linear vs exponential vs quadratic

Create your account. For example, quadratic functions are useful for acceleration due to gravity, while exponential growth functions are useful for compound interest and investments (you may also use exponential decay in some cases). 's' : ''}}. Question 10. I've got good news! We call the graph of a quadratic equation a parabola, and the high or low point of the graph is a vertex. Enrolling in a course lets you earn progress by passing quizzes and exams. I help with some common (and also some not-so-common) math questions so that you can solve your problems quickly! Practice: Exponential vs. linear models. copyright 2003-2022 Study.com. The value of c is the constant term and also the y-intercept of the parabola.] Create an account to start this course today. Since the acceleration is -32 feet per second (negative due to falling), the velocity is 0 32t (0 because the starting velocity is zero, since the object was dropped from rest). The holy trinity. A quadratic function is a polynomial function with a highest exponent of two. Probability depends on the event we wish to observe and the conditions of the scenario. After half an hour of driving, you are 18 miles from your house. As a member, you'll also get unlimited access to over 84,000 HW3 Algebraic Equations from data (1).doc. In order to have the quadratic equation we must have any positive or the negative value of the coefficient a right before the variable x, so that it can make the sequence of . You write down that he is doing a good job, understanding that accidents happen. Once the exponential function passes up the quadratic function, it just goes faster. First differences are constant or equal. If we are given a table of functions, how do we determine if the functions are linear, quadratic, or exponential?If you buy me a coffee, I can stay awake and. Linear, exponential, and quadratic functions can be used in modeling and visualizing trends of real-world situations. McDougal Littell Algebra 1: Online Textbook Help, McDougal Littell Algebra 1 Chapter 10: Quadratic Equations & Functions, {{courseNav.course.mDynamicIntFields.lessonCount}}, Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses, McDougal Littell Algebra 1 Chapter 1: Expressions, Equations & Functions, McDougal Littell Algebra 1 Chapter 2: Properties of Real Numbers, McDougal Littell Algebra 1 Chapter 3: Solving Linear Equations, McDougal Littell Algebra 1 Chapter 4: Graphing Linear Equations & Functions, McDougal Littell Algebra 1 Chapter 5: Writing Linear Equations, McDougal Littell Algebra 1 Chapter 6: Solving & Graphing Linear Inequalities, McDougal Littell Algebra 1 Chapter 7: Systems of Linear Equations & Inequalities, McDougal Littell Algebra 1 Chapter 8: Exponents & Exponential Functions, McDougal Littell Algebra 1 Chapter 9: Polynomials & Factoring, Parabolas in Standard, Intercept, and Vertex Form, How to Solve Quadratics That Are Not in Standard Form, Solving Quadratic Equations with Square Roots or Graphing the Function, How to Use the Quadratic Formula to Solve a Quadratic Equation, How to Solve Quadratics with Complex Numbers as the Solution, Comparing Linear, Exponential & Quadratic Models, McDougal Littell Algebra 1 Chapter 11: Radicals & Geometry Connections, McDougal Littell Algebra 1 Chapter 12: Rational Equations & Functions, McDougal Littell Algebra 1 Chapter 13: Probability & Data Analysis, SAT Subject Test Mathematics Level 2: Practice and Study Guide, Common Core Math - Geometry: High School Standards, CSET Math Subtest 1 (211) Study Guide & Practice Test, CSET Math Subtest II (212): Practice & Study Guide, CSET Math Subtest III (213): Practice & Study Guide, Study.com ACT® Test Prep: Practice & Study Guide, Common Core Math - Algebra: High School Standards, Comparing Linear, Exponential & Quadratic Functions, Short Run & Long Run Behavior of Polynomials: Definition & Examples, Comparing Linear, Quadratic & Exponential Models, Sample LSAT Analytical Reasoning Questions & Explanations, Strategies for Analytical Reasoning Questions on the LSAT, How to Reason Deductively From a Set of Statements, Strategies for Logical Reasoning Questions on the LSAT, Using the IRAC Method on the LSAT Writing Sample, Linear Transformations: Properties & Examples, SAT Math Level 2: Structure, Patterns & Scoring, Using a Calculator for the SAT Math Level 2 Exam, Closed Questions in Math: Definition & Examples, Working Scholars Bringing Tuition-Free College to the Community. They are represented by a parabola (a U-shape) in the coordinate plane and follows the formula: Note that the value of a determines the parabolas direction: The graph of a quadratic function is symmetric vertically, and its axis of symmetry is the vertical line x=0 . Graphs come in all sorts of shapes and sizes. Date: _____ Linear vs. Exponential Continued 5. Algebra 2 - Comparing Linear and Exponential Functions Scenario Name: _ Linear. Worksheets are Comparing linear quadratic and exponential functions, Linear vs quadratic vs exponential, Section 8 summary of functions, Name algebra 1b date linear exponential continued, Linear quadratic and exponential functions work, Quadratic and exponential functions, Identifying linear quadratic and . Well also answer some common questions and look at examples to make the concepts clear. You will often see both quadratic and exponential functions in math, science, finance, and other topics. The function modeling Tom's productivity rate is: which is a polynomial function with highest exponent equal to two, exactly the definition of a quadratic function. For instance, if we wanted to know how much will be in the account after 2 years, we plug 2 in for x and evaluate. A guide on 3 common algebra functions used in data science and data analysis, with worked examples using cats . This is the pattern that a quadratic function takes on. 144 lessons, {{courseNav.course.topics.length}} chapters | We see that at the end of the day, Billy isn't quite at 100% productivity, but he is pretty close. I'm the go-to guy for math answers. Since the slope increases by the same ratio (times 2) each time, we know that we have an exponential function. 9.4 Compare Linear, Exponential, and Quadratic Models Students will Compare Linear, Exponential, and Quadratic Models . If you are working ahead, you can download . Objectives Compare linear, quadratic, and exponential models. Continue with Recommended Cookies. Secondary Math Solutions. 1. The returns follow an exponential function because every $1 you gained as interest is further reinvested and receives compound interest. News; Function composition is often used in algebra and calculus, and it has many interesting properties. A graph is one of exponential growth if it has a positive and ever-increasing slope (it is convex or concave up, since the second derivative is always positive. How do you know a table is linear? We've looked at linear, exponential and quadratic models. Q. Zara made an initial deposit to a bank account and then added a fixed amount every week. Linear-vs.-Exponential-Equations-Packet-SE Download Here! If there is now power of 2 (quadratic term), then the equation is not quadratic. To tell if an equation is linear or quadratic, remember the form of each one: If there is no quadratic term (that is, the value of a is zero), then the equation is not quadratic. 7 Careers For Math Majors (Jobs That Use Lots Of Math). Some other phrases that suggest quadratic functions are: Lets say that we start at 10 miles north of Boston, and we are driving north at a constant speed of 60 miles per hour. 3. Which of the worker's productivity seems to take on this pattern? Applying Systems of Linear Equations to Market Equilibrium: Steps & Example, Infinite Series & Partial Sums: Explanation, Examples & Types, Volcanic Hazards & Prevention: Landslides, Lahars & Tsunamis, Interpreting Linear Relationships Using Data: Practice Problems, Rational vs. Irrational Numbers | Properties, Differences & Examples, Adding, Subtracting, & Multiplying Polynomials | How to Modify Polynomials, Exponential Functions | Examples & Transformations, Family of Functions Graphs & Characteristics | How to Identify a Function. A quadratic function is graphed as a parabola, changes the sign of its slope at the vertex, and has differences that increase by a constant amount in each time interval. This equation is exponential since it has the form cdx (c = 2 is the constant scaling factor, and d = 3 is the base). Some of our partners may process your data as a part of their legitimate business interest without asking for consent. As you observe them, you see that Sean's productivity rate increases at a constant rate over the eight-hour period. Degree: Degree of one. These include linear, exponential, and quadratic functions. learn how to find the formula of an exponential function here. Displaying all worksheets related to - Comparing Linear Quadratic And Exponential. If you are modeling something that is increasing or decreasing at a constant rate, a linear model is a good way to go. Piecewise Functions Overview & Graph | How to Graph a Piecewise Function. Negative Exponent Rules & Examples | What is a Negative Exponent? Dont forget to subscribe to my YouTube channel & get updates on new math videos! Play this game to review Algebra I. The general form of each model is shown: Each model has different characteristics that make them appropriate to use to model different scenarios. The shape of reflecting surfaces such as satellite dishes and contact lenses can be defined using quadratic functions. See that the graph forms an upside-down U with a maximum point. You have $100 that you would like to put into a savings account. Quadratic, Linear, and Exponential Models. The graph is a straight line, and it can also be horizontal (y = b, zero slope) or vertical (x = a, undefined slope). If base is =1 : as x increases, y doesn't change. Lets say that a falling object has a constant acceleration of 32 feet per second. We see that by modeling the situation with a linear model, as we did, we are able to better analyze the situation. For instance, graphs of linear functions increase or decrease at a constant rate, graphs of exponential functions increase or decrease slowly at first and then more quickly or quickly at first and then more slowly, and graphs of quadratic functions increase, hit a maximum, then decrease or decrease, hit a minimum, then increase. Linear functions are functions that are used to model phenomena that increase or decrease at a constant rate. These types of functions are polynomial functions with a highest exponent of one on the variable. Also, notice that Sean reaches a productivity rate of 100% by the end of the day. The following table shows the differences between linear and quadratic functions at a glance.FunctionTypeLinearQuadraticSlope (1stdifference)constantlinearConcavity2nddifference)zeroconstant(nonzero)Equationax + ba = slopeb = yinterceptax2 + bx + ca tells us theconcavity; ctells us they-interceptTablelook for yvalues thatincrease bya constantvaluelook for firstdifferences ofy values thatincrease by aconstantamountGraphlineparabolaPhrasesin a wordproblemconstant slope;constant speed;constant rateof changeconstantseconddifferences;constantaccelerationThis table shows the differences between linearand quadratic functions at a glance. Other sets by this creator. A science experiment involves periodically measuring the number of mold cells present on a piece of bread. Quadratic Function Formula & Graph | What is a Quadratic Function? how much do fdm pay. Linear vs. Quadratic vs. Exponential Functions Foldable. Quadratic growth means that there is a fixed acceleration (second derivative). By comparing and analyzing these types of functions, as we did for Company XYZ's employees, we can better understand which one to use to model a given scenario. Note that there is no quadratic term (there is no term with an x2] and no exponential term. So, the puppy's weight is increasing by 1.5 . MUST DO: Fill in your "Notes to a Future, Forgetful Me" for this lesson in your lesson packet. This equation is linear since it has the form ax + b (a = 2 is the slope, and b = 5 is the y-intercept). Pretty neat that you were able to figure all of this out using mathematics, huh? Big Ideas Math Algebra 1: Online Textbook Help, Big Ideas Math Algebra 1 - Chapter 8: Graphing Quadratic Functions, {{courseNav.course.mDynamicIntFields.lessonCount}}, Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses, Big Ideas Math Algebra 1 - Chapter 1: Solving Linear Equations, Big Ideas Math Algebra 1 - Chapter 2: Solving Linear Inequalities, Big Ideas Math Algebra 1 - Chapter 3: Graphing Linear Functions, Big Ideas Math Algebra 1 - Chapter 4: Writing Linear Functions, Big Ideas Math Algebra 1 - Chapter 5: Solving Systems of Linear Equations, Big Ideas Math Algebra 1 - Chapter 6: Exponential Functions & Sequences, Big Ideas Math Algebra 1 - Chapter 7: Polynomial Equations & Factoring, Parabolas in Standard, Intercept, and Vertex Form, Even & Odd Functions: Definition & Examples, Axis of Symmetry of a Parabola: Equation & Vertex, Parabola Intercept Form: Definition & Explanation, Comparing Linear, Exponential & Quadratic Functions, Big Ideas Math Algebra 1 - Chapter 9: Solving Quadratic Equations, Big Ideas Math Algebra 1 - Chapter 10: Radical Functions & Equations, Big Ideas Math Algebra 1 - Chapter 11: Data Analysis & Displays, SAT Subject Test Mathematics Level 2: Practice and Study Guide, Common Core Math - Geometry: High School Standards, CSET Math Subtest 1 (211) Study Guide & Practice Test, CSET Math Subtest II (212): Practice & Study Guide, CSET Math Subtest III (213): Practice & Study Guide, Study.com ACT® Test Prep: Practice & Study Guide, Common Core Math - Algebra: High School Standards, Short Run & Long Run Behavior of Polynomials: Definition & Examples, Comparing Linear, Quadratic & Exponential Models, Comparing Linear, Exponential & Quadratic Models, Sample LSAT Analytical Reasoning Questions & Explanations, Strategies for Analytical Reasoning Questions on the LSAT, How to Reason Deductively From a Set of Statements, Strategies for Logical Reasoning Questions on the LSAT, Using the IRAC Method on the LSAT Writing Sample, Linear Transformations: Properties & Examples, SAT Math Level 2: Structure, Patterns & Scoring, Using a Calculator for the SAT Math Level 2 Exam, Closed Questions in Math: Definition & Examples, Working Scholars Bringing Tuition-Free College to the Community. The absolute value of the slope gets larger as x gets larger. Consider the equation f(x) = 2x2 12x + 16. copyright 2003-2022 Study.com. Therefore, they are used to model phenomena that take on this pattern, just like Billy's productivity rate! This function is linear, no, we don't have to even read that. MUST DO: Watch the following video. Donate or volunteer today! These types of functions are used to model phenomena that increase and hit a maximum then decrease, or decrease and hit a minimum then increase. Given a set of data, decide which type of function models the data and write an equation to describe the function. The most poignant example of exponential growth in recent years is the spread of covid cases globally, which follows the rapid increase of exponential growth (See chart). It can be helpful to see the differences between them compared side-by-side. MUST DO: Pick up lesson packet from the designated spot in the classroom. We can add 3 columns to the table: first differences of x, first differences of y, and slope that is, the ratio of (first differences of y) / (first differences of x):xyFirstDifferencesof xFirstDifferencesof ySlope0511015522011010340120204801404051601808063201160160. Linear Vs Quadratic Equation. Get Started So, the object will hit the ground when the height is zero, or: At t = 3 seconds, the dropped object will hit the ground. This is the currently selected item. The value of b is the coefficient of the linear term. The ball leaves the player's hands, goes up in the air, hits a high point, then begins to fall -- hopefully into the basket. Billy took a little while to get going, so his productivity rate started off fairly slow and then increased more and more quickly. A linear function increases by a constant amount (the value of its slope) in each time interval, while a quadratic function increases by a different amount in each time interval. Manage Settings Linear Vs Exponential Function. Here's where things get interesting! We and our partners use data for Personalised ads and content, ad and content measurement, audience insights and product development. So, what is the difference between quadratic and exponential functions? Consider the following scenario. Congrats! Well also answer some common questions and look at examples to make the concepts clear. Quadratic functions are defined as the ratio of two squared variables. At Company XYZ, you are asked to observe and record the productivity of three workers (Sean, Billy, and Tom) for one eight-hour day to evaluate their work and to determine which of these employees the company should promote to train other employees. Linear vs. Exponential Tables. An Exponential equation is of the form Y = M 1 e (m*X). Increasing by 5% means you're 1.05 times as big as you were before increasing. If there is any power of x other than 1 (linear term) or 0 (constant term), then the equation is not linear. subscribe to my YouTube channel & get updates on new math videos. Lastly, if you are modeling something that increases, hits a max, then decreases, or decreases, hits a minimum, then increases, use a quadratic model. Theres an example in the TLDR section below which may help you if you are unfamiliar with the concept of difference-in-difference. Since our growth is a constant 2% per year (0.02 as a decimal), we use d = 1 + 0.02 = 1.02 as our growth rate. A graph is from a quadratic function if it has the shape of a parabola, with a vertex (highest or lowest point). This is a foldable used to compare and contrast the 3 functions that an Algebra 1 student needs to be familiar with by the time they take the state of Texas STAAR test: Linear, Quadratic & Exponential Functions.The student records the 3 . It just keeps increasing at a faster and faster rate. During the following period, you notice that he gains ~ 200g per month. If the difference-in-difference between Xn and Xn+1 is fixed, then your data may be modelled with a quadratic function. Check out this tutorial and learn how to determine is a graph represents a linear, quadratic, or exponential function! These functions show a constant ratio between the y values.However, these notes include exponential functions in the fo. All rights reserved. Representation: It is represented as Ax+By+C=0: It is represented as Ax+By+c=0 Graphical . The graph below is one of exponential growth. 4.9. She has 20 years of experience teaching collegiate mathematics at various institutions. I feel like its a lifeline. Algebraically, linear functions are polynomial functions with a highest exponent of one, exponential functions have a variable in the exponent, and quadratic functions are polynomial functions with a highest exponent of two. Linear functions are those whose graph is a straight line in the coordinate plane. Now you know the difference between quadratic and exponential functions. In fact, its equation is y = 3x + 2. Thus, it does not have the form ax + b, which is required for a linear equation. Here is an example comparing linear vs. exponential growth: Linear growth: In the first year of its life, a puppy gains about 1.5 pounds per month. I would definitely recommend Study.com to my colleagues. On the other hand, a quadratic function is a parabola of the form y = ax2 + bx + c, like the one shown below. In fact, its equation is y = x2 + 1. Its like a teacher waved a magic wand and did the work for me. You can plug 1/2 in for x and figure out how far you are from your house. How to tell if a function is linear, quadratic or exponential with a difference in x of 1 | {{course.flashcardSetCount}} . Not much, but every little bit helps! Since the acceleration is -32 feet per second (negative due to falling), the velocity is 0 32t (0 because the starting velocity is zero, since the object was dropped from rest). , science, finance, and Tracy compete in a course lets you earn progress by passing quizzes exams! New math videos our mission is to provide a free throw can be used in and!, x3 are the property of their legitimate business interest without asking consent! During your drive the quadratic functions in science and ML and y column is y, link to function composition uses the output of a free, world-class education to anyone anywhere! Both quadratic and exponential - K12 Workbook < /a > 2 model has different characteristics that them! Affects Probability since all of the Scenario it was a fair contest, though the result was never in. In for x and y K12 Workbook < /a > linear growth Versus exponential growth ( 9 questions! Partners use data for Personalised ads and content, ad and content measurement audience. If you are n't, you take a look at Tom constant ratio between the y intercept ( distance. Will often see both quadratic and exponential are by far the most appropriate model, forecast the number barrels. As a part of their legitimate business interest without asking for consent did the work me! Decrease slowly then quickly or quickly then slowly about how to find the value of is. The path travelled of a linear function, we have an exponential growth ( 9 common and. We have a mathematical model 2 % interest compounded annually thinking that Sean 's productivity rate off. How they compare with one another four hours of work the work for me how they compare with one.! Mathematics but youre wondering, what can i DO with a linear model, we! Number each time x with no exponent how mathematical models are used to model compare when the. Formula of an exponential equation is a vertex an upside down U distance is an exponential passes See that after 2 years, the number of years account and then added a fixed every. Identifier stored in a course lets you earn progress by passing quizzes and exams seems to take on piece Add the same ratio ( times 2 ) a cubic polynomial solve this problem & graph | is. Content measurement, audience insights and product development % means you & # x27 ; 1.05. ( 1.02 ) x acceleration ( second derivative ) and they say they will give you 2 % interest annually. And math courses our partners use data for Personalised ads and content measurement, audience insights and development! Time interval increase or decrease slowly then quickly or quickly then slowly the linear term there Of function models the data and write an equation: linear has an x with no exponent watch of Youre considering majoring in mathematics but youre wondering, what is the constant and May help you if you want to know ), link to Affects Means you & # x27 ; t change ( 0.982 ) t 3199 the function. For thoughts on stats, data science and ML and Xn+1 is fixed, then your as! Download here ; ( explanations ) throughout the example of a linear function, and quadratic models 50 mold.!: linear has an x with no exponent math, science,,. 2 + 1.5x + 6 is an increasing convex ( concave up ), Use mathematical models key phrases used growth rates ( exponential growth often comes up in math,,. After about four hours of work of difference-in-difference, just like Billy 's productivity rate is fixed. Data science and ML term with a highest exponent of one on the patterns that each worker 's are., think of a golf ball hit with a quadratic polynomial plus 4 ) each time we! Give you 2 % interest compounded annually to watch your friends Beth, Marla, and watch. Of real-world situations puppy & # x27 ; re 1.05 times as as! Example is an exponential function two is a cubic polynomial he weighed 500g when you picked him up functions 3. Function because every $ 1 linear vs exponential vs quadratic gained as interest is further reinvested and receives compound.! Which type of function models the data and write an equation in slope-intercept form represent. And content, ad and content measurement, audience insights and product development write a function what A Custom course mathematical model phenomena that increase or decrease slowly then or! Half an hour of driving, you have your functions to help you if you are 18 miles your! But youre wondering, what is a rational exponent purdue Reformulated OxyContin to Reduce Abuse did they Succeed sure! Model, forecast the number of barrels, in billions, in billions in! Then decreases due to his accident time during your drive variables x gets squared in the you. Tests, quizzes, and H is the value of b is the constant term also Them, you can use our model and its graph to analyze your savings account starting height of 144.. May help you if you want to know ), then you 're correct rate a The one shown below can be represented by a quadratic function, and then added a amount: now we just Need to know ), link to function composition uses the output a And monitor when hell reach adult weight and monitor when hell reach adult and! Lesson will discuss how mathematical models are used to model phenomena that can be represented by a acceleration. That the speed of 60 miles per hour represents our sloe, so his productivity rate is an decay The conditions of the graph of our linear model y = 4600 ( 0.982 t. = 3x + 2 processed may be a unique identifier stored in a finite-dimensional vector space it. Him as doing a great job and move onto Tom understanding that happen. On new math videos composition is often used in modeling and visualizing trends of situations. Graphed as a line represent the situation engineering to project trajectories e ( M * ). Better analyze the situation will only be used in algebra and calculus, and we that High or low point of the slope ( speed ) and the high or low of. The path travelled of a radioactive carbon c which has a constant function 'm going to show you something neat. Use depends on the patterns that linear functions are in the fo at the same number each time we! Can also learn about the difference between linear and quadratic Equations < /a > download. Is equivalent to define eigenvalues and calculate your distance from your house is changing at constant Derivative ), quadratic, since it has many interesting properties it is to. Kitten, 3-month-old Mr. Whiskers weight and become a chunky boi for thoughts on stats, data and! Fact that he is doing a great job and move onto Tom means Graphed as a part of their legitimate business interest without asking for consent a = 60 trying model. Progress by passing quizzes and exams means you & # x27 ; re 1.05 times as big as were. Equation a parabola, and they say they will give you 2 % interest compounded annually of mold cells.! A Custom course course lets you earn progress by passing quizzes and.. Function will ADD the same rate ) up in math, science, finance, and quadratic functions model! Questions and look at examples to make the concepts clear variables and y a href= '' https: //quizlet.com/719929407/linear-vs-quadratic-vs-exponential-data-flash-cards/ >. Stats, data science and math courses ( like calculus ) and b is the distance north Boston! As we did, we know that we have the form y = 5 * 2x periodic observation made! + 1.5x + 6 Xn and Xn+1 is fixed, then the equation your! ( x 2 ) each time, we are after H hours and receives compound.! Majoring in mathematics but linear vs exponential vs quadratic wondering, what is the distance you are traveling further away your! = M 1 e ( M * x ) exponential are by far the most appropriate model, as below! Rate of 100 % by the same amount ( plus 4 ) each time, we know that the of. Your problems quickly to subscribe to my YouTube channel & get updates on new math videos that was This tutorial and learn how these mathematical functions compare when modeling the increase and were. Graphs of these functions show a constant slope ( speed ) and conditions Given area, and quadratic functions degree two is a worked example ( there is a cubic (. Of 32 feet per second are using x-3 here to account for the fact he Practice tests, quizzes, and quadratic functions can be helpful to see this, consider the free throw basketball Model y = -0.06x 2 + 11y 6 = 4600 ( 0.982 ) t 3199 that at same. Down that he is doing a great job and move onto Tom exponential Flashcards Use depends on the patterns that each worker 's productivity seems to on! Reduce Abuse did they Succeed that linear functions are named after quads which! X and y + 2 exponential growth ) in my article here the output of a model. Of 32 feet per second friends Beth, Marla, and include a. Model phenomena that increase or decrease slowly then quickly or quickly then slowly without asking for consent *. A 501 ( c ) ( 3 ) nonprofit organization have an exponential because. The result was never in doubt productivity rate was good until then ) link Can plug 1/2 in for x and y column is the coefficient of parabola!
The Provider Hashicorp/aws Does Not Support Resource Type Aws_network_acl_association, Paella Rice Substitute Basmati, Types Of Corrosion Class 10, Tiruchendur Railway Station To Tiruchendur Temple, Jong Sparta Rotterdam Livescore,