Math · Interactive
Exponential Function
An exponential function y = a·bˣ multiplies by its base every step, so it grows or decays fast. Drag a and the base b below to reshape the curve and watch the y-intercept, asymptote, and behavior update live. Free to use, and exportable into your slides.
Drag the a and b sliders to reshape the curve. Open fullscreen ↗
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What is an exponential function?
An exponential function has the form y = a · b^x, where a is the initial value and b is the base, a positive number not equal to 1. Unlike a linear function that adds the same amount each step, an exponential function multiplies by the base b for every increase of 1 in x. That repeated multiplication is why exponential curves start slowly and then rise (or fall) dramatically. Drag the a and b sliders in the grapher above to see the curve reshape.
Exponential growth versus decay
When the base b is greater than 1 the function shows exponential growth: the output keeps multiplying and the curve sweeps upward. When the base is between 0 and 1 the function shows exponential decay: each step multiplies by a fraction, so the curve falls toward zero. The value a stretches the curve vertically and, if it is negative, reflects it below the x-axis. The grapher labels the behavior as growth or decay as you drag.
The y-intercept and the horizontal asymptote
Every basic exponential function passes through the point (0, a), because any base raised to the power 0 equals 1, so y = a · 1 = a. That makes the y-intercept easy to read straight from the equation. The curve also has a horizontal asymptote at y = 0: it gets closer and closer to the x-axis on one side without ever touching it. The grapher marks the y-intercept and reports the asymptote.
Where exponential functions appear
Exponential models describe compound interest, population growth, radioactive decay, and the cooling of a hot object. In each case a quantity changes by a constant percentage over equal time steps, which is exactly what multiplying by a fixed base does. Reading the base tells you the growth or decay rate per step: a base of 1.05 is 5 percent growth, and a base of 0.5 halves the value each step.
Frequently asked questions
What is an exponential function?
An exponential function has the form y = a · b^x with a base b that is positive and not equal to 1. It multiplies by b for each step in x, producing growth when b > 1 and decay when 0 < b < 1.
What is the difference between exponential growth and decay?
Growth happens when the base b is greater than 1, so the output keeps multiplying and the curve rises. Decay happens when the base is between 0 and 1, so each step multiplies by a fraction and the curve falls toward zero.
What is the y-intercept of an exponential function?
The y-intercept is (0, a), because b^0 = 1, so y = a · 1 = a. In the standard form y = a·b^x the coefficient a is exactly the value where the curve crosses the y-axis.
Does an exponential function have an asymptote?
Yes. The basic exponential function y = a·b^x has a horizontal asymptote at y = 0. The curve approaches the x-axis on one side, getting arbitrarily close without ever reaching it.
How do you find the growth rate from the base?
Subtract 1 from the base and read it as a percentage. A base of 1.08 means 8 percent growth per step, while a base of 0.9 means a 10 percent decrease per step.
How do I use this exponential grapher?
Drag the a and b sliders to change the function. The curve redraws instantly and the panel shows the y-intercept, whether it is growth or decay, the asymptote, and the rate per step. You can also export the graph to use in your own slides.