Graph PlottersExplore exponential growth and decay with a live equation, horizontal asymptote, and growth-rate summary.
Hover to inspect coordinates. Drag the graph to pan the view or use the wheel and zoom controls to explore the function.
ABOUT THE TOOL
Exponential functions model repeated percentage change. A positive B creates growth, while a negative B creates decay.
HOW TO USE IT
An exponential graph changes by a constant multiplicative factor rather than a constant additive amount. This calculator uses y = A e^(Bx + C) + D. A controls scale and reflection, B controls growth or decay rate, C shifts the exponent, and D sets the horizontal asymptote.
When B is positive, the curve grows as x increases; when B is negative, it decays. The horizontal asymptote is y = D. Exponential functions are defined for every real x, but their values can become very large, so the zoom controls are useful for wide coefficient ranges.
Try eˣ, 2eˣ, e^(−x), and shifted examples. Change B to compare slow and fast growth, then hover over the curve to read exact values at selected x-coordinates.
An exponential function changes by a constant factor over equal x-intervals, unlike a linear function that changes by a constant difference. In y = A e^(Bx + C) + D, B determines growth or decay, A controls scale and reflection, C shifts the input, and D is the horizontal asymptote. The curve approaches the asymptote but does not normally cross it. These features are useful when modeling population, finance, depreciation, and other repeated percentage changes.
For y = 2e^(0.5x) − 1, the function grows because B is positive, the horizontal asymptote is y = −1, and the y-intercept is 1 because 2e⁰ − 1 = 1. Enter the equation, use zoom controls if the growth leaves the first view, and check those two values with the graph cursor. Changing B to a negative value turns the same family into decay.
For y = 500(1.04)^t, the initial value is 500 and the factor 1.04 means a 4% increase per time unit. After two units, the value is 500(1.04)² = 540.8. The graph rises slowly at first and then more quickly because each increase is applied to the current amount. Use the calculator to compare this curve with a decay factor such as 0.96.
A linear model adds the same amount each step, while an exponential model multiplies by the same factor. For example, 100 + 10t adds 10 each time, but 100(1.1)^t grows by 10% of the current amount. The curves may look similar over a short window, so inspect several x-values and use the graph’s wider view before deciding which model matches a situation.
Explore related tools as you study functions: start with the online graphing calculator to plot a general equation, then use the linear graph calculator for constant rates and the slope calculator for rise over run. When a curve meets the x-axis, check the x intercept calculator; for parabolas, compare the quadratic graph calculator and vertex calculator. Use the cubic graph calculator for polynomial end behavior, the sine graph calculator and cosine graph calculator for periodic models, and the phase shift calculator for horizontal movement. The function transformations calculator compares parent and transformed curves, while the absolute value graph calculator explains V-shaped graphs. For growth and inverse relationships, try the exponential growth graph calculator, reciprocal graph calculator, and log graph calculator. For asymptote and periodic-function practice, use the tangent graph calculator, secant graph calculator, and cosecant graph calculator to compare branches, gaps, and repeating behavior.
COMMON QUESTIONS
Exponential growth occurs when a quantity changes by the same percentage over equal intervals.
B controls the growth or decay rate in e^(Bx + C).
For A e^(Bx + C) + D, the horizontal asymptote is y = D.
For b^x, a base b greater than 1 gives growth and a base between 0 and 1 gives decay. In e^(Bx), the sign of B gives the same distinction.
For y = A e^(Bx + C) + D, the horizontal asymptote is y = D.
It can if a vertical shift and scale place the curve across y = 0. The unshifted positive exponential itself stays above zero.