A decay function is an exponential function in Honors Pre-Calculus that decreases by a constant factor over equal intervals. Its graph falls quickly at first, then levels off toward a horizontal asymptote.
A decay function is an exponential function in Honors Pre-Calculus where the output gets smaller by the same multiplier each time x increases by 1. The general form is f(x) = a(b)^x, with 0 < b < 1. The number a is the starting value, and b is the decay factor.
That “less than 1” base is what makes the graph shrink instead of grow. If b = 0.8, then each step to the right keeps 80% of the previous value. If b = 0.5, the quantity is cut in half each step, which is why a graph can fall very fast at first.
The curve does not drop in a straight line. Exponential decay changes by a constant percent, not a constant amount. That means the graph is steep near the y-axis, then flattens out as x gets larger. In most class problems, that flattening shows up as a horizontal asymptote, often the x-axis when there is no vertical shift.
A compact example is f(x) = 200(0.6)^x. The initial value is 200, so f(0) = 200. After one step, the value is 120, then 72, then 43.2. Each output is 60% of the previous one, which is the pattern you look for when identifying decay.
A common mistake is mixing up decay factor and decay rate. If a problem says something decreases by 15% each period, the factor is 0.85, not 0.15. Another mistake is assuming decay means negative outputs. The function can stay positive while still decreasing, because the outputs are shrinking, not necessarily below zero.
Decay functions show up every time Honors Pre-Calculus asks you to connect an equation to a shrinking pattern. Once you recognize the base is between 0 and 1, you can tell the graph is decreasing without plotting every point.
This term also connects directly to exponential modeling, which is a big part of the unit on graphs of exponential functions. You may be asked to read a table, write an equation from context, or explain why one graph drops faster than another. Decay factor tells you the speed of the decrease, while the initial value tells you where the graph starts.
The concept comes up in half-life problems too. If a substance loses half of its amount every fixed time period, that is a special kind of decay function. In those problems, you are not just graphing a curve, you are tracking repeated multiplication and interpreting what the exponent means in the situation.
Decay functions also prepare you for later work with logarithms and transformations. When you can see how a graph approaches an asymptote and how a base below 1 changes the direction of the curve, you are building the same kind of function sense that shows up again in algebra, precalculus, and calculus.
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view galleryExponential Function
A decay function is one specific kind of exponential function. The whole family has the variable in the exponent, but decay happens when the base is between 0 and 1. That is what makes the graph decrease instead of increase. If you can spot the exponential form, you can decide whether it is growth or decay by checking the base.
Half-Life
Half-life is a real-world decay pattern where a quantity drops to half its amount after each fixed time interval. In Pre-Calculus, half-life problems are usually modeled with an exponential decay function. The half-life gives you the time step, and the equation tells you how much is left after several steps.
Asymptote
A decay graph often approaches a horizontal asymptote as x increases. In the simplest case, the asymptote is y = 0, which means the function keeps getting smaller but does not hit the axis right away. If the function is shifted up or down, the asymptote moves too.
Compound Interest Formula
Compound interest usually uses growth, but the same exponential setup shows up in reverse when money loses value, like depreciation. The formula structure is similar, so recognizing decay helps you see when a situation is multiplying by a factor less than 1 instead of more than 1.
A graphing question may give you a table, a formula, or a context and ask whether the pattern is decay. You identify decay by checking for a base between 0 and 1, then use the initial value and factor to sketch or evaluate the function. If the problem mentions percent decrease, convert it to a multiplier first, like 12% decay becoming 0.88. On quizzes and problem sets, you may also explain the horizontal asymptote or compare two decay rates to decide which quantity drops faster.
Decay and growth both use exponential functions, but they behave in opposite directions. Growth has a base greater than 1, so the graph rises as x increases. Decay has a base between 0 and 1, so the graph falls and levels off.
A decay function is an exponential function with a base between 0 and 1.
The initial value is the y-intercept, and the decay factor tells you how much of the quantity remains each step.
Decay means constant percent decrease, not constant amount decrease.
The graph usually falls quickly at first and then approaches a horizontal asymptote.
Percent decrease must be converted into a multiplier before you write the equation.
It is an exponential function that decreases by a constant factor over equal intervals. The standard form is f(x) = a(b)^x with 0 < b < 1, so the graph starts at a and then drops as x increases.
Check the base. If the base is between 0 and 1, the function is decay. You can also look for a graph that moves downward as you go right and levels off toward a horizontal asymptote.
The decay factor is the multiplier left after each step, like 0.85. The decay rate is the percent decrease, like 15%. Those are related, but they are not the same number.
Half-life problems track how long it takes a quantity to be cut in half repeatedly. You plug the half-life into the exponential decay model as the time interval, then use the exponent to find how much remains after a given time.