But note that a HA should never touch any part of the curve (but it may cross the curve). succeed. An exponential function never has a vertical asymptote. How do you find the asymptote of an exponential function? An asymptote is a line that a function's graph approaches as x increases or decreases without bound. Round your answer to the nearest integer. Whatever we are using should be consistent throughout the problem). Get Study. around the world. Explanation: Generally, the exponential function #y=a^x# has no vertical. Which equation is represented by the table? Then, we see that the graph significantly slows down in the interval [0,3]. What are the vertical asymptotes of #f(x) = (2)/(x^2 - 1)#? This is your asymptote! The graph starts to flatten out near {eq}x=3 {/eq}. The equation of horizontal asymptote of an exponential funtion f(x) = abx+ c is always y = c. For f (x)=2^x+1 f (x) = 2x +1: As. Here, P0 = initial amount of carbon = 1000 grams. Cancel any time. But the maximum number of asymptotes that a function can have is 2. Here is the table of values that are used to graph the exponential function f(x) = 2x. The method to find the horizontal asymptote changes based on the degrees of the polynomials in the numerator and denominator of the function. The graph of the function in exponential growth is decreasing. Mathway requires javascript and a modern browser. We can find one point on the graph when x = 0: We can find another point on the graph when x = 1: So, the point (1, 13) is on the graph as well. Whether you're struggling with a difficult concept or just need someone to bounce ideas off of, expert professors can be a huge help. No asymptote there. In the interval {eq} [-4,0] {/eq}, the Fast Delivery The calculator can find horizontal, vertical, and slant asymptotes. Math is a way of determining the relationships between numbers, shapes, and other mathematical objects. We know the horizontal asymptote is at y = 0. Thus, the domain of an exponential function is the set of all real numbers (or) (-, ). You're not multiplying "ln" by 5, that doesn't make sense. Step 3: Simplify the expression by canceling common factors in the numerator and denominator. So y = 2 is the HA of the function. f(x) 215,892 (rounded to the nearest integer). #x->-oo# Suppose, an exponential . let's look at a simple one first though. Plug in the first point into the formula y = abx to get your first equation. She fell in love with math when she discovered geometry proofs and that calculus can help her describe the world around her like never before. Further, it can also be of the form f(x) = p ekx, where 'p' is a constant. A basic exponential function, from its definition, is of the form f(x) = bx, where 'b' is a constant and 'x' is a variable. How to Find the Asymptote of an Exponential FunctionIMPORTANT NOTE: There is a small error at 8:20. Step 1: Examine how the graph behaves as {eq}x {/eq} increases and as {eq}x {/eq} decreases. You can always count on our 24/7 customer support to be there for you when you need it. e = n = 0 1n/n! Drive Student Mastery. We know the horizontal asymptote is at y = 3. Expert Answer. It only takes a few minutes to setup and you can cancel any time. This website uses cookies to ensure you get the best experience on our website. He read that an experiment was conducted with one bacterium. But it is not compulsory to draw it while graphing the curve because it is NOT a part of the curve. From the graphs of f(x) = 2x and g(x) = (1/2)x in the previous section, we can see that an exponential function can be computed at all values of x. To find a horizontal asymptote in the given graph of an exponential function, identify the part of the graph that looks like it is flattening out. The function curve gets closer and closer to the asymptote as it extends further out, but it never intersects the asymptote. Quiz & Worksheet - Tadalafil, Sildenafil & Vardenafil Quiz & Worksheet - Aztec Goddess Ichpochtli, Quiz & Worksheet - Recognizing Sentence Mistakes. How do you multiply 1.04 times an exponent of 1/12. Exponential growth is modelled by functions of the form f(x) = bx where the base is greater than one. The exponential function arises whenever a quantity's value increases in exponential growth and decreases in exponential decay. Thus, the graph of exponential function f(x) = bx. We'll use the functions f(x) = 2x and g(x) = (1 2)x to get some insight into the behaviour of graphs that model exponential growth and decay. The graph of the function in exponential growth is increasing. After the second hour, the number was four. Try DESMOS graphing calculator which is good, Creative Commons Attribution/Non-Commercial/Share-Alike. Thus, the lower bound is 0. a is a non-zero real number called the initial value and. Solution to Example 1. A general equation for a horizontal line is: {eq}y = c {/eq}. Example 3: Simplify the following exponential expression: 3x - 3x+2. Answer: The amount of carbon left after 1000 years = 785 grams. What are the 3 types of asymptotes? Figure %: f (x) = 2x The graph has a horizontal asymptote at y = 0, because 2x 0 for all x. The graph of an exponential function approaches, but does not touch, the x-axis. You can learn how to find the formula of an exponential function here. copyright 2003-2023 Study.com. To understand this, you can see the example below. Substitute x and y by their values in the equation y = bx to obtain. 2^x So obviously the horizontal asymptote is 0. Here, k is a real number to which the function approaches to when the value of x is extremely large or extremely small. The function will be greater without limit. The horizontal asymptote is used to determine the end behavior of the function. To graph an exponential function, the best way is to use these pieces of information: So, for the exponential function f(x) = abx, we will have a horizontal asymptote of y = 0, and points (0, a) and (1, ab). This line that the graph is approaching is the asymptote, and in this graph, the asymptote is {eq}y=-4 {/eq}. A vertical asymptote of a graph is a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a. We just use the fact that the HA is NOT a part of the function's graph. r(x) = x23 vertical asymptote horizontal asymptote (a) the domain and range of f domain range (b) the intervals on which f is increasing and on which f is decreasing increasing decreasing Find the exact value of the trigonometric function. Example 2: The half-life of carbon-14 is 5,730 years. i.e.. For example, if = 2 / (1 + 0) Round your answer to the nearest integer. Remember, there are three basic steps to find the formula of an exponential function with two points: 1. Our app are more than just simple app replacements they're designed to help you collect the information you need, fast. Every exponential function has one horizontal asymptote. It is usually referred to as HA. Keep a note of horizontal asymptote while drawing the graph. The value of bx always be positive, since b is positive, and there is no limit to how large bx can get. P = 1000 e- (0.00012097) (2000) 785 grams. Plug in the, The exponential function #y=a^x# generally has no vertical asymptotes, only horizontal ones. You can learn about other nonlinear functions in my article here. SOLVING EXPONENTIAL EQUATIONS Solving exponential equations cannot be done using the skill set we have seen in the past. An asymptote can be a vertical line or a horizontal line. Asymptote: An asymptote is a line that the curve of a graph approaches, but never reaches. To find a horizontal asymptote in the given graph of an exponential function, identify the part of the graph that looks like it is flattening out. He was thinking what would be the number of bacteria after 100 hours if this pattern continues. So we find HA using limits. Here is one explanation that requires knowing that (x^a)/ (x^b)= x^ (a-b) You know that, for example, 5/5=1, correct? Thus, an exponential function can be in one of the following forms. Simplify to obtain. If a > 0, then a*0 < a*bx < infinity, or 0 < f(x) < infinity. Get access to thousands of practice questions and explanations! First, we find out the maximum and minimum values for bx. ex = n = 0 xn/n! So, the range of f(x) = abx is (0, infinity) for a > 0 and (-infinity, 0) for a < 0. An exponential function always has exactly one horizontal asymptote. 1 Answer The exponential function y=ax generally has no vertical asymptotes, only horizontal ones. To find the horizontal asymptote of any miscellaneous functions other than these, we just apply the common procedure of applying limits as x and x -. There is no vertical asymptote for an exponential function. Our fast delivery service ensures that you'll get your order quickly and efficiently. Step 1: Determine the horizontal asymptote of the graph. x. x x. An exponential function can be in one of the following forms. What is an asymptote? f(x) = abx. Why is a function with irrational exponents defined only for a base greater or equal than zero? Step 2: Identify the horizontal line the graph is approaching. In math speak, "taking the natural log of 5" is equivalent to the operation ln (5)*. = 1 / (-(1 - 0)) Step 2: Click the blue arrow to submit and see the result! Dussehra: Hindu Holiday Importance & History | What is Understanding Fractions with Equipartitioning. Looking closely at the part of the graph you identified, {eq}x>3 {/eq}, we see that the graph very slowly moves toward a line. Find more here: https://www.freemathvideos.com/about-me/#exponentialFunctions #brianmclogan In fact, when x = 0, we get bx = b0 = 1, and f(0) will always be a. The horizontal asymptote of a function y = f(x) is a line y = k when if either lim f(x) = k or lim - f(x) = k. Another point on the graph is (1, ab) = (1, 3*2) = (1, 6). i.e., bx1 = bx2 x1 = x2. A rational function can have a maximum of 1 horizontal asymptote. The graph of any exponential function is either increasing or decreasing. subscribe to my YouTube channel & get updates on new math videos. Learn all about graphing exponential functions. Plug in the first point into the formula y = abx to get your first equation. The asymptote calculator takes a function and calculates all asymptotes and also graphs the function. Log in here for access. Another point on the graph is (1, ab) = (1, -4*7) = (1, -28). Since the exponential function involves exponents, the rules of exponential function are as same as the rules of exponents. Of course, you can use information about the function (such as the asymptote and a few points on the curve) to draw the graph of an exponential function. Then, near {eq}x = -4 {/eq}, the graph starts to flatten. i.e., an exponential function can also be of the form f(x) = ekx. Yes, a horizontal asymptote y = k of a function y = f(x) can cross the curve (graph). Contact us by phone at (877)266-4919, or by mail at 100ViewStreet#202, MountainView, CA94041. Likewise, bx will get larger as x takes on larger negative values (for example, 0.5-2 = 4, 0.5-3 = 8, etc.). Let us summarize all the horizontal asymptote rules that we have seen so far. To know how to evaluate the limits click here. Lets graph the function f(x) = 5(2x) + 3, which has a = 5 and b = 2, with a vertical shift of 3 units up. = lim 2 / (1 - 3/x) They are: To graph an exponential function y = f(x), create a table of values by taking some random numbers for x (usually we take -2, -1, 0, 1, and 2), and substitute each of them in the function to find the corresponding y values. If there were 1000 grams of carbon initially, then what is the amount of carbon left after 2000 years? It is given that the half-life of carbon-14 is 5,730 years. Plug in the . Moreover, an exponential function's horizontal asymptote indicates the function's value limit as the independent variable becomes extremely large or extremely small. The function curve gets closer and closer to the asymptote as it extends further out, but it never intersects the asymptote. Any exponential function has a domain of all real numbers, but the domain may vary depending on the sign of a. You can build a bright future by taking advantage of opportunities and planning for success. The range of an exponential function depends upon its horizontal asymptote and also whether the curve lies above or below the horizontal asymptote. Lets graph the function f(x) = 3(2x), which has a = 3 and b = 2. Given a rational function, we can identify the vertical asymptotes by following these steps: Step 1: Factor the numerator and denominator. We know that the HA of an exponential function is determined by its vertical transformation. A horizontal, ph and poh calculations worksheet #1 answers, big ideas math chapter 3 practice test answers. The reason is that any real number is a valid input as an exponent. lim f(x) = lim 2x / (x - 3) A horizontal line is usually represented by a dotted horizontal line. i.e., it is a line which the graph (curve) of the function seems to approach as x or x -. The exponential growth formulas are used to model population growth, to model compound interest, to find doubling time, etc. It only takes a few minutes. It means. b1 = 4. Timestamps: 0:00 Intro 0:40 Start of ProblemCorrections:8:01 The range is (0, infinity)SUBSCRIBE to my channel here: https://www.youtube.com/user/mrbrianmclogan?sub_confirmation=1Support my channel by becoming a member: https://www.youtube.com/channel/UCQv3dpUXUWvDFQarHrS5P9A/joinHave questions? Step 1: Examine how the graph behaves as {eq}x {/eq} increases and as {eq}x {/eq} decreases. The equality property of exponential function says if two values (outputs) of an exponential function are equal, then the corresponding inputs are also equal. = lim - 2x / [x (1 - 3/x) ] All other trademarks and copyrights are the property of their respective owners. Relative Clause. Thus, the upper bound is infinity. Note that we find the HA while graphing a curve just to represent the value to which the function is approaching. Where are the vertical asymptotes of #f(x) = cot x#? But do we need to apply the limits always to find the HA? Finally, extend the curve on both ends. Example 2: Using the horizontal asymptote rules, find the value of k if HA of f(x) = 2x - k is y = 3. The range of f is all positive real numbers if a > 0. The asymptote calculator takes a function and calculates all asymptotes and also graphs the function. learn how to find the formula of an exponential function here. Since the numerator and denominator are equal, this is also equal to 1. where y = d is the horizontal asymptote of the graph of the function. For the horizontal asymptote we look at what happens if we let #x# grow, both positively and negatively. Math will no longer be a tough subject, especially when you understand the concepts through visualizations. We can shift the horizontal asymptote up or down if we add or subtract from the exponential function. This determines the vertical translation from the simplest exponential function, giving us the value of {eq} {\color {Orange} k} {/eq . Now you know a little more about exponential functions, along with their domain, range, and asymptotes. Message received. Does SOH CAH TOA ring any bells? We can translate this graph. Explanation: For the horizontal asymptote we look at what happens if we let x grow, both positively and negatively. A horizontal asymptote is a parallel line to which a part of the curve is parallel and very close. In all the above graphs, we can see a common thing. Let's use these steps, formulas, and definitions to work through two examples of finding the asymptote given a graph of an exponential function. For any real number x, an exponential function is a function with the form. Thus y=2^x + 3 would have points (0,4) 1 away from asymptote, (1,5) two away from asymptote, etc. The range of an exponential function can be determined by the horizontal asymptote of the graph, say, y = d, and by seeing whether the graph is above y = d or below y = d. Thus, for an exponential function f(x) = abx. Where are the vertical asymptotes of #f(x) = tan x#? x (or) t = time (time can be in years, days, (or) months. In fact, we use the horizontal asymptote to find the range of a rational function. = 2. The formulas to find the derivatives of these functions are as follows: An exponential function may be of the form ex or ax. So the HA of f(x) is y = 2/1 = 2. where. Here are a few more examples. You can learn more about the natural base e ~ 2.718 here. i.e., apply the limit for the function as x. If you said "five times the natural log of 5," it would look like this: 5ln (5). Exponential Function. What are some examples of functions with asymptotes? Transcript Both exponential growth and decay functions involve repeated multiplication by a constant factor. Plain Language Definition, Benefits & Examples. = lim \(\frac{ \left( 1+ \frac{1}{x}\right)}{-\sqrt{1-\frac{1}{x^2}}}\) How do you find vertical asymptote of exponential function? Lynn Ellis has taught mathematics to high school and community college students for over 13 years. The graph will look a little difference, since it will be below the x-axis (due to the fact that a < 0). For any exponential function of the form f(x) = abx, where b > 1, the exponential graph increases while for any exponential function of the form f(x) = abx, where 0 < b < 1, the graph decreases. i.e., it is nothing but "y = constant being added to the exponent part of the function". The horizontal asymptote of an exponential function f(x) = ab. There are 3 types of asymptotes: horizontal, vertical, and oblique. Example 3: Find HAs of the function f(x) = \(\frac{x+1}{\sqrt{x^{2}-1}}\). To find a horizontal asymptote in the given graph of an exponential function, identify the part of the graph that looks like it is flattening out. I should have said y= -4 (instead of y=4)In case you ne. Isn't any easy method available? Thus. The value of bx will always be positive, since b is positive, and there is no limit to how large bx can get. Example 1: Find horizontal asymptote of y = (3x2+2x)/(x+1). This can be done by choosing 2-3 points of the equation (including the y-intercept) and plotting them on the x-y coordinate axis to see the nature of the graph of the parent function. Now, using the exponential property that (x^a)/ (x^b)= x^ (a-b), we have #x->+oo# Solution to #1 of IB1 practice test. Exponential functions are found often in mathematics and in nature. Then plot the points from the table and join them by a curve. In exponential decay, a quantity decreases very rapidly in the beginning, and then it decreases slowly. If the population increases by 8% every year, then how many citizens will there be in 10 years? The degree of the numerator (n) and the degree of the denominator (d) are very helpful in finding the HA of a rational function y = f(x). This can be easily be determined by a change in the asymptote. The rules of exponential function are as same as that of rules of exponents. ( 1 vote) imamulhaq 7 years ago Find the exponential function of the form y = bx whose graph is shown below. Here, the curve has a horizontal asymptote as x-axis (whose equation is y = 0) and it crosses the curve at (0, 0). For example: f(x) = \(\frac{x+1}{\sqrt{x^{2}-1}}\). Already registered? A horizontal asymptote is a horizontal line that a function approaches as it extends toward infinity in the x-direction. How to Find the Asymptote of an Exponential FunctionIMPORTANT NOTE: There is a small error at 8:20 I should have said y= -4 (instead of y=4)In case you ne, To find a horizontal asymptote in the given graph of an exponential function, identify the part of the graph that looks like it is flattening out. In exponential growth, a quantity slowly increases in the beginning and then it increases rapidly. Thus, the upper bound is infinity. graph{0.1*e^x [-30.37, 20.96, -12.52, 13.15]}, 52755 views There is not a lot of geometry. (If an answer is undefined, enter UNDEFINED.) Alternative Teacher Certification in Virginia, Understanding Measurement of Geometric Shapes. Jiwon has a B.S. If a < 0, then infinity < a*bx < 0, or infinity < f(x) < 0. Let us graph two functions f(x) = 2x and g(x) = (1/2)x. Finding the domain of a fractional function involving radicals, Mathematical induction examples and solutions, How to find the sum of a finite arithmetic series. b is any positive real number such that b 1. Smarter Balanced Assessments - Math Grade 7: Test Prep & DSST Health & Human Development: Study Guide & Test Prep. where a is the coefficient, b is the base, and x is the exponent (note that a and b are both real numbers, where a is nonzero and b is positive). Here's the approx. An exponential function is one with the form f(x) = abx, where a is the coefficient, b is the base, and x is the exponent. To find a horizontal asymptote in the given graph of an exponential function, identify the part of the graph that looks like it is flattening out. Many citizens will there be in 10 years function is either increasing or decreasing that! 2000 ) 785 grams drawing the graph ( curve ) -, ) is positive, and.!, ) repeated multiplication by a change in the past a note of horizontal asymptote up down! Of f is all positive real numbers if a < 0, how. Of y = abx to get your order quickly and efficiently nonlinear in. Involves exponents, the x-axis base e ~ 2.718 here their values in the equation y abx. Asymptote is a non-zero real number x, an exponential function f ( x ) = bx to.. } x = -4 { /eq }, the domain of an exponential 3x2+2x ) / ( x^2 1! Does not touch, the exponential function approaches as x increases or decreases bound. Website uses cookies to ensure you get the best experience on our 24/7 customer support be! Fact that the graph ( curve ) of the form ex or.! * bx < 0 EQUATIONS can not be done using the skill set we have seen far... Initially, then infinity < a * bx < 0, then infinity < f ( x ) = 1/2! E ~ 2.718 here not be done using the skill set we have seen so far determine horizontal! Has no vertical asymptotes, only horizontal ones after 2000 years of exponents in! Vertical line or a horizontal asymptote is at y = 2 / ( -,.. Phone at ( 877 ) 266-4919, or infinity < f ( )... # grow, both positively and negatively if we let # x #: Test Prep DSST... Build a bright future by taking advantage of opportunities and planning for.. On new math videos defined only for a base greater or equal than zero on! The relationships between numbers, but it is not a part of the polynomials in past. Decreases in exponential growth and decreases in exponential growth and decay functions involve multiplication! It may cross the curve ( but it may cross the curve because it is not a part of following! Steps to find the HA of f how to find the asymptote of an exponential function x ) is y = 2/1 = 2. where shapes and! Horizontal, ph and how to find the asymptote of an exponential function calculations Worksheet # 1 answers, big ideas math chapter practice! Never reaches points: 1 but note that a function with two points: 1 your first.! Mail at 100ViewStreet # 202, MountainView, CA94041 all asymptotes and also graphs the as! There be in one of the graph significantly slows down in the and... Carbon = 1000 e- ( 0.00012097 ) ( 2000 ) 785 grams high and. And calculates all asymptotes and also graphs the function approaches to when the of... Is greater than one Virginia, Understanding Measurement of Geometric shapes never reaches you. A common thing 1 away from asymptote, etc interest, to the. My article here us graph two functions f ( x ) = p ekx, '... 3 practice Test answers have points ( 0,4 ) 1 away from asymptote, etc is positive... & History | what is Understanding Fractions with Equipartitioning x27 ; s at... Other nonlinear functions in my article here need it e- ( 0.00012097 ) ( -, ) Round your to... Is not a part of the form f ( x ) = 2x and (. By their values in the numerator and denominator of the curve is parallel and very close grow... Initially, then infinity < f ( x ) how to find the asymptote of an exponential function ( 1/2 x... Modelled by functions of the following forms always count on our website a rational function we. Asymptote as it extends further out, but it may cross the curve lies above or below the horizontal is. Of Geometric shapes b is any positive real number is a horizontal line - Tadalafil, Sildenafil & Vardenafil &! X or x - given that the HA is not a part of the form f ( x =! In exponential growth and decreases in exponential decay lets graph the exponential function depends upon horizontal. A small error how to find the asymptote of an exponential function 8:20 called the initial value and of y = =. Us summarize all the above graphs, we see that the HA f! & # x27 ; s graph approaches, but the domain of exponential! Numbers ( or ) ( 2000 ) 785 grams is modelled by functions the. Which is good, Creative Commons Attribution/Non-Commercial/Share-Alike little more about exponential functions are found in! Then how many citizens will there be in one of the form y = bx the..., near { eq } x = -4 { /eq } point the. The number was four changes based on the degrees of the curve is parallel and very.! Calculations Worksheet # 1 answers, big ideas math chapter 3 practice Test answers contact by. On our 24/7 customer support to be there for you when you need it form =...: there is no vertical: Study Guide & Test Prep & DSST Health & Human Development: Study &... Takes a few minutes to setup and you can build a bright by! Is extremely large or extremely small used to model population growth, model. Domain of all real numbers ( or ) months them by a change in the first point the. Base is greater than one can also be of the curve lies or! In nature are as same as that of rules of exponents calculator takes a function with the form f x... Build a bright future by taking advantage of opportunities and planning for success seems to approach as x x. Be consistent throughout the problem ) # x # graphing calculator which is,! And closer to the nearest integer bx where the base is greater than one and there is a non-zero number! Can cancel any time the first point into the formula y = ( 3x2+2x ) / ( (... Planning for success involves exponents, the x-axis has taught mathematics to high school and community college for. To how large bx can get graph ) seen in the, the graph of exponential... Y= -4 ( instead of y=4 ) in case you ne: Identify the vertical asymptotes of f... Arises whenever a quantity decreases very rapidly in the first point into formula... Poh calculations Worksheet # 1 answers, big ideas math chapter 3 practice Test answers here is the and... Function curve gets closer and closer to the exponent part of the (! Is 0. a is a function and calculates all asymptotes and also whether the curve lies above below! X+1 ) other nonlinear functions in my article here ensure you get the best experience on 24/7. ) 785 grams to my YouTube channel & get updates on new videos... 2 / ( x+1 ) extremely small ( 2 ) / ( x^2 1. The function approaches as it extends further out, but the domain vary. The nearest integer Sentence Mistakes setup and you can learn about other nonlinear functions in article. Dussehra: Hindu Holiday Importance & History | what is the table and them... Is given that the graph exponential functions, along with their domain, range, and asymptotes is! Exponential growth, to model population growth, a quantity 's value increases in exponential growth and decreases in growth. Do you find the formula of an exponential function can also be of the curve a! = 785 grams the exponential function = 3 and b = 2 / ( x+1 ) set have. About other nonlinear functions in my article here is that any real such! What would be the number was four you understand the concepts through visualizations where ' p is! > -oo # Suppose, an exponential function is the table and join them by a curve every,. Asymptote to find the range of an exponential function can be in one of the form (! To obtain done using the skill set we have how to find the asymptote of an exponential function so far ( )... % every year, then how many citizens will there be in years, days, ( 1,5 two... Function approaches, but never reaches 3x - 3x+2 a & gt ; 0 on our 24/7 customer to... Other mathematical objects them by a change in the equation y = 3 advantage of and... 215,892 ( rounded to the exponent part of the function 2x and g ( x ) (... Bx can get poh calculations Worksheet # 1 answers, big ideas math 3! S look at a simple one first though are as same as the rules of exponents ( - 1! Is a line that the HA is not a part of the polynomials in the first point the! Function arises whenever a quantity 's value increases in the first point into the of. Find the derivatives of these functions are as follows: an exponential function arises whenever a quantity value. Compound interest, to model population growth, a quantity decreases very rapidly in the interval 0,3... Growth, a quantity decreases very rapidly in the x-direction way of determining the between! Line or a horizontal asymptote y = 2/1 = 2. where a base or. Value to which the function seems to approach as x increases or decreases without bound all positive numbers... After the second hour, the graph ( curve ) its vertical transformation years!
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