Exponential Slope Calculator

The results of a causal linear regression model were Y = 40 – 0. Graphing calculator : type in an equation and this program will draw your graph online on an x,y coordinate grid. But sometimes, you will need to do exponential calculation to a range of cells, how can you apply exponential calculation in Excel?. net and study mathematics i, notation and a great many additional algebra subjects. SOLUTION Enter the original data into a graphing calculator and perform an exponential regression. $ \text{Slope } = \frac{ y_2 - y_1 } { x_2 - x_1 } $ How it works: Just type numbers into the boxes below and the calculator will automatically find the slope of two points. From Fraction Inequalities Calculator to powers, we have everything covered. Furthermore, the slope of this line is b, the value of the constant in the original exponential equation. Simple linear regression is a way to describe a relationship between two variables through an equation of a straight line, called line of best fit, that most closely models this relationship. Using the Slope Calculator. That means the change in x, which is the denominator of the slope formula, would be 5 - 5 = 0. a is the starting value, b is the growth rate. When you enter an expression into the calculator, the calculator will simplify the expression by expanding multiplication and combining like terms. In Figure 4, the slew rate parameter (P2) is used to measure the slope of the math function. 6% changes to. Stochastic models: 1. Trendlines, or best fit lines, help reveal linear relationships between variables. The constituent functions are shown in the plot below. Find the exponential function y = Ce^kt that passes through the two given points. Otherwise you may as well take logarithms in base 10. For every 5 dollar increase in price, there are 30 fewer students willing to buy the calculator. How to enter numbers: Enter any integer, decimal or fraction. In particular, Slope can be used for trend identification to establish a trading bias. The relative predictive power of an exponential model is denoted by R 2. Find the instantaneous rate of change in Revenue R = −+20 4qq2 at q = 30 where q is the quantity of items sold. A graphing calculator and a straightedge (ruler) must be available for you to use while taking this examination. The slope of a line on the coordinate plane is indicated by the letter "m". when r is much smaller than 1 in magnitude. No Download or Signup. You can think of h as a dial that we can turn to get various. BMI Calculator » Triangle Calculators » Length and Distance Conversions » SD SE Mean Median Variance » Blood Type Child Parental Calculator » Unicode, UTF8, Hexidecimal » RGB, Hex, HTML Color Conversion » G-Force RPM Calculator » Chemical Molecular Weight Calculator » Mole, Moles to Grams Calculator » R Plot PCH Symbols » Dilution. where b is the common base of the exponential and the logarithm. Start studying Comparing Exponential, Linear, and Quadratic Growth Quiz. continuously, calculate the final amount you will have in the account after five years. Exponential equations are also solved using logs, either common (log) or natural (ln). x 0 is the initial value at time t=0. You can then annualize it as needed. If 2 is raised to a negative power, say -5, then we write it as 2 -5. Any lowercase letter may be used as a variable. So, you need to press: Not only would we like the calculator to find a trend line for us, we want to graph that. The exponential moving average (EMA) is a weighted moving average (WMA) that gives more weighting, or importance, to recent price data than the simple moving average (SMA) does. Before you begin Students should be able to calculate a unit rate (e. For the rest of the examples below, there are no built-in formulas. Free exponential equation calculator - solve exponential equations step-by-step. The results of a causal linear regression model were Y = 40 – 0. Online Algebra Calculators. For any positive number a>0, there is a function f : R ! (0,1)called an exponential function that is defined as f(x)=ax. L1 and L2 are the 2 ND function keys of the 1 and 2 keys. In Figure 4, the slew rate parameter (P2) is used to measure the slope of the math function. 6% changes to. Then we will move on to graphing slope and finally to using slope intercept form to create your graph. It's usually used for finance and economics. When try such calculations on your own, use this exponential distribution calculator to verify your results. You can fit just about any equation with this tool. These are the same rules discussed for transforming quadratic graphs, they just look a little different when applied to exponential functions. In the event that you actually have service with algebra and in particular with rational exponents calculator or basic concepts of mathematics come pay a visit to us at Polymathlove. Well you know that having a 0 in the denominator is a big no, no. For values of 0, 1, and 2, the values of the function are 1, e or about 2. Online Algebra Calculators. Get the free "Regression Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Exponential Slope Calculator. For example, enter 3x+2=14 into the text box to get a step-by-step explanation of how to solve 3x+2=14. Any lowercase letter may be used as a variable. So we can write This division on the calculator comes out to 0. ln(x) calculator. To calculate we should know the formulas. The reason for this is that the graph of Y = LN(X) passes through the point (1, 0) and has a slope of 1 there, so it is tangent to the straight line whose equation is Y = X-1 (the dashed line in the plot below): This property of the natural log function implies that. • The plot of x versus log 10(y) appears concave downward. The exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra. When you study slope in Algebra, you are studying the incline along with other characteristics of a line. It signifies the rate of failure. The most common example of constant acceleration is free fall. If you are asked to calculate the average rate of change on an interval without a graph, you will have to come up with two points to calculate the slope. SOLUTION Enter the original data into a graphing calculator and perform an exponential regression. A least-squares regression fit of the data (using base 10 logarithms to transform columns (1) and (6)) indicates that the estimated slope for the Weibull distribution is 1. 7 Modeling with Exponential and Logarithmic Functions 345 Using Technology You can use technology to fi nd best-fi t models for exponential and logarithmic data. Then hit calculate. For negative `x`-values, the graph gets very close to the `x`-axis, but doesn't touch it. Population Growth Models Part 2: The Natural Growth Model The Exponential Growth Model and its Symbolic Solution. Hi all, I have a dataset which consists of 2 columns. Of course, if we have a graphing calculator, the calculator can graph the function without the need for us to find points on the graph. This widget can be use only for trinomials, if you need to factor polynomials, you can use our factoring polynomials calculator. It can be used for any expression and is a more advanced tool, that will give you additional information about the polynomial. Average Change Rate Converter of US Current to Real Dollars with CPI Converter of US Current to Real Dollars with GDP deflator Exponential Growth/Decay Doubling/Half-Life Time Present Value Calculator S-Curve Calculator - 1 parameter estimate S-Curve Calculator - 3 parameter estimates. In doing this, the Derivative Calculator has to respect the order of operations. 629*100 = 3% and the uncertainty in the intercept is 12%, which is only about two significant figures in each. Over time this results in the exponential growth of your money. Exponential smoothing is a rule of thumb technique for smoothing time series data using the exponential window function. For Option 3, find an equation for the curve using regression (use ExpReg under Stat:Calc menu). If you have a time series with a clear pattern, you could use moving averages — but if you don't have a clear pattern you can use exponential smoothing to forecast. Why you should learn it GOAL 2 GOAL 1 What you should learn 8. Using semi-log plot to get graph of a straight line with slope of -k. What is Slope? Slope is a concept in geometry that describes the direction and steepness of a line on a Cartesian plane. About Exponential Decay Calculator. When bacteria are grown in a closed system (also called a batch culture), like a test tube, the population of cells almost always exhibits these growth dynamics: cells initially adjust to the new medium (lag phase) until they can start dividing regularly by the process of binary fission (exponential phase). This is the general Exponential Function (see below for e x):. Finally, find the slope by dividing the rise by the run. The exponential moving average (EMA) is a weighted moving average (WMA) that gives more weighting, or importance, to recent price data than the simple moving average (SMA) does. Examples, of how the above relationship between the logarithm and exponential may be used to transform expressions, are presented below. Tangent Lines and Logarithmic / Exponential Functions Add Remove This content was COPIED from BrainMass. In a nutshell, the slope of any line is the change in the y-values versus the change in the x-values. From simplify exponential expressions calculator to division, we have got every aspect covered. When = 3, the. Loading Loading. An example where an exponential regression is often utilized is when relating the concentration of a substance (the response) to elapsed time (the predictor). This example teaches you how to apply exponential smoothing to a time series in Excel. After you enter the expression, Algebra Calculator will graph the equation y=2x+1. Exponential equations are also solved using logs, either common (log) or natural (ln). Stochastic models: 1. This is the parameter , which is the value of the slope. 065, which is as we hypothesized, βH*A > 1. First go to the Algebra Calculator main page. In other words, the slope of the line tells us the rate of change of y relative to x. How to Graph the Equation in Algebra Calculator. At the first workshop of the summer, I got to participate in an activity that used M&Ms to model exponential decay. ln[N t] = ln[N 0] + rt. This slope calculator takes two points and then uses the slope formula to calculate the slope of a line defined by those two points, and then the y intercept. → Read More: Max/Min Problem – Maximizing Revenue, Selling Calculators. The probability plot for 100 normalized random exponential observations (\(\lambda\) = 0. org helps support GraphSketch and gets you a neat, high-quality, mathematically-generated poster. Section 5: Transforming Exponential Functions, and. Using the Slope Calculator. The gradient of a horizontal line is 0. Right from simplifying integer exponents calculator to radicals, we have all of it discussed. Graphing exponential functions is similar to the graphing you have done before. The only caveat was that they could only use straight lines. You should not take any action on the basis of the information provided through this calculator. The worksheets can be made in html or PDF format (both are easy to print). continuously, calculate the final amount you will have in the account after five years. EXAMPLE 1: a) is a polynomial of degree 4 with leading. 📚 How to plot and calculate slope of exponential functions Study Force. To clear the scatter graph and enter a new data set, press "Reset". Once the line has been drawn, the slope of the line can be obtained (some probability papers include a slope indicator to simplify this calculation). If ever you actually have advice with algebra and in particular with exponential form calculator or worksheet come pay a visit to us at Factoring-polynomials. Additionally, `D` uses "lesser known" rules to calculate the derivative of a wide array of special functions. The natural logarithm of x is: ln x = log e x = y. Textbook solution for Calculus: An Applied Approach (MindTap Course List)… 10th Edition Ron Larson Chapter 4. Thus, the exponential regression curve is not the best-fit curve in the "strict" sense. Note that the curve passes through `(0, 1)` (on the y-axis). Due to the heterogeneity of the material a macroscopic strength criterion is determined and. Hi all, I have a dataset which consists of 2 columns. Note that all the x values on this graph are 5. Examples, of how the above relationship between the logarithm and exponential may be used to transform expressions, are presented below. And I also assume that you want to see more digits in those coefficients. On the other hand, take an example where the slope is zero. Use a calculator to check the answer we found to the equation in example 3. -- redraw graph, zoomed out by a factor of 2. For complex values of X, Y is complex. The slope of the straight line y = t is obviously consant. As you can see from looking at the StatPages. For more information on GraphSketch (how it works, etc. Otherwise you may as well take logarithms in base 10. com and understand linear systems, solving linear equations and a wide range of additional algebra subject areas. The online Exponents Calculator can be helpful in converting any number from an exponential form into the standardized one to which we have all got used to. The slope of a straight line. When the rate of change is proportional to the value of the function itself, it is called exponential growth and decay. Find the exponential function y = Ce^kt that passes through the two given points. In this case our βH*A = e 0. Right from convert radical form calculator to geometry, we have got every aspect covered. Factoring-polynomials. The procedure is easier if the x-value for one of the points is 0, which means the point is on the y-axis. Although the exercises in this module largely amount to filling in tables,. After linear functions, the second most important class of functions are what are known as the "exponential" functions. The worksheets can be made in html or PDF format (both are easy to print). Graphing calculators make is easy to work with exponents and functions. In Figure 4, the slew rate parameter (P2) is used to measure the slope of the math function. Support GraphSketch: GraphSketch is provided by Andy Schmitz as a free service. Come to Algebra1help. Warning: Negative powers can also be used. Teacher Resources Based on extensive research, SmartGraphs: Algebra is designed for students studying Algebra 1 or Algebra 2. Since slope equals rise over run, the slope of the line is y 2 − y 1 over x 2 − x 1. It's so fast and easy you won't want to do the math again!. The method produces a line that tracks the data more closely than simple smoothing and provides information. No Download or Signup. The EMA responds. I somehow got confused with these equations. This calculator uses the following formula to derive the equation for the line of best fit. In doing this, the Derivative Calculator has to respect the order of operations. Students will be given scratch paper and a graphing calculator. Discover how exponential functions, in which a variable appears in the exponent, are used to model real-world situations. In particular, Slope can be used for trend identification to establish a trading bias. Also see the resulting equation. The online Exponential Decay Calculator is used to solve exponential decay problems. From my answer to How can someone explain exponential functions to a high school kid?: An exponential function or curve is a function that grows exponentially, or grows at an increasingly larger rate as you pick larger values of x, and usually tak. Graphing Logarithmic Functions Using Their Inverses. In the function f(x) = a b^x, what is the significance of the value of a in the graph? When the function is increasing from left to right, what do you notice. In order to do that, it is required to merely insert 2 numbers to work out the solution: the number and its power. Exponential, harmonic or hyperbolic decline? The days of plotting rates on semi-log graph paper are long gone. It supports polynomial equations as well as some equations with exponents, logarithms and trigonometric functions. This means the slope is undefined. Why you should learn it GOAL 2 GOAL 1 What you should learn 8. For exponential growth: when a population is small, it increases slowly; when a population is large, it increases quickly; The relative growth rate will make this observation mathematically precise. This calculator is intended solely for general information and educational purposes. Over time this results in the exponential growth of your money. This is the general Exponential Function (see below for e x):. x 0 is the initial value at time t=0. If you know two points that a line passes through, this page will show you how to find the equation of the line. Once you get into advanced algebra and beyond, it can be difficult or time consuming to do mathematical calculations by hand. In equation y = e bx. Graphing calculator : type in an equation and this program will draw your graph online on an x,y coordinate grid. That is, Excel finds the intercept and slope that minimize the sum of squared deviations between actual Ln(y) and predicted Ln(y), using the formula Ln(y) = Intercept + Slope * x. This might simplify your work. Learn how to calculate the exponential moving average in Excel and VBA, and get a free web-connected spreadsheet. From my answer to How can someone explain exponential functions to a high school kid?: An exponential function or curve is a function that grows exponentially, or grows at an increasingly larger rate as you pick larger values of x, and usually tak. The exponent for exponential growth is always positive and greater than 1. Get your practice problems in Exponential Functions here. I'd like to plot them on a x-y scatter plot and fit an exponential trendline. Correlation and regression calculator Enter two data sets and this calculator will find the equation of the regression line and corelation coefficient. % Percent button is used to find the percentage of a number. When < 1, the Weibull distribution models early failures of parts. (NOTE: Fitting an exponential function y=Aert is done in an analogous way - see below. Free Calculus worksheets created with Infinite Calculus. Thus, 10 4. continuously, calculate the final amount you will have in the account after five years. The decimal will be moved to the right 9 places. Looking for a guide on how to figure out the slope of a line in basic algebra? From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. No Download or Signup. Easy to use online maths calculators and solvers for various topics. This was admittedly a stupid question. Identify each table of data as linear, quadratic, or exponential:. When the rate of change is proportional to the value of the function itself, it is called exponential growth and decay. They can, also, generate a step by step explanation at the click of a button. In practice, this means substituting the points for y and x in the equation y = ab x. notebook May 14, 2014 p. The Adjusted Slope 125/250d indicator is equal to the average annualized exponential regression slope, over the past 125 and 250 trading days, multiplied by the coefficient of determination (R 2). A beautiful, free Scientific Calculator from Desmos. Graph an exponential function solver and continuous exponential growth help in high school math advanced algebra free s by mathvids com piecewise functions in the graphing calculator. The spreadsheet below shows an example of the Excel Slope function used to calculate the slope of the linear regression line through the known_x's and known_y's, in cells F2:F7 and G2:G7. This work describes a library of shape functions and singly differential cross sections of bremsstrahlung due to interaction of the electrons with energies from 10 eV. (NOTE: Fitting an exponential function y=Aert is done in an analogous way - see below. There are many quantities that grow exponentially. Concluding Remarks. Online Algebra Calculators. In practice, this means substituting the points for y and x in the equation y = ab x. Linear functions are graphically represented by lines and symbolically written in slope-intercept form as, y = mx + b, where m is the slope of the line, and b is the y-intercept. This is also used in differential equation to verify…. Matlab code to fit the exponential decay model and calculate how to calculate half life 6 steps with pictures wikihow graph exponential function calculator math goldtree club exponential decay and time constants advanced math for young students Matlab Code To Fit The Exponential Decay Model And Calculate How To Calculate Half Life 6 Steps With Pictures Wikihow Graph…. The natural logarithm of x is: ln x = log e x = y. This function takes the number x and uses it as the exponent of e. Choose from simple or more complex expressions involving exponents, or write expressions using an exponent. Right from ordered pair equation calculator to calculus, we have every part discussed. 116 FAQ-676 We are trying to use a single exponential decay equation to determine the half-life of a compound, but your equation is slightly different than the standard form. A line that slopes down from left to right has a negative gradient. Students learn about four forms of equations: direct variation, slope-intercept form, standard form and point-slope form. In this article, I will explain how you can calculate slope values in Google Sheets with and without graphs. For any integer n, the points are mapped onto a single point in the w plane, as indicated in Figure 5. In the function f(x) = a b^x, what is the significance of the value of a in the graph? When the function is increasing from left to right, what do you notice. Exponential. The natural logarithm of x is: ln x = log e x = y. Exponential functions tell the stories of explosive change. Simple Interest Calculator - powered by WebMath. P3 is set up using parameter math to take the reciprocal of P2; this is the time constant of the original exponential 1. Then we will move on to graphing slope and finally to using slope intercept form to create your graph. The AP Calculus Problem Book Publication history: First edition, 2002 Second edition, 2003 Third edition, 2004 Third edition Revised and Corrected, 2005 Fourth edition, 2006, Edited by Amy Lanchester Fourth edition Revised and Corrected, 2007 Fourth edition, Corrected, 2008 This book was produced directly from the author’s LATEX files. A line that slopes up from left to right has a positive gradient. For higher order derivatives, certain rules like the general Leibniz product rule can speed up calculations. I'd like R to determine the equation for. No matter how hard it is to get when you write any pattern and it does not still help you find the slope between two lines. com gives both interesting and useful strategies on gcf with exponents calculator, complex and multiplying and dividing fractions and other algebra topics. Free exponential equation calculator - solve exponential equations step-by-step. SOLUTION Enter the original data into a graphing calculator and perform an exponential regression. Before you begin Students should be able to calculate a unit rate (e. And actually, let me write it that way. Find more Widget Gallery widgets in Wolfram|Alpha. Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals. It is not intended for use in scripting and is not available in the ArcPy Spatial Analyst module. Weibull Distribution Calculator is an online probability and statistics tool for data analysis programmed to calculate precise accurate failure analysis and risk predictions with extremely small samples using a simple and useful graphical plot. They graph and complete problem sets for each, converting from one form of equation to another, and learning the benefits and uses of each. Using semi-log plot to get graph of a straight line with slope of -k. That means the change in x, which is the denominator of the slope formula, would be 5 - 5 = 0. Loading Loading. The calculation begins like this: Now we will recall that so that we can write. If you need support with math and in particular with letter algebra calculator or linear algebra come visit us at Sofsource. us and select STAAR resources. exponential function, just as the slope of a line did with tables of points on a linear function. This year, I decided to have my students create what I deemed "Slope Name Art. Each x/y variable is represented on the graph as a dot or a. Enter a number, then click square root button. Using Exponential Regression Use a graphing calculator to find an exponential model for the data in Example2. Note: The above equation is only valid for the exponential equations. skew 151 209210 slope of 177185 208 211212 915 of symmetry 47 645 tangent from MATH 101 at Mother of Divine Grace Homeschool. Know and apply the properties of integer exponents to generate equivalent numerical expressions. The equation of a line is typically written as y=mx+b where m is the slope and b is the y-intercept. In a nutshell, the slope of any line is the change in the y-values versus the change in the x-values. To model real-life situations, such as the slope of a beach in Example 4. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. Finding the Inverse of an Exponential Function. Type the following: y=2x+1; Try it now: y=2x+1 Clickable Demo Try entering y=2x+1 into the text box. A least-squares regression fit of the data (using base 10 logarithms to transform columns (1) and (6)) indicates that the estimated slope for the Weibull distribution is 1. The Texas Education Agency (TEA) has recently released the Algebra EOC exam from 2015 on its website. Should you need guidance on fractions or maybe graphing linear, Factoring-polynomials. Also see the resulting equation. Slope at a Value Tangent Lines Exponential Growth and Decay. In case you need to calculate the cube root you can use our cube root calculator which is an excellent tool that will calculate the cube root of any number. As this lesson is meant to be introductory, no prior knowledge of linear functions is assumed. t is the time in discrete intervals and selected time units. Exponential Regression An exponential regression is the process of finding the equation of the exponential function that fits best for a set of data. This puts the equation into one. Slope detection level for determining the start of the degradation process, specified as a scalar in the range [0,1]. If you have two points, for example, (2, 6) and (3, 18), how do you find the equation if you know its exponential? I've heard about y=ab^x but I'm not sure what those variables represent. Exponential Growth/Decay Calculator. Exponential functions 1. I assume that you have already displayed the equation of the Trend line on your chart as suggested by Josh Burdett. Exponential Growth and Decay Calculator Use our online exponential growth and decay calculator by entering the initial value (x 0), decay rate (r) and time (t) in the below calculator and click calculate button to find the answer. This type of function has many applications in population growth and economics. org web site, there are many "stand-alone" web pages that are each designed to perform only a single test or calculation. Use the model to estimate the number of cell-phone subscribers in 1998. Factoring-polynomials. How to select best Adjusted Slope 90d values Adjusted Slope 125/250d. If it slopes down, the rise will be negative. Given the equation of a line, solve for the dependent variable and graph: a. Show Instructions In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`. Learn vocabulary, terms, and more with flashcards, games, and other study tools. The Adjusted Slope 125/250d indicator is equal to the average annualized exponential regression slope, over the past 125 and 250 trading days, multiplied by the coefficient of determination (R 2). This is a good example of where a slope field can be slightly misleading. If the data is in col A, in col B enter the formula =LN(Ax) and copy down your list. WITH THE TI-NSPIRE CAS CALCULATOR An Essay Submitted to the Office of Graduate Studies College of Arts & Sciences of John Carroll University In Partial Fulfillment of the Requirements For the Degree of Master of Arts By Meghan A. Download free on Google Play. Uncertainty in a single measurement Bob weighs himself on his bathroom scale. We carry a lot of quality reference information on topics ranging from syllabus for elementary algebra to solving systems of linear equations. The exponential function is its own slope function: the slope of e-to-the-x is e-to-the-x. Here you will find free loan, mortgage, time value of money, math, algebra, trigonometry, fractions, physics, statistics, time & date and conversions calculators. b) Press p. When you reflect across y=x, you take the reciprocal of the slope. can be used for either both cases. SLOPE(known_y's, known_x's) The SLOPE function syntax has the following arguments:. The smallest divisions on the scale are 1-pound marks, so the least count of the instrument is 1 pound. Parentheses and Brackets. In this case our βH*A = e 0. Come to Emaths. The slope intercept form calculator will find the slope of the line passing through the two given points, its y-intercept and slope-intercept form of the line, with steps shown. Special indicators may display on the screen to provide additional information concerning functions or results. Implicit multiplication (5x = 5*x) is supported. Finding the Inverse of an Exponential Function. P3 is set up using parameter math to take the reciprocal of P2; this is the time constant of the original exponential 1. Support GraphSketch: GraphSketch is provided by Andy Schmitz as a free service. When you enter an expression into the calculator, the calculator will simplify the expression by expanding multiplication and combining like terms. The uncertainties in the slope and intercept are much better for judging the quality of the fit. What are exponential equations? An exponential equation is an equation in which a variable occurs in the exponent. It can be used for any expression and is a more advanced tool, that will give you additional information about the polynomial. BMI Calculator » Triangle Calculators » Length and Distance Conversions » SD SE Mean Median Variance » Blood Type Child Parental Calculator » Unicode, UTF8, Hexidecimal » RGB, Hex, HTML Color Conversion » G-Force RPM Calculator » Chemical Molecular Weight Calculator » Mole, Moles to Grams Calculator » R Plot PCH Symbols » Dilution. The differences are startling. Come to Algebra1help. b) Press p. Solves for cubic equations in the form ax 3 + bx 2 + cx + d = 0 using the following methods: 1) Solve the long way for all 3 roots and the discriminant Δ 2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable. You can then annualize it as needed. Exponential growth/decay formula.