# How do you graph exponential decay?

A quantity decades exponentially if it decreases in time at a rate that is linearly proportional to its value. In terms of equations, if *exponential decay constant* and it characterizes the decay.

The variables involved in this differential equation are separable. If we separate them, we get

where

This means that graphing the exponential decay turns into graphing an exponential function with negative exponent. For

We can plot

To get a visual intuition of what the differential equation

If we fix a point

So, the point we are searching for is the point

From the differential equation we get that

*[This argument is not rigorous: we should speak about* finite differences *and how they are related to differentials, but the core idea emerges anyway. Also notation is invented for the purpose of the argument.]
*

*If we repeat this operation for some points, we get the following picture. Note that vectors are normalized (i.e. made unitary dividing by their norm) and this particular plot is made fixing #k=0.2# (other positive values of #k# don't change the qualitative behavior).*

#y# has value #y_p# at time #t=t_p#. In the following picture I chose #(t_p,y_p)=(2,6)# and #k=0.2# as in the previous examples.

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To graph exponential decay, follow these steps:

- Choose a suitable scale for the x-axis (horizontal) and y-axis (vertical).
- Plot several points using values of x and y that satisfy the exponential decay equation, (y = a \cdot e^{-bx}), where (a) is the initial value, (b > 0) is the decay rate, and (x) is the independent variable.
- Connect the points smoothly with a curve.
- If necessary, label the axes and add a title to the graph for clarity.

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*Answer from HIX Tutor*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

*When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.*

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