Fun fact exponential growth. Find the population of the village in 2024.

Fun fact exponential growth. In order to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth. Matthew Island in the Bering Sea, illustrate how Exponential decay refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time. Thus, knowledge growth is likened to compound interest-the increase at any time is a fixed percentage of the current amount. Here is the prompt I have used: “100 hyper-specific, surprising, mind-blowing and niche things I absolutely need to know about exponential changes. Get ready to explore the possibilities that arise when growth becomes exponential. But after a period of very fast population Exponential growth is the increase in number or size at a constantly growing rate. Oct 20, 2019 · This is my very first exponential function activity that I ever created. Find the domain and the range of an exponential function using its graph and write them in interval notation. Apr 5, 2016 · How do we prepare for a future tracking to exponential trends, if we aren’t accustomed to thinking this way? Let’s start with the basics of exponential growth. We’ll take a look in this section at one of them: exponential growth. ". Whether you're curious about personal growth, historical events, or scientific discoveries, these facts will offer valuable insights. Solution: The time will be 4 years. 2. This exponential growth is evident in the fact that FIVB is the world’s largest international sporting federation. On the flip side, exponential decay is crucial in calculating the And, the beauty of e is that not only is it used to represent continuous growth, but it can also represent growth measured periodically across time (such as the growth in Example 1). 3 Exponential Functions Learning Outcomes Evaluate exponential functions. Use the fact that 10 years after 1982 the population increased by 10 million to find k to three decimal places. One The exponential growth calculator calculates the final value of some quantity, given its initial value, rate of growth, and elapsed time. For most of human history, the global population was a tiny fraction of what it is today. This exponential growth has had a profound impact on our planet and has led to numerous challenges and opportunities. (See this video for an illustration of the difference between additive and multiplicative growth. For example, when it grows at 3 times its size, but when it grows at 30% of its size. It can also be an idea becoming more complex, or a group of things getting larger. Increase is slow when numbers are low but rises sharply as numbers increase. The number \ (r\) is called the growth rate or decay rate of the function, and represents the percent change of the function as a decimal. Not all cases of growth at an always increasing rate are instances of exponential growth. Exponential growth With exponential growth, on the other hand, each step multiplies the total by a fixed amount. Apr 24, 2022 · One type of mathematical calculation that amazes me is something called exponential growth. Exponential growth is a pattern where a quantity increases at a rate proportional to its current value. Example: y = e x is a simple exponential function. It leads to rapid and dramatic increases over time, often seen in populations, investments, and technological advancements. We will be taking a look at some of the basic properties and graphs of exponential functions. In fact, it is the graph of the exponential function y = 2 x The general form of an exponential function is y = ab x. This article explores the definition of exponential growth, provides real-world examples, explains the formula to calculate it, and discusses its applications. [1][2][3] Dec 6, 2024 · In fact, it has more than doubled in the past 50 years alone. Imagine growth in a population of bacteria. . Another idea is to let them each pick a specific application of exponential growth that's interesting to them, have them research the topic, and have them write a short essay, make a PowerPoint presentation, or something similar explaining how exponential functions are useful in that specific context. From population growth and … Exponential functions exhibit exponential growth or decay, meaning that their values increase or decrease at an accelerating rate. Then, solve the function and get the answer! Jul 23, 2025 · Mathematics: Simplifying expressions and solving exponential and logarithmic equations. This module describes the history of exponential equations and shows how they are graphed. Unlike linear growth, which advances at a steady, predictable rate, exponential growth accelerates over time, doubling at regular intervals. An exponential model can be found when two data points from the model are known. But as it gets larger, it grows much faster. Define your variables and relationships. We can write the full value of e as 2. 2 5x), the exponential function will increase even more quickly. The situation is represented by the exponential growth formula P (t) = 1200 (1. Learn Exponential function facts for kidsThe natural logarithm is the opposite of an exponential function. Sketch a graph of an exponential function. May 17, 2013 · This entry was posted in Algebra, Exponents, Math in the Real World and tagged compound interest, earthquake scale, exponent powers, exponential decay, exponential decrease, exponential graphs, exponential growth, exponents, exponents compound interest, exponents in the real world, exponents music video, exponents rap, half life, how people use exponents, index numbers, index numbers in the How Do You Solve a Word Problem with Exponential Growth? If something increases at a constant rate, you may have exponential growth on your hands. Bacteria can multiply rapidly, showing exponential growth. Growth happens all around us, in many different ways. Sample problems, including a look at the growth rate of the reindeer population on St. We'll break down how small, consistent increases can lead to This lesson explores the mathematical concepts of exponential growth and decay, emphasizing their real-world applications, particularly in finance and biology. 6 is an approximation to 10 1. So, exponentiation is repeated multiplication in the same way that linear growth is repeated addition. Understand the exponential formula along with examples and FAQs. Jan 22, 2024 · The growth you get from addition is a lot slower, which is why it's easy to underestimate the speed of exponential growth. If the argument is further scaled by a positive number greater than 1 (eg. Find the equation of the asymptote of an exponential function. How many flies will be there after days? The inverse function is the natural logarithm ln (x); because of this, some old texts [3] refer to the exponential function as the antilogarithm. The scenario in the India population example is diferent because we have a percent The general exponential function, where the base is not necessarily , is among the most useful of mathematical functions. Below is a graph of f (x) = 2 -x. The is called the rate of growth. In exponential growth, a population’s per capita (per individual) growth rate stays the same regardless of the population size, making it grow faster and faster until it becomes large and the resources get limited. 7182818284590452353602874713527 . May 21, 2018 · *exponential growth* A form of population growth [1] in which the rate of growth is related to the number of individuals present. Fun facts about exponential functions: We don't allow \ (a = 1\) because \ (1^x\) is always 1, giving a super boring constant function \ (f (x) = 1\). Whether you’re an investor looking to harness its power or a student trying to understand this fundamental concept, this article will guide you through it. Part one: Two ways to understand exponential growth Here’s how I learnt exponential growth in maths class: when the speed of growth is proportional to the size of the population, that’s exponential growth. While linear regression is a commonly and widely used tool in modeling and data analysis, some data sets are better modeled by non-linear equations. 10. We will construct two functions. Exponential decay refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time. 3: Exponential Growth Exponential growth is a transformative force, reshaping industries, societies, and even the trajectory of humanity itself. Fun facts about exponential functions: The domain of an exponential function is all real numbers. The following formula is used to illustrate continuous growth and decay. We don't allow negative bases because raising negative numbers to even (and some Mar 23, 2025 · Understanding exponential growth is essential for modeling investments, populations, and other phenomena that grow at an accelerating rate over time. Part 1: Use some of the information to find the decay rate of radium. Exponential growth and exponential decay are two of the most common applications of exponential functions. Part 2: Answer the question using the rest of the given information. One of the best examples of exponential growth in real life can be seen by looking at the multiplication of bacteria in a culture. The exponential growth model is also known as the Malthusian growth model. The mathematical representation of exponential growth allows Explore real-life examples of exponential growth, from technology and population trends to finance and nature, revealing their profound impacts on our world. Oct 31, 2024 · These 125 cool random fun facts will blow your mind. The same formula, x (t) = ae kt can be used in the same manner as the example above. That's a bit like exponential growth! This type of growth happens when the speed at which something changes is directly related to how much of it there already is. Example: If a population of rabbits doubles every month, we would have 2, then 4, then 8, 16, 32, 64, 128, 256, etc! Exponential growth is the inverse of logarithmic growth. Exponential Growth – Examples and Practice Problems Exponential functions can be used to model population growth scenarios or other situations that follow patterns with growth at fixed rates. . Sometimes the term exponential function is used more generally for functions of the form cbx, where the base b is any positive real number, not necessarily e. Over the last few centuries, the human population has gone through an extraordinary change. Table of Contents: Definition Oct 20, 2024 · Discover 30 fascinating facts about growth, from personal development to economic expansion, and unlock the secrets to thriving in various aspects of life. From tiny cells to entire populations, everything can Aug 5, 2025 · There are a couple of other interesting things to note about exponential models: The value is the value at the start of the exponential growth (or decay). Exponential Function Reference This is the general Exponential Function (see below for e x): f (x) = a x a is any value greater than 0 Properties depend on value of "a" When a=1, the graph is a horizontal line at y=1 Apart from that there are two cases to look at: World Population - Live Update In the following interactive, you can choose different years and see the population and growth rates for that year, and see how the population was growing (or will grow) on a typical day in that year. Here is a problem to try. A function is evaluated by solving at a specific input value. It's about an increase in something. In the exponential decay of g(x) g (x), the function shrinks in half every time you add one to its input x x. Mar 4, 2025 · In this blog post, we'll dive into 37 fascinating facts about exponential growth, shedding light on its impact in various fields like biology, technology, and finance. Understanding how to solve exponential functions and represent them through tables, graphs, and equations can help us make sense of exponential growth or decay in various scenarios. ) An important feature of exponential growth is that it speeds up as time goes on. 9 likes, 6 comments - triviaquest_channel on April 11, 2024: "Today's Interesting Fact: If you could fold a piece of paper 42 times, it would reach the moon due to exponential growth in its thickness. In this tutorial, learn how to turn a word problem into an exponential growth function. Euler's number, e plays a key role in calculating compound interest continuously over time. Jul 23, 2025 · An interesting fact about Euler's number e is that it appears in a famous equation called Euler's identity: eiπ + 1 = 0. This kind of single cell life propagates via binary Nov 1, 2024 · Discover 37 fascinating facts about personal growth that can inspire and guide you on your journey to self-improvement and success. Knowing … Identifying exponential Functions When exploring linear growth, we observed a constant rate of change—a constant number by which the output increased for each unit increase in input. 30 Exponential growth in bacteria. An exponential function represents the relationship between an input and output, where we use repeated multiplication on an initial value to get the output for any given input. May 2, 2014 · Today I'm sharing a creative way to use candy (in this case Skittles) for a modeling exponential growth and decay activity in Algebra.   Having already worked with each of these functions gives us an advantage. Exponential growth happens when an initial population increases by the same … While exponential functions exhibited fast growth (or decay), logarithmic functions exhibit slow growth (or decay) as \ (x \rightarrow \infty\). This scenario is called exponential growth. Characteristics of Graphs of Exponential Functions Learning Outcomes Determine whether an exponential function and its associated graph represents growth or decay. Exponential Functions The exponential function with base \ (a\) is defined by \ ( f (x) = a^x\), for all real numbers \ ( x\), where \ ( a > 0\) and \ (a \neq 1\). In 1800, there were one billion people. 71828. Exponential functions are often used to model things in the real world, such as populations, radioactive materials, and compound interest. Many natural patterns, like snowflakes and coastlines, can be described using exponents. implies the endless nature of the value. Such as always doubling. Discover what is: Exponential growth, its characteristics, applications, and real-world examples in data science. I'm not a physicist/geologist, but I assume that 31. 121 random fun facts that will blow your mind Our collection of the best interesting trivia covers animals, biology, geography, space and much more The annual growth rate is 1. Where . Explore the fascinating world of exponential growth through a classic story of a clever mathematician and a king's chessboard. This means that the graph rapidly decreases towards 0 as x increases. See exponential growth for this usage. Area The area up to any x-value is also equal to ex : An Interesting Property Just for fun, try "Cut Up Then Multiply" Let us say that we cut a number into equal parts and then multiply those parts together. Exponential equations are indispensable in science since they can be used to determine growth rate, decay rate, time passed, or the amount of something at a given time. They all pass through the point \ ( (1,0)\), which makes sense because \ ( \log_a ( 1) = 0 \) translates to the fact \ ( a^0 = 1\) for any \ (a\). Jul 30, 2024 · Forbes Home’s solar energy facts that may surprise you details more interesting facts such as how solar energy is cheaper than coal, diesel, nuclear, natural gas and fossil fuel and is four Mar 25, 2025 · Discover 32 fascinating facts about exponential sums, exploring their mathematical significance, applications, and intriguing properties. Graph exponential functions by creating a table of values. Exponential growth and decay show up in a host of natural applications. There are so many interesting exponential function situations! It’s been tried, tested and tweaked. Definition: Exponential Function An exponential function is a function that can be written \ (f (x) = a (1+r)^x\) for some numbers \ (a\) and \ (r\). Dec 8, 2024 · Section 1. Jan 19, 2023 · An exponential function is a mathematical function used to calculate the exponential growth or decay of a given set of data. Systems that exhibit exponential growth follow a model of the form [latex]y= {y}_ {0} {e}^ {kt}. For values of the base between 0 and 1, such as f (x) = 0. How fast is technology growing? Discover 58 key stats on AI, IoT, mobile, and data that reveal the rapid pace of tech transforming our world. In terms of energy release, an increase by 1 on the Richter scale corresponds to an increase in energy by a factor of 31. 7. - YouTube We will concentrate on two basic types of models in this section: exponential growth and exponential decay. Exponential functions can grow or decay very quickly. For example, if you have a small amount, it grows slowly. Exponential Growth and Decay Exponential growth can be amazing! The idea: something always grows in relation to its current value, such as always doubling. This is due to the fact that each increment in the exponent leads to a multiplication by the base, resulting in a rapid change in the function's value. Get ready to see change in a whole new light and learn how to embrace it with confidence! Sep 23, 2020 · We will see in this article that exponential growth doesn’t have a speed: it is the way the speed keeps changing that is important. For example the function grows at an ever increasing rate, but is much slower than growing exponentially. Today there are more than 8 billion of us. For example, in the equation f (x) = 3x + 4, the slope tells us the output increases by 3 each time the input increases by 1. com Exponential growth is the inverse of logarithmic growth. Jan 20, 2020 · In fact, we can use the Exponential Growth and Decay Formula to find snow depth levels, the magnitude of a star, how temperature affects a body, or how a fast-food chain expands its business as Khan Academy nicely shows. These functions include the functions of the form ⁠ ⁠, which is exponentiation with a fixed base ⁠ ⁠. Learn Growth facts for kidsGrowth means getting bigger or changing over time. There are formulas that can be used to find solutions to most problems related to exponential growth. You’ll discover Home Depot’s origins, values, operational metrics, global footprint, future outlook and more. Simply put, exponential growth is when a volume or quantity increases over time by a percentage Feb 27, 2025 · Discover 34 fascinating facts about exponential growth and decay, and understand their impact on science, technology, and everyday life. An exponential function is a mathematical function, which is used in many real-world situations. The Richter scale is exponential, but an increase in 1 only means that the amplitude measured by a Wood-Anderson seismometer has increased by a factor of 10. 4%. It’s not the first lesson I teach when I’m starting this content, but it’s my favorite. 5 Must Know Facts For Your Next Test In exponential growth, even small percentage increases can lead to large absolute increases over time due to the compounding effect. In mathematics, the exponential function is a function that grows quicker and quicker. Part 1: Find the decay rate of radium. Exponential functions are used to model relationships with exponential growth or decay. We will also discuss what many people consider to be the exponential function, f (x) = e^x. In other words, [latex] {y}^ {\prime }=ky. Finance: Calculating compound interest, where exponential growth describes accumulated interest over time. Physics and Chemistry: Modeling exponential growth and decay, calculating sound intensity in decibels, and describing radioactive decay. In this section, we will learn techniques for solving exponential functions. Oct 23, 2021 · Posts about exponential growth written by Anthony Ambrosius Aurelius Apr 2, 2025 · 1. Simply put, exponential growth is when a volume or quantity increases over time by a percentage Mar 4, 2025 · In this blog post, we'll dive into 37 fascinating facts about exponential growth, shedding light on its impact in various fields like biology, technology, and finance. This is why \ (e\) appears so often in modeling the exponential growth or decay of everything from bacteria to radioactivity. The figure below is an example of exponential growth. For example, understanding exponential growth can help you appreciate how compound interest works in a savings account, allowing your money to grow over time. This rapid and efficient method of reproduction allows for exponential growth of bacterial populations. In this section, we will explore using exponential functions be used to describe a relationship between two quantities where the rate of change is not constant and a linear function would not suffice. When is positive, the model is asymptotic to for large negative values of . For example, exponential functions can be used to calculate changes in population, loan interest charges, bacterial growth, radioactive decay or the spread of disease. 5 and there's some square-cube Exponential decay Exponential decay is more or less the same as exponential growth, except that the exponential function decreases over time rather than increases. Explore the concept of exponential growth, its real-life examples, and implications across technology, population dynamics, and sustainable strategies for the future. It is mainly used to find the exponential decay or exponential growth or to compute investments, model populations and so on. Exponential growth and decay: Exponential functions of the form f (x) = a^x and f (x) = e^kx are widely used to model growth and decay in various fields such as biology, economics, and physics. The exponential growth function A=25ekt describes the population of this country t years after 1982. Whenever an exponential function is decreasing, this is often Feb 1, 2025 · This aligns with insights from articles on and . What is Exponential Growth? Exponential growth refers to an increase that occurs at a rate proportional to the current value, resulting in a rapid escalation over time. 5: Exponential Growth and Decay is shared under a not declared license and was authored, remixed, and/or curated by Isabel K. 32 Fractals in nature. Population growth is one of the most important topics we cover on Our World in Data. Find the population of the village in 2024. The only difference is that the growth factor, k, is a negative value. If 50 grams are present now, how much will be present in 630 years? Solution: There is a two part process to this problem. This type of growth is described mathematically by an exponential function. In Algebra 2, the exponential e will be used in situations of continuous growth or decay. Mar 10, 2025 · This exponential growth in AI-powered deception represents one of the most concerning downsides of advancing AI capabilities, forcing organizations to invest heavily in detection technologies and May 17, 2013 · This entry was posted in Algebra, Exponents, Math in the Real World and tagged compound interest, earthquake scale, exponent powers, exponential decay, exponential decrease, exponential graphs, exponential growth, exponents, exponents compound interest, exponents in the real world, exponents music video, exponents rap, half life, how people use exponents, index numbers, index numbers in the Mar 8, 2025 · Here are some examples. Oct 28, 2017 · Example: The half life of radium is 1690 years. This can be a physical change, like a plant getting taller or a person growing up. Apr 9, 2024 · Exponential growth is a powerful concept in finance and mathematics. 31 Exponential decay in radioactive materials. Read Carefully: The time in this problem is "the number of years since 2020". Use these interesting fun facts to learn something new today! They're sure to amaze adults and kids alike. Exponential growth can quickly surpass linear growth; this is why it’s important to understand and anticipate potential impacts on systems or populations. Exponential growth occurs when a function's rate of change is proportional to the function's current value. [/latex] In exponential growth, the rate of growth is proportional to the quantity present. One of the most prevalent applications of exponential functions involves growth and decay models. Since we are using an exponential model for 07 - What is an Exponential Function? (Exponential Growth, Decay & Graphing). The general formula is: y (x) = abkx Notice the variable x on the right hand side is part of the exponent, hence " exponent ial". The next growth we will examine is exponential growth. If F(t) represents the size at time t, the exponential function, or law, may be expressed as F(t) =aebt (1) Exponential growth is a process that increases quantity over time at an ever-increasing rate. Jan 20, 2025 · This blog post dives into 32 fascinating facts about change that will help you understand its nature and impact. In this article, you will learn about exponential function formulas, rules, properties, graphs, derivatives, exponential series and examples. This phenomenon is often represented mathematically by the equation (y = a(1 + r)^t), where (y) is the final amount, (a) is the initial amount, (r) is the The exponential function is occasionally called the natural exponential function, matching the name natural logarithm, for distinguishing it from some other functions that are also commonly called exponential functions. Real-Life Examples Exponential functions are super useful! They help us understand things that grow or shrink very quickly. For us to gain a clear understanding of exponential growth, let us contrast exponential growth with linear growth. Exponential Growth more Where a value increases in proportion to its current value. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. 6. 3 x, the Mar 4, 2025 · In this blog post, we'll dive into 37 fascinating facts about exponential growth, shedding light on its impact in various fields like biology, technology, and finance. Simply put, exponential growth is when a volume or quantity increases over time by a percentage Apr 16, 2025 · Discover 36 fascinating facts about growth, from biological processes to environmental impacts, and how they shape life on Earth. It helps you find the power you need to raise 'e' to, to get a certain number. An exponential model can be found On the one hand, this fact more than any other helps justify the focus of many growth models on the balanced growth path, a situation in which all economic variables grow at constant exponential rates forever. What is exponential growth? What is exponential growth? Whenever a quantity is increasing or growing rapidly as a result of a constant rate of growth applied to it, that quantity is experiencing exponential growth. Exponential growth can be amazing! The idea: something always grows in relation to its current value, such as always doubling. Basically this lesson is a collaborative activity Exponential Functions The exponential function with base \ (a\) is defined by \ ( f (x) = a^x\), for all real numbers \ ( x\), where \ ( a > 0\) and \ (a \neq 1\). [/latex] Key Concepts An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent. In this article, we will explore 20 unbelievable facts about human population growth. The value of Euler's number e determines the rate of growth or decay, which is an essential parameter in many applications. This will cause the function to decrease rather than increase with increasing t. Modeling exponential growth in practical scenarios So, how do you model exponential growth to predict things like product adoption? Start by identifying the key drivers—user acquisition rate, retention rate, and viral coefficient. 6: Exponential and Logarithmic Equations Uncontrolled population growth can be modeled with exponential functions. The exponential growth function A -25ekt describes the population of this country t years after 1982. What is exponential growth? Unlike linear growth, which results from repeatedly adding a constant, exponential growth is the repeated multiplication of a constant. Oct 2, 2023 · The list below will expand your understanding of exponential changes in clear and simple terms. Radioactive substances decay at an exponential rate. There are several non-linear equations that can be used to model growth. Growing Fast: Exponential Growth You can see exponential growth in many places, like Aug 25, 2025 · Exponential function facts. Exponential decay occurs in the same way when the growth rate is negative. Bacteria are single-celled microorganisms that cannot be seen by the naked eye. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. Introduction & Motivation: Exponential functions are found all around us in real-life situations. Nov 16, 2022 · In this section we will introduce exponential functions. An exponential function is a function of the form f (x) = a b x, f (x) = a⋅bx, where a a and b b are real numbers and b b is positive. Fact 8#: The unprecedented growth Over the last decade, volleyball has undergone unprecedented growth, and it now stands at being among the big five international sports along with soccer, cricket, field hockey, and tennis. More precisely, it is the function [math]\displaystyle { \exp (x) = e^x } [/math], where e is Euler's constant, an irrational number that is approximately 2. In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. In the exponential growth of f(x) f (x), the function doubles every time you add one to its input x x. An exponential model can be found when the growth rate and initial value are known. Apr 2, 2025 · Understanding exponential functions is crucial for anyone looking to comprehend the world around them, from finance to biology. Oct 16, 2024 · Learn about the AI trends that will determine the state of technology, business and society in the upcoming years. 6. So e is perfect for natural growth, see exponential growth to learn more. In the case of a discrete domain of definition with equal intervals it is also called geometric growth or geometric decay (the function values form a geometric progression). A negative argument results in exponential decay, rather than exponential growth. The exponential growth formula is used in finding the population growth, finding the compound interest, and finding the doubling time. Solving Exponential Functions: Exponential functions have the general form: ( y = ab^x ), where ( a ) and ( b ) are Aug 25, 2024 · Join me as a veteran retail analyst to explore 25+ impressive facts and statistics highlighting the exponential growth behind this iconic brand. Here are ten important facts about exponential functions that highlight their significance in both theoretical and practical applications. Napier was from Scotland, and his work was published in 1614, while Burgi, a native of Switzerland, developed his work in 1620. 014) t, where t is the number of years since 2020. See full list on mathsisfun. The presence of this doubling time or half-life is characteristic of exponential functions, indicating how fast they grow or decay. Therefore May 20, 2025 · Exponential functions were created by two men, John Napier and Joost Burgi, independently of each other. Linear growth occur by adding the same amount in each unit of time. This guide provides a comprehensive overview of the concept, including key formulas, practical examples, and expert insights. This activity is the ultimate collaborative and differentiable activity. After days there are flies, and after days there are flies. Another application of the exponential function is exponential decay, which occurs in radioactive decay and the absorption of light. It is used to represent exponential growth, which has uses in virtually all scientific disciplines and is also prominent in finance. Darcy. The number \ ( e\) is thought of as the base that represents the growth of processes or quantities that grow continuously in proportion to their current quantity. This page titled 2. Jul 11, 2023 · Reproduction in Full Swing Eubacteria reproduce through binary fission, a process where a single bacterium divides into two identical daughter cells. If \ (a > 1\), the function is increasing (graph going uphill from left to right). wpy istx oywrp kypui laqnngt iczcay grpli xadyfuc mruqkiv tvxrdht