How do you graph exponential decay?

Answer 1

A quantity decades exponentially if it decreases in time at a rate that is linearly proportional to its value. In terms of equations, if #y# decays exponentially, then #dy/dt = - k y#, where #k>0# is the exponential decay constant and it characterizes the decay.

The variables involved in this differential equation are separable. If we separate them, we get #1/y dy = -k dt#. Now we can integrate:
#ln(y)=-kt+tilde{c}#
where #tilde{c}# is a constant. In the end: #y(t)=ce^{-kt}#, where #c=e^{tilde{c]}>0#.

This means that graphing the exponential decay turns into graphing an exponential function with negative exponent. For #t=0# (the instant in which the decay starts), we get that #y_0=y(0)=c#. So the parameter #c# is the #y#-intercept of the function: it represents the value of our quantity #y# at the instant in which the decay starts.
We can plot #y(t)# for some values of #c#. In the following plot #k=0.2# and different colors represent graphs for #c=10#, #c=5#, #c=2#, #c=1# and #c=0.5# (from the top to the bottom).


To get a visual intuition of what the differential equation #dy/dt=-ky# really means, let's consider the #(t,y)#-plane. We are indeed interested in the behavior of the quantity #y# in time #t#.
If we fix a point #(t_p,y_p)# on the plane, we are stating that the quantity #y# has the value #y=y_p# at time #t=t_p#. We want to represent the decay, so we ask ourselves where would the point #(t_p,y_p)# "decay" after a "bit" of time. This "bit" can be thought as infinitesimally small: we denote it by #(dt)_p#. In this infinitesimal time, the quantity #y# changes by an infinitesimally small amount #(dy)_p#.
So, the point we are searching for is the point #(t_p+(dt)_p,y_p+(dy)_p)#, which is the point that is going to describe the decay an infinitesimal amount of time after #t_p#. To represent this information we draw an arrow (namely a vector) in #(t_p,y_p)#, pointing in the direction of #(t_p+(dt)_p,y_p+(dy)_p)#. The vector's direction is given by the difference
#(t_p+(dt)_p,y_p+(dy)_p)-(t_p,y_p)=((dt)_p,(dy)_p)#
From the differential equation we get that #((dt)_p,(dy)_p)=((dt)_p,-ky_p(dt)_p)#. Now we can choose the size of the arrow by setting #(dt)_p# as small as we like.
[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.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To graph exponential decay, follow these steps:

  1. Choose a suitable scale for the x-axis (horizontal) and y-axis (vertical).
  2. 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.
  3. Connect the points smoothly with a curve.
  4. If necessary, label the axes and add a title to the graph for clarity.
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
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.

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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7