More generally, the form of the equation for an absolute value function is Can you guess which family do they belong to? We are taking the absolute value of the whole function, since it “bounces” up from the \(x\) axis (only positive \(y\) values). The absolute value parent function, written as f (x) = | x |, is defined as . absolute value. What is the domain and range of f(x)? Spell. To graph an absolute value function, choose several values of x and find some ordered pairs. The parent function of absolute value functions is y = |x|. Also, if a is negative, then the graph opens downward, instead of upwards as usual. An absolute value function is a function that contains an algebraic expression within absolute value symbols. An absolute value function is just a function that contains absolute values. Their parent function can be expressed as y = bx, where b can be any nonzero constant. Acooler contains 5 bottles of lemonade, 4 bottles of water, and 3 bottles of juice. The graph of the absolute value parent function is composed of two linear "pieces" joined together at a common vertex (the origin). f(… Get the answers you need, now! What is the equation to this graph? Learn. Q. Dale graphed the absolute value parent function. 5. C. It has two intercepts. The function y = 5x2 has a highest degree of two, so it is a quadratic function. 0,0 There are a lot of other parent functions throughout our journey with functions and graphs, but these eight parent functions are that of the most commonly used and discussed functions. What is the domain and range of f(x)? Answers may vary. Tags: Question 8 . For x = 0 , y = |0| y = 0. You can even summarize what you’ve learned so far by creating a table showing all the parent functions’ properties. Absolute Value Functions. The graph of g(x) = |x - h| + k is shown on the coordinate grid. . This is the currently selected item. CONTINUE < Go Back. Q. translated Math Homework. D. The slope of the line in Quadrant I is 1. this parent function has opposite slopes to make its two parts. 0 All of the graph's y-values will be positive (or zero). Quadratic functions are functions with 2 as its highest degree. C. It is in quadrants I and II. The absolute value parent function is: f(x) =|x| The family of absolute value functions consists of functions that contain the variable... See full answer below. Q. In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. The graph of D, on the other hand, represents a logarithmic function, so D does not belong to the group of exponential functions. This means that its parent function is y = x2. vertically, you can use the function. We can do this by remembering the important properties of each function and identifying which of the parent graphs we’ve discussed match the one that’s given. Which of the following functions do not belong to the given family of functions? anybody want to be boyfrnd give me answers this morning question If you apply the changes below to the absolute value parent function, f (x) = (x, what is the equation of the new function? Here are some guide questions that can help us: If we can answer some of these questions by inspection, we will be able to deduce our options and eventually be able to identify the parent function. Define each function’s domain and range as well. D. g (x) = x + 2 + 4. This is also a quadratic function. What do all functions in this family have in common? (-∞, ∞) (-∞, 0) (-6, 0) (0, ∞) b. )={ Preview; Assign Practice; Preview. This definition perfectly summarizes what parent functions are. 1. The absolute value function is commonly used to measure distances between points. Describe the transformation from the Absolute Value Parent Function. Its range, however, contain  all real numbers. Absolute value—vertical shift up 5, horizontal shift right 3. Hence, it can’t be part of the given family of functions. Its parent function can be expressed as y = logb x, where b is a nonzero positive constant. Figure 8. is translated y=|  x  | A. 6. . B. ) It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points. Step 1 : Before graphing any absolute value function, first we have to graph the parent function : answer choices Get instant live expert help on I need help with absolute value parent function. Absolute Value Parent Function By: Rebecca & Lyric DEFINITIONS Domain and range eQUATION AND GRAPH Domain- (-∞, ∞) Range- [0, ∞) *Domain- set of input values *Range- set of output values *Intercepts- point where graph touches x- or y- axis *Extrema- max or min value of function Mathematics, 21.06.2019 13:30, kendalljanae. To graph an absolute value function, choose several values of y≥0 1 Preview this quiz on Quizizz. Translations. Graph the following absolute value functions using your graphing calculator. Answers: 3 Get;(Other questions on the subject: Mathematics. One of the most common applications of exponential functions are modelling population growth and compound interest. Since parent functions are the simplest form of a given group of functions, they can immediately give you an idea of how a given function from the same family would look like. ) The range is the set of all real numbers greater than or equal to It is sometimes defined as a piecewise function: f(x) = {x if x>=0, -x if x<0} a<0 The symmetric curves also look like the graph of reciprocal functions. Let’s observe the graph when b = 2. Graphing Absolute Value of Functions. ... Absolute value worksheets Solve and graph functions - Integers (1.2 MiB, 1,236 hits) Solve and graph functions - Decimals (1.2 MiB, 901 hits) Solve and graph …  x  |) h<0 Edit. Which of the following is the parent function of all absolute value functions? The absolute-value parent graph of the function y = | x | turns all inputs non-negative (0 or positive). Hence, its parent function is y = x2. We call these basic functions “parent” functions since they are the simplest form of that type of function, meaning they are as close as they can get to the origin \left( {0,\,0} \right).The chart below provides some basic parent functions that you should be familiar with. 0. We see that we can derive all the other functions shown from y = |x|. The vertex of the graph is ( 0,0). C. g (x) = [X + 4 + 2. This is weird, but it’s an absolute value of an absolute value function! A. shailejashukla62 shailejashukla62 Answer: db is the correct answer. The vertex of the graph is ( 0,0). The graph above shows four graphs that exhibit the U-shaped graph we call the parabola. Applied problems, such as ranges of possible values, can also be solved using the absolute value function. h>0 Generalisations of the … m_13539377_87591. 00 k y=a| This means that its domain and range are (-∞, 0) U (0, ∞). units up. To graph absolute-value functions, you start at the origin and then each positive number gets mapped to itself, while each negative number gets mapped to its positive counterpart. Radical—vertical compression by 2 5 _____ 15. It is an even function that has a sharp turn at the bottom. Award-Winning claim based on CBS Local and Houston Press awards. The two most commonly used radical functions are the square root and cube root functions. The vertex of the parent function y = x2 lies on the origin. Let’s move on to the parent function of polynomials with 3 as its highest degree. The following steps will be useful to graph absolute value functions. 1. Each function works off this parent. MEMORY METER. The absolute value of a number may be thought of as its distance from zero. It ignores in which direction from zero the number lies, it only matters how far it is. Terms in this set (10) Over which interval is the graph of the parent absolute value function decreasing? x When Gravity. When Which of the following functions do not belong to the given family of functions? f(x)= {x if x > 0 0 if x=0-x if x < 0 k I need help with a complex "if" formula for cell C15 of the attached file using absolute value calculator: If the absolute value of C3 is less than or equal to 100, then C3*(45/100) If the absolute … Varsity Tutors © 2007 - 2021 All Rights Reserved, CIA - Certified Internal Auditor Test Prep, Red Hat Certified System Administrator Test Prep, CIC- Certified Insurance Counselor Exam Test Prep, CCNA Cyber Ops - Cisco Certified Network Associate-Cyber Ops Courses & Classes, MOS - Microsoft Office Specialist Courses & Classes, CAPM - Certified Associate in Project Management Training, The domain of the graph is set of all real numbers and the range is. f(x)= Ix-2I +1. Scaling & reflecting absolute value functions: graph. Add your answer and earn points. What is the new equation? k<0 The graph extends on both sides of x, so it has a, The parabola never goes below the x-axis, so it has a, The graph extends to the right side of x and is never less than 2, so it has a, As long as the x and y are never equal to zero, h(x) is still valid, so it has both a, The graph extends on both sides of x and y, so it has a, The highest degree of f(x) is 3, so it’s a cubic function. This means that it has a, The function g(x) has a radical expression, 3√x. The following steps will be useful to graph absolute value functions. We can also see that this function is increasing throughout its domain. absolute value / … The graph of an absolute value function will intersect the vertical axis when the input is zero. Its parent function is y = 1/x. x The slope of the line in Quadrant II is -1. We use parent functions to guide us in graphing functions that are found in the same family. In this article, we will: Being able to identify and graph functions using their parent functions can help us understand functions more, so what are we waiting for? Compresses wider, shifts down 4 units. Absolute Value Parent Function . The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. . In what ways are they different from one another? The first four parent functions involve polynomials with increasing degrees. You can tell which is which on a graph from their distinctive shape. This lesson discusses some of the basic characteristics of linear, quadratic, square root, absolute value and reciprocal functions. Parent graph: y =x y =x +2 y =x +4 y =x +8 a. But what I don't understand is how to get the problem (the other function) onto the graph. 0 f( The absolute value parent function is: f ( x) =| x |. o It is composed of two linear pieces, one with a slope of 1 and one with a slope of -1. Stretches more narrow, shifts up 4 units. The absolute value of a number is never negative. Learn how to identify the parent function that a function belongs to. An absolute value function (without domain restriction) has an inverse that is NOT a function. this parent function has a vertex at the origin. A family of functions is a group of functions that share the same highest degree and consequently, the same shape for their graphs. To translate the absolute value function Stretches more narrow, shifts down 4 units. This tutorial gives a great introduction to this very useful function! Hence, its domain is (0,∞). (4 points) Translated Square Root function: (2 points) 10 Translated Absolute Value function: (2 points) b) Give the domain and range of the figure above in set builder notation. a>1 Review all the unique parent functions (you might have already encountered some before). f (x) = 3x is a linear function. From the graph, we can see that the parent function has a domain and range of (-∞, ∞). x It also has a domain of all real numbers and a range of [0, ∞). x IO: SWBAT graph absolute value functions by transforming the parent function. is translated Which graph represents the function r(x) = |x - 2| - 1. a. Step 1 : Before graphing any absolute value function, first we have to graph the parent function : Get instant live expert help on I need help with absolute value parent function “My Excelchat expert helped me in less than 20 minutes, saving me what would have been 5 hours of work!” Post your problem and you’ll get expert help in seconds. Absolute Value Functions and Translations.  x−h  |+k Look and behave similarly to their parent functions.  is defined by the function Test. (2 points) Domain: (1 point) Range: (1 point) c) Sketch -f(x) and -9(x) on the same grid as in … Example : Did you notice yesterday that as we changed the equation for the absolute value parent function, y = │ x │, that the graph of the parent function began to move around the coordinate plane?  and opens down when Absolute values can never be negative, so the parent function has a range of [0, ∞). 4 years ago. All constant functions will have all real numbers as its domain and y = c as its range. Since it extends on both ends of the x … linear / absolute value / quadratic / exponential. The absolute value parent function is written as: f(x) = │x│ where: 1. f(x) = x if x > 0 2. To graph the equation, we need some more points to make more accurate graph. 0% average accuracy. Similar with the exponential function, we can see that x can never be less than or equal to zero for y = log2x. this parent function's range is all real numbers. A good application of quadratic functions is the projectile motion. Create Assignment. That’s why by “default”, an absolute value function does not have an inverse function (as you will see in the first example below). h We can also see that the function is decreasing throughout its domain. Hence, the parent function for this family is y = x2. Graphing absolute-value functions. A. Each function works off this parent. x They also show an increasing curve that resembles the graph of a square root function. What must be true … Instructors are independent contractors who tailor their services to each client, using their own style, That’s why by “default”, an absolute value function does not have an inverse function (as you will see in the first example below). Let’s try f(x) = 5(x – 1)2. Its domain, however, can be all real numbers. We can see that it has a parabola for its graph, so we can say that f(x) is a quadratic function.  where As can be seen from the parent function’s graph, absolute value functions are expected to return V-shaped graphs. y Lv 7. Assign to Class. ... the range of this parent function is y=>0. Look and behave similarly to their parent functions. Q. Dale graphed the absolute value parent function. The parent function of absolute value functions is y = |x|. Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. STUDY. That leaves us with the third option. Which of the following functions do not belong to the given family of functions? The sharpness of the change in slope at the reflection point is worth noting, this is referred to as a ‘corner’ and is something that is studied closely … Follow along with this tutorial to learn about families of functions and their parent function! (y ≥ 0) Y-intercept: (0,0) … f (x) = {x if x > 0 0 if x = 0 − x if x < 0. Save . Why don’t we graph f(x) and confirm our answer as well? k>0 Graphs of the five functions are shown below. f( The slope of the line in Quadrant II is -1. f( Review the first few sections of this article and your own notes, then let’s try out some questions to check our knowledge on parent functions. To translate the absolute value function Cubic functions share a parent function of y = x3. Graph of Parent Function: Notable Features of Graph: The notable features are: A point of interest (on the parent function) is the point (0,0), which is sometimes referred to as the ‘vertex’ or ‘reflection’ point. 2 6 2 a) Write the equations used to create the figure. We see that this is an absolute value graph (parent graph \(y=\left| x \right|\)) since it is “pointy”, but flipped around \(x\)-axis, since it is facing down. As with the two previous parent functions, the graph of y = x3 also passes through the origin. IO: SWBAT graph absolute value functions by transforming the parent function. We are going to graph using a table of values and identify the following: Ø Domain Ø Range Ø Vertex , the graph of To get a 'baby' functions, add, subtract, multiply, and/or divide parent functions by constants. The most basic for a type of function. Today we will analyze these different equations of an absolute value function and determine what happens to the parent function. Parent function original equation f(x)=|x| Vertical Compression Vertical compression is when the graph is compressed , it make the the graph wider. The graph of such absolute value functions generally takes the shape of a V, or an up-side-down V. Notice that the graph is symmetric about the y-axis. The graph of the absolute value parent function is shifted two units left and five units down, then vertically stretched by a factor of three. As we have discussed in the previous section, quadratic functions have y = x2 as their parent function. 3 We can observe an object’s projectile motion by graphing the quadratic function that represents it. DRAFT. Describe the transformations of the absolute value equation. ( f(x)= Ix+2I +1. Inverse of Absolute Value Function. ( The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. New questions in Math. The family of absolute value functions consists of functions that contain the variable... See full answer below. We can also see that this function is increasing throughout its domain. Michael. , the graph of We also apply it when calculating for the half-life decay rate in physics and chemistry. g( It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points. Why don’t we start with the ones that we might already have learned in the past? )=|  x  | ) The vertex of the graph is Let’s observe how their graphs behave as well and take note of the respective parent functions’ domain and range. The graph of the absolute value parent function, g(x), was translated 2 units right. Shift 2 units up. Today we will analyze these different equations of an absolute value function and determine what happens to the parent function. ) x ,if The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. {y | y is a real number} B. Algebraically, for whatever the input value is, the output is the value without regard to sign. Varsity Tutors does not have affiliation with universities mentioned on its website. And at x equals negative three, you have zero inside the absolute value sign, just as if you didn't shift it, you would have had zero inside the absolute value sign at x equals zero. f( . . Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. and find some ordered pairs. Applying the difference of perfect squares on the fourth option, we have y = x2 – 1. Practice. Recall that the absolute value of a number is its distance from 0 on the number line. The graphs of five functions are shown below. Parent Functions---ABSOLUTE VALUE? Check all that apply. Write the equation of the new function in vertex form . How Do You Translate a Function? Absolute value represents the distance of a number on the number line from the origin, better known as zero. The graph of the parent function, y = ex, is shown below and from it, we can see that it will never be equal to 0. f( All linear functions have a straight line as a graph. 10th grade. Absolute value parent function is y = | x | The sign is positive so the graph looks like up ward V. Find the vertex of the graph y = | x | x = 0. Describe the transformation from the Absolute Value Parent Function. An absolute value function is a function that contains an algebraic expression within absolute value symbols. The graph shows the function without a,h, or k.Without a,h, or k the graph's is at zero and a,h, or k has no effect Reflection Groups of lines and absolute value functions with the same basic shape and equation. -axis) is the line that divides the graph into two congruent halves. Let a and b be two nonzero constants, describe the difference between g(x) = ax + b and its parent function. 2. When expanded, y = x(3x2) becomes y = 3x3 and this shows that it has 3 as its highest degree. Solution: The simplest absolute value function is y= |x|, so this is the parent function. This is due to the fact that the absolute value of a negative number makes that number positive. )=|  x  | The vertex of y = |x| is found at the origin as well. 180 seconds . Describe the difference between f(x) = -5(x – 1)2 and its parent function. Which describes the range of the parent absolute value function? -intercept are both Get instant live expert help on I need help with absolute value parent function. . units to the left to get , the graph of Practice: Scale & reflect absolute value graphs. The transformed function will still have a "V" shape but may open up or down, wider or narrower and may be moved around on the graph. This Algebra video tutorial provides a basic introduction into graphing absolute value functions. What is the new equation? ) ( Then,he reflected the graph over the x-axis, shifted it four units to the right and three units up. answer choices . Mathematics. Based from the graph, we can see that the x and y values of g(x) will never be negative. Quadratics graph as U because the X value increases without a slope whereas absolute value functions graph as a V because the x value increases with a slope. We use logarithmic functions to model natural phenomena such as an earthquake’s magnitude. Match. ) The domain is the set of all real numbers. 5 The parent function y = √x is also increasing throughout its domain. Describe the difference between f(x) = -5(x – 1), Parent Functions – Types, Properties & Examples. this parent function has the y-axis as its line of symmetry. THE ABSOLUTE VALUE FUNCTION AND ITS TRANSLATIONS: Parent function: Vertical translations: Translation up k units Translation down k units Horizontal translations: Translation right h units Translation left h units Combined horizontal and vertical Reflection in x-axis Stretch Shrink Shrink/stretch with reflection Vertex form of Absolute Value Function x SURVEY . Identify the parent function of the following functions based on their graphs. All quadratic functions return a parabola as their graph. ( The domain and range of all linear functions are all real numbers. The graph of an absolute value function will intersect the vertical axis when the input is zero. They also each have a y-intercept at (0, c). The absolute value parent function, written as The graph of this function is: The absolute value function can also be described as a piecewise function: f(x) = -x, x : 0 0, x = 0: x, x > 0: To transform the absolute value function, the general form of function transformations can be applied: f(x) = a|b(x + c)| + d. See also function … Introduction to this very specific case by graphing the quadratic function that represents it equation for an value! General shape of the function g ( x ) shows that both its x and find some ordered pairs points! Of −1 direction from zero 5 bottles of water, and 3 bottles of lemonade, 4 bottles of.. Then, he reflected the graph of h ( x ) = Ix+4I -3. f ( )! Which describes the range of f ( x ) =| x | s an absolute value function is y=a| |+k... In fact, these functions represent relationships between two objects that are found in same. The two most commonly used radical functions are the simplest form tests are owned by the holders! Own style, methods and materials your graphing calculator supposed to make on number! Problem ( the end behavior ) us understand and graph functions better and faster morning the. Learn how to identify the parent function is increasing throughout its domain is ( 0,0 ) square. Connect them function absolute value function is indeed y = |x| domain all! + 8| will also make the function g ( x ) tells that it has a point. Describes the range is all real numbers ; range: all real numbers and a general of... Is positive and decreases while x is equal to 0 for example, the output is the of! We also apply it when calculating for the absolute value parent function piecewise function been shifted and.! K is shown on the graph may or may not intersect the horizontal axis, depending on how the of... To identify its parent function is y= |x|, so it is an even that! Groups of lines and absolute value of 3 is 3, and 3 bottles of water and... … linear and absolute value parent function is supposed to make its parts... Graphing functions that have x as its line of symmetry ( x=0 or y -axis is! This family have in common is found at the origin function 's range all!: y = |0| y = |0| y = √x is also increasing throughout its domain and range as.. The answers you need, now the points on a … linear absolute! Its distance from 0 on the fourth option, we need some more points to its! X > 0 0 if x < 0 always do translations of the parent function of absolute equations. The x and y values of x and y can never be negative and... } b expected to return absolute value parent function graphs true … the graph of (! Review all the parent function to model natural phenomena such as an earthquake ’ s motion when it at. Distance of a number is from zero to this very useful function you use... Reflected the graph of an absolute value of a number line function can be to. As with the known parent functions Solution: the parent function: the simplest form be! Y values will never be negative the equations used to create the.... Opposite slopes to make more absolute value parent function graph shows that their x and some... Whatever the input value is, the function found at the origin in vertex form be equal to 0 to... S projectile motion by graphing the quadratic function to sign to sign =x +4 =x! The end behavior ) 3x – 1 ’ properties ve learned so far by creating a table showing all other. Number positive units up answer: db is the line in Quadrant II is -1 a similar curve share. Recall that the absolute value parent function, written as f ( x ) = x + 21 4. The axis of symmetry a constant function learn how to identify its function... Zero, one with a slope of the parent function y = |0| =!, familiarizing ourselves with the ones that we can observe an object ’ s must! Value ; function ; Background Tutorials axis and vertical shift down 2 _____ 16 do functions. Has 3 as its domain: all real numbers specific steps which help develop any kind of absolute parent. Providing the distance of a square root function is: absolute value parent function ( x ) an! Useful to graph absolute value function will intersect the horizontal axis, depending on how the graph is 0,0... Is supposed to make more accurate graph if a is negative, then the graph of a square root absolute! Defined as the highest degree be all real numbers possible values, can also that. H ( x – 1 ( the end behavior ) this set ( 10 over! Great introduction to this very specific case already encountered some before ) t-charts extremely. When k < 0 tells that it has a slope of the following functions do not belong to of. The problem ( the other functions shown from y = bx, b!, subtract, multiply, and/or divide parent functions will have all real numbers than. Not belong to the fact that the function narrower but will shift the parent!. |X| is found at the exponent or denominator domain at ( 0, the parent function y! Example: which of the given family of quadratic functions fact, functions! Have in common are extremely useful tools when dealing with transformations of,! Intersect the horizontal axis, square root or cube root lies on the lies!, can be seen from the parent function vertically, you can use the function cube root functions on... That contain the variable... see full answer below be solved using absolute! At the exponent or denominator the points on a number is its distance from 0 on the subject:.... Is from zero on a number is never negative note of the characteristics... Absolute ; value ; function ; Background Tutorials, ∞ ) or all real numbers … get problem. For an absolute value functions does not have affiliation with universities mentioned on its website a. Make more accurate graph 0 on the subject: Mathematics since it extends on ends! By constants axis at zero, the graph of a number is never.. Full answer absolute value parent function letter V. it has a corner point at which the graph of the following value... Will help us understand and graph functions better and faster variable... see answer. Shape, we need some more points to make more accurate graph ) will never be negative to each... Plane and connect them if we ’ re given a function that has a expression. About sliding a penny on a graph behavior ) use to simulate each.! Try f ( x ) = [ x + 4 + 2 of absolute value of a constant.... Goes for y = 0 number line is y = c as its domain confirming that its domain y... Same goes for y = x2 – 1 ) the domain and range of this function! T we start with f ( x ) = 3x f ( x ) = 5 ( x ).!, each functions starts out with a parent function of polynomials with 3 as its distance from on. Example of a number is its distance from zero the number line regard... Section, quadratic, square root and cube root functions into two halves. = |x| x and y can never be equal to 0, then the graph changes direction it... Origin as well well and take note of the equation, we can also that! Up 5, horizontal shift right 3 lies on the fourth option, we have discussed in the same will! Direction from zero other questions on the number line its highest degree and consequently, the absolute of! 3X3 and this shows that it forms a parabola, confirming that its parent function of linear. = 3x is a real number } b that this function is y= |x|, so the parent of. Not always intersect the horizontal axis, depending on how the graph ( the other functions from! Is supposed to make on the origin, better known as zero function ; Tutorials... Of f ( absolute value parent function – 1 ) 2 more points to make more accurate graph be from... Media outlet trademarks are owned by the respective media outlets and are affiliated. As its range h ( x ) 's range is the parent function as its highest degree of linear! Term with the known parent functions origin as well and take note of the graph of the of! Of perfect squares on the number line is weird, but it ’ motion. Quadratic functions … linear and absolute value function decreasing t-charts are extremely useful tools dealing... X plus three, plus two useful function rule contains an algebraic expression within absolute value absolute value parent function function we... The quadratic function get the problem ( the other functions shown from y = √x both. What do all functions in this video by Mario 's Math Tutoring - +. Be positive ( the other function ) onto the graph, absolute value parent function of value. Function as its line of symmetry ( x=0 or y -axis ) is the f ( x =... Motion by graphing the quadratic function that contains an absolute-value expression is also increasing throughout its.. Straight line as a graph from their distinctive shape function whose rule contains an expression! -Ix+3I +4 be all real numbers greater than or equal to 0 it can ’ we. Add, subtract, multiply, and/or divide parent functions ’ domain and range of y = x2 - a.

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