=3x+2f(x) = 3x + 2)
    • Degree 2: quadratic functions (f(x)=x2−4x+3f(x) = x^2 - 4x + 3)
    • Degree 3: (f(x)=2x3−x+1f(x) = 2x^3 - x + 1)
  • Leading coefficient is the coefficient of the highest degree term

Manipulation of polynomials

  • Evaluate by substituting xx value and simplifying
  • Add, subtract, and multiply polynomials to create new polynomial functions ()
  • Factor polynomials into lower-degree polynomials
    • Factor out (GCF)
    • Factor by grouping
    • Factor using special patterns
      • : a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
      • Sum and : a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2-ab+b^2) and a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2+ab+b^2)
  • Find or where function equals zero
    • : degree nn polynomial has nn complex roots (including repeated)
    • lists possible rational roots
    • Use or to find roots and factorize completely

Advanced Polynomial Concepts

  • : Combining two or more functions to create a new function
  • : Using technology and analytical methods to sketch polynomial functions
  • : Understanding the role of imaginary roots in polynomial equations
  • : Solving and graphing inequalities involving polynomial expressions

Key Terms to Review (61)

A_n: The term a_n, also known as the nth term, is a fundamental concept in mathematics that appears in various contexts, including power functions, polynomial functions, arithmetic sequences, and geometric sequences. It represents the value of a particular term or element within a sequence or function, where the subscript 'n' denotes the position or index of that term within the sequence.
Axes of symmetry: Axes of symmetry are lines that divide a figure into two mirror-image halves. In hyperbolas, these axes typically refer to the transverse and conjugate axes.
Coefficient: A coefficient is a numerical or constant factor that multiplies a variable in a term of an algebraic expression or equation. For example, in the term $5x^2$, 5 is the coefficient.
Complex Numbers: Complex numbers are a mathematical concept that extend the real number system by including the imaginary unit, denoted as $i$, which is defined as the square root of -1. They are used to represent quantities that have both magnitude and direction, and are essential in various areas of mathematics, including algebra, calculus, and physics.
Constant Function: A constant function is a function where the output value is the same for any input value. Regardless of the input, the function always returns the same constant value, making it a special type of linear function and polynomial function.
Cube root: A cube root of a number is a value that, when multiplied by itself three times (cubed), gives the original number. Mathematically, if $x^3 = a$, then $x$ is the cube root of $a$, denoted as $\sqrt[3]{a}$.
Cubic Function: A cubic function is a polynomial function of degree three, where the highest exponent of the variable is three. Cubic functions have a distinctive S-shaped curve and can exhibit a variety of behaviors, including having one, two, or three real zeros, depending on the coefficients of the function.
Cubic functions: A cubic function is a polynomial function of degree three, typically expressed in the form $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are constants and $a \neq 0$. These functions can have up to three real roots and exhibit distinct characteristics such as points of inflection.
Decreasing linear function: A decreasing linear function is a linear function where the value of the function decreases as the input increases. It has a negative slope.
Degree: The degree of a polynomial is the highest power of the variable in its expression. It determines the most significant term when expanding or simplifying the polynomial.
Degree: In mathematics, the term 'degree' refers to the measure of a polynomial or the measure of an angle. It is a fundamental concept that underpins various topics in algebra, trigonometry, and calculus, including polynomials, power functions, graphs, and trigonometric functions.
Difference of Cubes: The difference of cubes is a special case of polynomial factorization where a polynomial expression can be factored by recognizing the difference between two cubes. This factorization technique is useful in solving certain types of equations and understanding the behavior of power functions and polynomial functions.
Difference of squares: The difference of squares is a specific type of polynomial that takes the form $a^2 - b^2$, which can be factored into $(a + b)(a - b)$. It is based on the property that the product of a sum and a difference of two terms results in the difference of their squares.
Difference of Squares: The difference of squares is a special type of polynomial expression where the terms are the difference between two perfect squares. This concept is particularly important in the context of factoring polynomials, working with rational expressions, solving quadratic equations, and understanding the properties of power functions and polynomial functions.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
End Behavior: The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity. It describes the limiting values or patterns that the function exhibits as it extends towards the far left and right sides of its graph.
Even function: An even function is a function $f(x)$ where $f(x) = f(-x)$ for all $x$ in its domain. This symmetry means the graph of an even function is mirrored across the y-axis.
Even Function: An even function is a mathematical function where the output value is the same regardless of whether the input value is positive or negative. In other words, the function satisfies the equation f(x) = f(-x) for all values of x in the function's domain.
Exponentiation: Exponentiation is the mathematical operation of raising a number or expression to a power. It represents the repeated multiplication of a base number by itself a specified number of times, known as the exponent. This fundamental concept is central to understanding power functions and polynomial functions in mathematics.
F(x): f(x) is a function notation that represents a relationship between an independent variable, x, and a dependent variable, f. It is a fundamental concept in mathematics that underpins the study of functions, their properties, and their applications across various mathematical topics.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
Function Composition: Function composition is the process of combining two or more functions to create a new function. The output of one function becomes the input of the next function, allowing for the creation of more complex mathematical relationships.
Fundamental Theorem of Algebra: The Fundamental Theorem of Algebra states that every non-constant polynomial equation with complex number coefficients has at least one complex number solution. It is a foundational result in algebra that connects the properties of polynomials to the nature of complex numbers.
General form of a quadratic function: The general form of a quadratic function is expressed as $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants and $a \neq 0$. This representation is crucial for solving quadratic equations and analyzing their properties.
Graphing Techniques: Graphing techniques refer to the methods and strategies used to visually represent mathematical functions, relationships, and data on a coordinate plane or graph. These techniques allow for the effective visualization and analysis of various types of functions, including power functions and polynomial functions.
Greatest common factor: The greatest common factor (GCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It is useful in simplifying fractions, factoring polynomials, and solving equations.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides two or more integers without a remainder. It is a fundamental concept in algebra that is essential for understanding real numbers, factoring polynomials, working with rational expressions, solving quadratic equations, and analyzing power and polynomial functions.
Inverse of a rational function: The inverse of a rational function is a function that reverses the effect of the original rational function. It essentially swaps the dependent and independent variables, solving for the input in terms of the output.
Irrational Function: An irrational function is a function that contains variables with irrational exponents or irrational coefficients. These functions cannot be expressed exactly using rational numbers and often involve the use of irrational constants like pi or the square root of 2. Irrational functions are important in the study of power functions and polynomial functions, as they introduce new mathematical properties and challenges.
Leading coefficient: The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial function.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It is the first, or leading, coefficient in the standard form of a polynomial expression. The leading coefficient plays a crucial role in understanding and analyzing various topics in college algebra, including polynomials, quadratic equations, functions, and more.
Leading term: The leading term of a polynomial is the term with the highest power of the variable. It determines the end behavior of the polynomial function.
Linear Function: A linear function is a mathematical function that represents a straight line on a coordinate plane. It is characterized by a constant rate of change, known as the slope, and can be expressed in the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
Long division: Long division is a method used to divide polynomials by another polynomial of lesser or equal degree. It involves repeated division, multiplication, and subtraction to obtain the quotient and remainder.
Long Division: Long division is a step-by-step procedure for dividing one polynomial by another, where the divisor is of a higher degree than the dividend. It involves repeatedly subtracting multiples of the divisor from the dividend until the remainder is of a lower degree than the divisor.
Odd function: An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. Graphically, odd functions exhibit symmetry about the origin.
Odd Function: An odd function is a function that satisfies the property $f(-x) = -f(x)$ for all $x$ in the domain of the function. This means that the graph of an odd function is symmetric about the origin, with the graph being a reflection across both the $x$-axis and the $y$-axis.
Polynomial Arithmetic: Polynomial arithmetic refers to the fundamental operations performed on polynomial expressions, including addition, subtraction, multiplication, and division. These operations are essential for manipulating and simplifying polynomial functions, which are a crucial component of power functions and polynomial functions.
Polynomial function: A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It can be expressed in the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ where $a_i$ are constants and $n$ is a non-negative integer.
Polynomial Function: A polynomial function is an algebraic function that can be expressed as the sum of a finite number of non-negative integer powers of a variable, with coefficients. Polynomial functions are a fundamental concept in algebra and are closely related to topics such as power functions, polynomial division, and the zeros of polynomial functions.
Polynomial inequalities: Polynomial inequalities are mathematical expressions that involve polynomials and establish a relationship of inequality, such as greater than, less than, greater than or equal to, or less than or equal to. Understanding polynomial inequalities allows for the exploration of the regions where the polynomial function takes on specific values, leading to graphical representations and solutions that identify intervals of interest on the number line. They often require analysis of the polynomial's roots and its behavior at these critical points to determine where the inequality holds true.
Power function: A power function is a function of the form $f(x) = ax^n$ where $a$ and $n$ are constants, $a \neq 0$, and $n$ is a real number. Power functions are a basic type of polynomial function when $n$ is a non-negative integer.
Power Function: A power function is a mathematical function that involves raising a variable to a constant power. These functions are characterized by their ability to model exponential growth or decay, and they play a crucial role in understanding polynomial functions.
Quadratic Function: A quadratic function is a polynomial function of degree two, where the highest exponent of the independent variable is two. Quadratic functions are widely used in various mathematical and scientific applications, including physics, engineering, and economics.
Range: In mathematics, the range refers to the set of all possible output values (dependent variable values) that a function can produce based on its input values (independent variable values). Understanding the range helps in analyzing how a function behaves and what values it can take, connecting it to various concepts like transformations, compositions, and types of functions.
Rational Function: A rational function is a function that can be expressed as the ratio of two polynomial functions. It is a function that can be written in the form $f(x) = P(x)/Q(x)$, where $P(x)$ and $Q(x)$ are polynomial functions and $Q(x)$ is not equal to zero.
Rational Root Theorem: The Rational Root Theorem is a fundamental principle in the study of polynomial functions that provides a way to determine the possible rational roots of a polynomial equation. It helps simplify the process of finding the roots or zeros of a polynomial by narrowing down the potential solutions.
Roots: Roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as solutions or zeros of the equation.
Roots: In mathematics, the term 'roots' refers to the solutions or values of a polynomial equation that make the equation equal to zero. Roots are an essential concept in various topics related to polynomial functions and equations, including quadratic equations, power functions, and the graphs of polynomial functions.
Sum of Cubes: The sum of cubes is a mathematical expression that represents the sum of the cubes of two or more numbers. It is an important concept in various areas of mathematics, including factoring polynomials, solving certain types of equations, and understanding the behavior of power functions and polynomial functions.
Symmetry: Symmetry is the quality of being made up of exactly similar parts facing each other or around an axis. It is a fundamental concept in mathematics and geometry that describes the balanced and harmonious arrangement of elements in an object or function.
Synthetic division: Synthetic division is a simplified method of dividing polynomials where only the coefficients are used. It is particularly useful for dividing by linear factors of the form $x - c$.
Synthetic Division: Synthetic division is a shortcut method used to divide a polynomial by a linear expression of the form $(x - a)$. It allows for the efficient computation of polynomial division, providing a streamlined approach to determining the quotient and remainder of the division process.
Term of a polynomial function: A term of a polynomial function is an expression consisting of a coefficient and a variable raised to a non-negative integer exponent. Each term in the polynomial is separated by addition or subtraction.
Turning point: A turning point is a point on the graph of a polynomial function where the graph changes direction from increasing to decreasing or vice versa. It occurs at local maxima or minima.
Turning Point: A turning point is a critical moment or event that marks a significant change or shift in direction, often serving as a pivotal point that can alter the course of something. This term is particularly relevant in the context of analyzing the behavior and characteristics of various mathematical functions, including polynomials, power functions, and parabolas.
Volume of a sphere: The volume of a sphere is the measure of the amount of space inside the sphere, calculated using the formula $V = \frac{4}{3} \pi r^3$, where $r$ is the radius.
Zeros: Zeros of a polynomial function are the values of the variable that make the function equal to zero. In other words, they are the solutions to the equation $P(x) = 0$.
Zeros: Zeros, also known as roots, are the values of the independent variable that make a function equal to zero. They are the points where the graph of a function intersects the x-axis, representing the solutions to the equation f(x) = 0.
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Glossary
Glossary
5.3 Graphs of Polynomial Functions