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Solving Polynomial Equations in Excel. Given the polynomial function \(f(x)=x^44x^245\), determine the \(y\)- and \(x\)-intercepts. The \(x\)-intercepts are \((3,0)\) and \((3,0)\). $, $ Solution. Math can be a difficult subject for some students, but with a little patience and practice, it can be mastered. Use Figure \(\PageIndex{4}\) to identify the end behavior. To derive the equation of a function from a table of values (or a curve), there are several mathematical methods. Calculate the slope of the line and equation of the line with given coordinates of. Free time to spend with your family and friends. \[ \begin{align*} f(0) &=(0)^44(0)^245 \\[4pt] &=45 \end{align*}\]. Any lowercase letter may be used as a variable. This polynomial is considered to have two roots, both equal to 3. How To: Given two points on the curve of an exponential function, use a graphing calculator to find the equation. In L1, enter the x-coordinates given. Visualize the exponential function that passes through two points, which may be dragged within the x-y plane. Example \(\PageIndex{8}\): Determining the Intercepts of a Polynomial Function. A power function contains a variable base raised to a fixed power (Equation \ref{power}). Rewrite the equation. A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. $50 = c \cdot a^5 \\ We can combine this with the formula for the area A of a circle. In some cases, linear algebra methods such as Gaussian elimination are used, with optimizations to increase speed and reliability. 2. Sorry I didn't mean the notation but if there were any actual steps that I could have skipped, which you both helped with so thanks. Math can be confusing, but there are ways to clarify questions and get the answers you need. This function will be discussed later. In both cases, you could divide your first equation by the second one (or vice versa) and then take ln on both, Write a power function y 5 axb whose graph passes through (3, 2) and (6, 9). This is called an exponential function, not a power function. Power function calculator two points - Analyzes the data table by power regression and draws the chart. Suppose a certain species of bird thrives on a small island. The REAL Answer To The Viral Chinese Math Problem How Old Is The Captain?. ln(1600) = ln( c ) + rln(10) the video describes how to find exponential function from given two points of the function As \(x\) approaches positive infinity, \(f(x)\) increases without bound; as \(x\) approaches negative infinity, \(f(x)\) decreases without bound. A polynomial equation is a combination of variables and coefficients with arithmetic operations. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. 5 Examples of Solving Equations in Excel. The best way to protect your data is to keep it secure. general form of a polynomial function: \(f(x)=a_nx^n+a_{n-1}x^{n-1}+a_2x^2+a_1x+a_0\). Identify the term containing the highest power of \(x\) to find the leading term. We can see these intercepts on the graph of the function shown in Figure \(\PageIndex{12}\). Enter x and y and this calculator will solve for the exponent n using log (). Describe the end behavior of the graph of \(f(x)=x^8\). The exponent of the power function is 9 (an odd number). In both cases, you could divide your first equation by the second one (or vice versa) and then take ln on both, To find an exponential function, f(x)=ax f ( x ) = a x , containing the point, set f(x) f ( x ) in the function to the y y value 25 25 of the point, Application of integral calculus in engineering, Best way to respond to interview questions, Compound inequality with no solution example, Distribution of the sample mean calculator, Find the area of the region bounded by the given curves. Polynomial Equation Solver ln(50) = ln( c ) + rln(5) \\ Get help from our expert homework writers! Learn more about Stack Overflow the company, and our products. The leading coefficient is the coefficient of that term, 4. you can remove the extra step when evaluating c, you only need to do it once. As \(x{\rightarrow}{\infty}\), \(f(x){\rightarrow}{\infty}\); as \(x{\rightarrow}{\infty}\), \(f(x){\rightarrow}{\infty}\). The square and cube root functions are power functions with fractional powers because they can be written as \(f(x)=x^{1/2}\) or \(f(x)=x^{1/3}\). It only takes a minute to sign up. Is it suspicious or odd to stand by the gate of a GA airport watching the planes? STEP 1 Substitute the coordinates of the two given points into y 5. order now. Power Function Calculator. In algebra, a quadratic equation is any polynomial equation of the second degree with the following form: ax 2 + bx + c = 0 where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. \[\begin{align*} 0&=-4x(x+3)(x-4) \\ x&=0 & &\text{or} & x+3&=0 & &\text{or} & x-4&=0 \\ x&=0 & &\text{or} & x&=3 & &\text{or} & x&=4 \end{align*}\]. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. It solves all my mah problems and explains them clearly I'm grateful to this app. Given the function \(f(x)=0.2(x2)(x+1)(x5)\), express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function. A log is the inverse of an exponent. Pi. $. There are many forms that can be used to provide multiple forms of content, including sentence fragments, lists, and questions. Determine that linear model, e.g., with the method of least squares. Read More The steps seem to be good. \frac{ln(50) - ln(1600)}{ln(5) - ln(10)} = r I think you realized the issue of factoring by your question near the end. It is because the numerator and denominator are equal. Wolfram|Alpha is a great tool for finding polynomial roots and solving systems of equations. The constant and identity functions are power functions because they can be written as \(f(x)=x^0\) and \(f(x)=x^1\) respectively. Describe in words and symbols the end behavior of \(f(x)=5x^4\). The Equation of a Line Calculator is an online tool that shows the slope and equation of a line, for the given input. Equation of a line given two points. 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. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. 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. The steps seem to be good. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 1600 = c \cdot a^{10} Thanks for the developers for creating this beautiful and very useful app. The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). Here are some examples illustrating how to formulate queries. A power function is a variable base raised to a number power. 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. BYJU'S two-point form calculator makes it simple to find the slope of a line if the coordinates of the two points are given. In symbolic form, we would write, \[\begin{align*} \text{as }x{\rightarrow}-{\infty},\;f(x){\rightarrow}{\infty} \\ \text{as }x{\rightarrow}{\infty},\;f(x){\rightarrow}-{\infty} \end{align*}\]. Where: c = Coefficient. Midpoint of two points. $, $ Determine whether the constant is positive or negative. Mathepower calculates your function by solving equation systems. Example \(\PageIndex{11}\): Drawing Conclusions about a Polynomial Function from the Graph. But the updates are pretty quick and decent. The point corresponds to the coordinate pair in which the input value is zero. Identify end behavior of power functions. Based on this, it would be reasonable to conclude that the degree is even and at least 4. If you don't know how, you can find instructions. 1600 = c \cdot 10^r Class 9 science chapter 2 extra questions and answers, Marginal probability examples with solutions. Each \(a_i\) is a coefficient and can be any real number. . Identify the degree, leading term, and leading coefficient of the polynomial \(f(x)=4x^2x^6+2x6\). Dead laser accurate camera with a freaking university level calculator. There are multiple ways to reduce stress, including exercise, relaxation techniques, and healthy coping mechanisms. Press [STAT]. As the input values \(x\) get very small, the output values \(f(x)\) decrease without bound. Absolutley amazing app and would definitely recommend. Example \(\PageIndex{10}\): Determining the Number of Intercepts and Turning Points of a Polynomial. Which of the following functions are power functions? Entertainment-- I'm looking at data from the coronavirus outbreak. Why does Mister Mxyzptlk need to have a weakness in the comics? The behavior of a graph as the input decreases without bound and increases without bound is called the end behavior. Given the function \(f(x)=0.2(x2)(x+1)(x5)\), determine the local behavior. Obtain the general form by expanding the given expression for \(f(x)\). The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. If you answer the question with 5+5+5=15, you would be wrong. If you have two points, (x1, y1) and (x2, y2), you can define the exponential function that passes through these points by substituting them in. Legal. The math equation is simple, but it's still confusing. y = 6x2 ln(x), y = 24 ln(x), How to find length of square with only diagonal, How to make a data chart in google sheets, Solve the word problem using the rdw strategy. In most cases, you'll do this by entering the first number, pressing the carrot ( ^) button, and entering the number to which you want to raise the first number. Connect and share knowledge within a single location that is structured and easy to search. 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. \(\PageIndex{5}\): Given the polynomial function \(f(x)=2x^36x^220x\), determine the \(y\)- and \(x\)-intercepts. \Rightarrow c = \frac{1600}{1024} = \frac{25}{16} Describe the end behavior and determine a possible degree of the polynomial function in Figure \(\PageIndex{8}\). By taking the time to explain the problem and break it down into smaller pieces, anyone can learn to solve math problems. You can find a base-10 log using most scientific calculators. The graph has 2 \(x\)-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Press ENTER or = to solve the equation. When one piece is missing, it can be difficult to see the whole picture. The degree of a polynomial function helps us to determine the number of \(x\)-intercepts and the number of turning points. Display output to. Identify the degree, leading term, and leading coefficient of the following polynomial functions. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. This page titled 3.3: Power Functions and Polynomial Functions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. In symbolic form, we could write, \[\text{as } x{\rightarrow}{\pm}{\infty}, \;f(x){\rightarrow}{\infty} \nonumber\]. Which of the following are polynomial functions? For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems. A Parametric Equation Calculator works by solving the parametric equation's algebra using arbitrary values for the parameter serving as the independent variable, Plane and Parametric Equations in R 3 Calculator: Given a vector A and a point (x,y,z), this will calculate the following items: 1) Plane Equation passing, 90 level how to solve inequalities using power of quotient, Class 11 trigonometry previous year questions, Critical values for a left tailed z test calculator, How to find the length of an arc of a circle with radius, How to identify the domain and range of a quadratic function, Slader an introduction to mathematical statistics and its applications, Slope as rate of change algebra 1 homework answers, Solve cubic equation calculator with steps, Solve my algebra word problem step by step free. To determine when the output is zero, we will need to factor the polynomial. Share Cite Follow answered Nov 11, 2012 at 15:09 Bhavish Suarez 664 1 7 15 Add a comment You must log in to answer this question. Add texts here. Free functions calculator - explore function domain, range, intercepts, extreme points and asymptotes step-by-step. An amazing app that gives you the correct answer every time. Power function from points calculator - We discuss how Power function from points calculator can help students learn Algebra in this blog post. A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. If you are still unsure, ask a friend or teacher for help. It is possible to find the equation of a power function from its graph or from any two points on the graph. Easy to use, no glitches, very few ads. The function for the area of a circle with radius \(r\) is, and the function for the volume of a sphere with radius \(r\) is. As the input values \(x\) get very large, the output values \(f(x)\) increase without bound. We can see these intercepts on the graph of the function shown in Figure \(\PageIndex{11}\). The mathematics power calculator is a simple maths calculator that tells you how much a number equals in its exponential form. Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage. We can use this model to estimate the maximum bird population and when it will occur. As with all functions, the \(y\)-intercept is the point at which the graph intersects the vertical axis. Example \(\PageIndex{9}\): Determining the Intercepts of a Polynomial Function with Factoring. Our math homework helper is here to help you with any math problem, big or small. To find an exponential function, f (x) = ax f ( x) = a x, containing the point, set f (x) f ( x) in the function to the y y value 25 25 of the point, and set x x to the x x value 2 2 of the point. $, $ If has degree , then it is well known that there are roots, once one takes into account multiplicity. Example \(\PageIndex{2}\): Identifying the End Behavior of a Power Function. 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). Since in the equation y = 0.1349x^0.9719, the exponent is so close to one, it looks like for every increase of one unit in x, y increases by a little less than 0.1349 units. The reciprocal of a number is a number which when multiplied with the actual number produces a result of 1 For example, let us take the number 2. Power function calculator with points can help students to understand the material and improve their grades. Without graphing the function, determine the maximum number of \(x\)-intercepts and turning points for \(f(x)=10813x^98x^4+14x^{12}+2x^3\). The general rule is that for any n given points there is a function of degree whose graph goes through them. Also, when we multiply the reciprocal with the original number we get 1 1 2 2 = 1 1 2 2 = 1 We can also use this model to predict when the bird population will disappear from the island. Possibly one of the most useful apps out there. Two point slope form equation. I almost fail math and don't understand what teacher said, i ever never facing problems in maths solution's. Given a polynomial function, determine the intercepts. 10/10 Way quicker than typing it into the website. Ratings 15447+ Completed orders Find the Exponential Function (2,25) Tool to find the equation of a function from its points, its coordinates x, y=f(x) according Power (Including Inverse and nth . With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Can Martian Regolith be Easily Melted with Microwaves. For example, to calculate 2 2, you would type in 2^2 and then press ENTER or =. There are at most 12 \(x\)-intercepts and at most 11 turning points. $, $ What can we conclude about the polynomial represented by the graph shown in Figure \(\PageIndex{15}\) based on its intercepts and turning points? The first two functions are examples of polynomial functions because they can be written in the form of Equation \ref{poly}, where the powers are non-negative integers and the coefficients are real numbers. The \(x\)-intercepts are the points at which the output value is zero. This calculator solves equations that are reducible to polynomial form. How to find exponential model from two points - Given, two points, (x1,y1) and (x2,y2) and y=aebx. \[\begin{align*} f(x)&=x^44x^245 \\ &=(x^29)(x^2+5) \\ &=(x3)(x+3)(x^2+5) Power regression Calculator Home / Mathematics / Regression Analyzes the data table by power regression and draws the chart. This is called the general form of a polynomial function. 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. The graph of the polynomial function of degree \(n\) must have at most \(n1\) turning points. $, $ How would I write a power function through the two points (1,9. Can airtags be tracked from an iMac desktop, with no iPhone? These examples illustrate that functions of the form \(f(x)=x^n\) reveal symmetry of one kind or another. Use the Power Function Calculator and Chart Maker to evaluate and chart a power function of the general form: f (x) = c * x ^b. Now, using the exponential property that (x^a)/ (x^b)= x^ (a-b), we have Step-by-step Assuming you want a sentence related to the background information: The best way to learn something new is to break it down into small, manageable steps. Guidelines for interpreting correlation coefficient r . 20 years old level / A teacher / A researcher / Useful /. To improve your math performance, practice regularly and persistently. $, $ In order to better understand the bird problem, we need to understand a specific type of function. The numerals a, b, and c are coefficients of the equation, and they represent known numbers. I have a problem where I'm asked to determine the constants of exponential and power functions that go throughboth points (5, 50) and (10, 1600). Power Function Examples & Equation (9/4) = Aek and (9/32) = Ae2k are the 2 equations needed to determine the two parameters. These methods are carefully designed and chosen to enable Wolfram|Alpha to solve the greatest variety of problems while also minimizing computation time. More than just an app, Tinder is a social platform that allows users to connect with others in their area. Our math solver offers professional guidance on How to find a power function given two points every step of the way. This app has helped me check my son's homework and explain to him where he made his mistakes and how to fix it to get the right answer. Set up the equation so that you are taking the log of both sides. 1. Tool to find the equation of a function from its points, its coordinates x, y=f(x) according Power (Including Inverse and nth Root) using Curve Fitting. Trigonometry. If you have two points, (x 1, y 1) and (x 2, y 2 ), you can define the exponential function that passes through these points by substituting them in the equation y = ab x and solving for a and b. Why is 5+5+5=15 on Common Core math tests wrong. Math is a process of finding solutions to problems. Free exponential equation calculator - solve exponential equations step-by-step. It is possible to find the equation of a power function from its graph or from any two points on the graph. We can describe the end behavior symbolically by writing, \[\text{as } x{\rightarrow}{\infty}, \; f(x){\rightarrow}{\infty} \nonumber\], \[\text{as } x{\rightarrow}-{\infty}, \; f(x){\rightarrow}-{\infty} \nonumber\]. It has the shape of an even degree power function with a negative coefficient. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 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. Example \(\PageIndex{12}\): Drawing Conclusions about a Polynomial Function from the Factors. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Use this step-by-step Exponential Function Calculator, to find the function that describe the exponential function that passes through two given points. Write a power function y 5 axb whose graph passes through (3, 2) and (6, 9). Because a polynomial is a function, only one output value corresponds to each input value so there can be only one \(y\)-intercept \((0,a_0)\). In L2, enter the corresponding y-coordinates. An example of how to solve for a power function given two data points on the curve. 25 = a2 25 = a 2 Solve the equation for a a. \end{align*}\], \[\begin{align*} x3&=0 & &\text{or} & x+3&=0 & &\text{or} & x^2+5&=0 \\ x&=3 & &\text{or} & x&=3 & &\text{or} &\text{(no real solution)} \end{align*}\]. \Rightarrow r = 5 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. The quadratic and cubic functions are power functions with whole number powers f(x) = x2 and f(x) = x3. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Each product \(a_ix^i\) is a term of a polynomial function. Math can be confusing, but there are ways to clarify questions and get the answers you need. In particular, we are interested in locations where graph behavior changes. Is a PhD visitor considered as a visiting scholar? Enter some points / maxima / minima / slopes etc. \Rightarrow c = \frac{50}{32} = \frac{25}{16} Is this integration approximation method known/used? The leading term is \(3x^4\); therefore, the degree of the polynomial is 4. A polynomial function of \(n^\text{th}\) degree is the product of \(n\) factors, so it will have at most \(n\) roots or zeros, or \(x\)-intercepts. Example \(\PageIndex{6}\): Identifying End Behavior and Degree of a Polynomial Function. 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. . In both cases, you could divide your first equation by the second one (or vice versa) and then take ln on both sides. As \(x\) approaches positive infinity, \(f(x)\) increases without bound. By taking the time to explain the problem and break it down into smaller pieces, anyone can learn to solve math problems. We write as \(x,\) \(f(x).\) As \(x\) approaches negative infinity, the output increases without bound. Exponential Function Calculator Use this step-by-step Exponential Function Calculator, to find the function that describe the exponential function that passes through two given points. Wolfram|Alpha is a great tool for finding the domain and range of a function. The radius \(r\) of the spill depends on the number of weeks \(w\) that have passed. Given the function \(f(x)=4x(x+3)(x4)\), determine the local behavior. The behavior of the graph of a function as the input values get very small \((x{\rightarrow}{\infty})\) and get very large \(x{\rightarrow}{\infty}\) is referred to as the end behavior of the function. Uh oh! Distance between two points. Definite integral involving negative infinity, Solving an expression containing two added exponential functions, Ensure exponential function passes through two specific points. Some functions are limited now because setting of JAVASCRIPT of the browser is OFF. example. these look correct. Solve math questions To solve a math equation, you need to find the value of the variable that makes the equation true. This too is typically encountered in secondary or college math curricula. Each piece of the equation fits together to create a complete picture. The leading term is \(0.2x^3\), so it is a degree 3 polynomial. It also factors polynomials, plots polynomial solution sets and inequalities and more. To determine its end behavior, look at the leading term of the polynomial function. The procedure to use the vertex calculator is as follows: Step 1: Enter the values "a", "b", "c" in the respective input field Step 2: Now click the button "Solve" to get the coordinates Step 3: Finally, the vertex coordinates will be displayed in the output field What is Meant by Vertex Coordinates? Exponential Function Calculator. a = 5,5 a = 5, - 5 Click on the "Calculate" button to compute the quadratic regression equation. We are also interested in the intercepts. There are many ways to improve your writing skills. Love it, super helpful, especially with trigonometric equations, no ads( not really sure why there would be) and no long waiting. rev2023.3.3.43278. :), Exponential and power functions through two points, We've added a "Necessary cookies only" option to the cookie consent popup. As a result, Wolfram|Alpha also has separate algorithms to show algebraic operations step by step using classic techniques that are easy for humans to recognize and follow. a nonzero real number that is multiplied by a variable raised to an exponent (only the number factor is the coefficient), a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph, the highest power of the variable that occurs in a polynomial, the behavior of the graph of a function as the input decreases without bound and increases without bound, the term containing the highest power of the variable. \Rightarrow c = \frac{2}{125} Example \(\PageIndex{4}\): Identifying Polynomial Functions. The value of a is 0.05. Common core math wrong answers is a software program that supports students solve math problems. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools. We Answer! Lets use simpler terms (2^2*2^3)^3*2^2 = (4*8)^3*4 = 131072. Systems of linear equations are often solved using Gaussian elimination or related methods. Since the numerator and denominator are equal, this is also equal to 1. 1600 = 1024c Example: So e.g. Find the highest power of \(x\) to determine the degree function. Its population over the last few years is shown in Table \(\PageIndex{1}\). Because of the end behavior, we know that the lead coefficient must be negative. ln(50) = ln( c ) + 5ln(a) \\ The term containing the highest power of the variable is called the leading term. Math can be a difficult subject for many people, but it doesn't have to be! A Parametric to Cartesian Equation Calculator is an online solver that only needs two parametric equations for x and y for conversion. \Rightarrow ln(\frac{50}{1600}) = -5ln(a) { "3.00:_Prelude_to_Polynomial_and_Rational_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.01:_Complex_Numbers" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.02:_Quadratic_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.03:_Power_Functions_and_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.04:_Graphs_of_Polynomial_Functions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "3.05:_Dividing_Polynomials" : "property get [Map 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Behavior of Power Functions, Identifying the Degree and Leading Coefficient of a Polynomial Function, Identifying End Behavior of Polynomial Functions, Identifying Local Behavior of Polynomial Functions, source@https://openstax.org/details/books/precalculus, status page at https://status.libretexts.org.