Difference between revisions of "Intro to Power Functions"
| Line 1: | Line 1: | ||
A monomial, also called '''power product''', is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, <math>x^2yz^3=xxyzzz</math> is a monomial. The constant <math>1</math> is a monomial, being equal to the empty rpdouct and to <math>x^0</math> for any variable <math>x</math>. If only a single variable <math>x</math> is considered, this means that a monomial is either <math>1</math> or a power <math>x^n</math> of <math>x</math>, with <math>n</math> a positive integer. If several variables are considered, say, <math>x, y, z,</math> then each can be given an exponent, so that any monomial is of the form <math>x^a y^b z^c</math> with <math>a,b,c</math> non-negative integers (taking note that any exponent <math>0</math> makes the corresponding factor equal to <math>1</math>). | A monomial, also called '''power product''', is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, <math>x^2yz^3=xxyzzz</math> is a monomial. The constant <math>1</math> is a monomial, being equal to the empty rpdouct and to <math>x^0</math> for any variable <math>x</math>. If only a single variable <math>x</math> is considered, this means that a monomial is either <math>1</math> or a power <math>x^n</math> of <math>x</math>, with <math>n</math> a positive integer. If several variables are considered, say, <math>x, y, z,</math> then each can be given an exponent, so that any monomial is of the form <math>x^a y^b z^c</math> with <math>a,b,c</math> non-negative integers (taking note that any exponent <math>0</math> makes the corresponding factor equal to <math>1</math>). | ||
| − | A power function | + | A power function is a function that can be represented in the form |
| − | <math> f(x) = kx^p | + | <math> f(x) = kx^p </math> |
| − | where <math>k</math> and <math>p</math> are real numbers, and <math>k</math> is known as the coefficient. | + | where <math>k</math> and <math>p</math> are real numbers, and <math>k</math> is known as the coefficient. We can also think of a power function as a monomial function; that is, a power function takes the form <math> y = f(x) </math>, where <math> f(x) </math> is a single-variable monomial. |
==Resources== | ==Resources== | ||
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Intro_to_Polynomial_and_Power_Functions/MAT1053_M2.1Intro_to_Power_Functions_and_Polynomial_Functions.pdf Intro to Power Functions and Polynomial Functions], Book Chapter | * [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Intro_to_Polynomial_and_Power_Functions/MAT1053_M2.1Intro_to_Power_Functions_and_Polynomial_Functions.pdf Intro to Power Functions and Polynomial Functions], Book Chapter | ||
| + | * [https://openstax.org/books/college-algebra/pages/5-2-power-functions-and-polynomial-functions Power Functions and Polynomial Functions | ||
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Intro_to_Polynomial_and_Power_Functions/MAT1053_M2.1Intro_to_Power_Functions_and_Polynomial_FunctionsGN.pdf Guided Notes] | * [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Intro_to_Polynomial_and_Power_Functions/MAT1053_M2.1Intro_to_Power_Functions_and_Polynomial_FunctionsGN.pdf Guided Notes] | ||
Revision as of 13:40, 4 October 2021
A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, is a monomial. The constant is a monomial, being equal to the empty rpdouct and to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^0} for any variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . If only a single variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is considered, this means that a monomial is either Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} or a power Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^n} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} a positive integer. If several variables are considered, say, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y, z,} then each can be given an exponent, so that any monomial is of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^a y^b z^c} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a,b,c} non-negative integers (taking note that any exponent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} makes the corresponding factor equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} ).
A power function is a function that can be represented in the form
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = kx^p }
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} are real numbers, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is known as the coefficient. We can also think of a power function as a monomial function; that is, a power function takes the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = f(x) } , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } is a single-variable monomial.
Resources
- Intro to Power Functions and Polynomial Functions, Book Chapter
- [https://openstax.org/books/college-algebra/pages/5-2-power-functions-and-polynomial-functions Power Functions and Polynomial Functions
- Guided Notes
