Suppose you have a substance with a half-life of 1 million years that decays according to
y = y0 e − kt
Make a graph of the percent of material y as a function of the time in years.
To graph this formula, we need to find the value of y0 and k from the given information. We’ll do this in
several steps before putting into our graphing calculator.
Find the value of y0: The constant y0 always corresponds to the initial amount of the substance. In
many cases you’ll be given an amount in grams that was originally present. In this case, y0 is that
original amount. But this problem is a little different. Since y corresponds to the percent of the material,
y0 corresponds to the original percent of material present. Usually we would assume this amount is
100% so y0 = 100.
Use the half-life to find k: To find the value of k, you need to understand what the term half-life means.
A half-life is the amount of time it takes for the amount of material to reduce by half. The easiest way to
use this information is to note that if the original percent of material is 100 percent, then at t = 1 million
the value for y = 50 percent. Putting these values into the equation above yields
50 = 100e − k ⋅1
Now we can solve for k.
To do this, isolate the exponential by dividing both sides by 100. This leaves us with
50
= e− k ⋅1
100
1
= e− k
2
Convert this equation to its logarithm form to yield
ln ( 12 ) = −k
Multiplying both sides by -1 gives us k:
k = − ln ( 12 ) ≈ 0.6931
Write the equation: Now that you have found the values for y0 and k, we can put them into the
equation and write down the relationship between t and y:
y = 100e
− ln( 12 ) t
You can use the approximate value for the logarithm,
y = 100e −0.6931t
but your y values will only be approximate.
Use your calculator to graph the function: We’ll graph this function in the window [0, 10] by [0,
100]. This corresponds to a horizontal window of 0 to 10 million years (many half-lives) and a vertical
window of 0 to 100 percent.
1. Turn your calculator on by pressing É.
2. To enter the equation, press o.
3. Enter the equation above using x in place of t. You’ll need to press
yμ to get the exponential and μ for the logarithm.
4. To change the window, press p.
5. Enter the values shown to the right to accommodate a [0,10] by [0, 100]
window.
6. Press s to see the function.
7. To insure that this is indeed the correct graph, let’s check a point
corresponding to the half-life. Press r.
8. Enter the point X = 1 by pressing a ÀÍ.
9. As you would hope, after one half-life, 50 percent of the material is left.
This verifies that you have entered the correct function.
Use Excel to graph your function: You can get a similar graph in Excel. To do this you’ll need to
create two columns of data corresponding to the t and y values on the graph and then make a scatter plot.
1. Open Excel.
2. In cell A1 enter a 0. Press Enter on
your keyboard.
3. In cell A2, enter a 1. Press Enter on
your keyboard.
4. Click on cell A1. While holding the
left mouse button, drag the cursor to
select cells A1 and A2 as shown on
the right.
5. Release the mouse button.
6. Move your mouse over the fill handle.
When your mouse is over the handle,
it will change to a plus.
7. Hold down your left mouse button
and drag the fill downwards. As you
do this, you’ll see Excel indicate what
will be filled in each cell. Cells A1
and A2 establish the pattern of the fill.
Since we listed 0 and 1, the fill will
continue in increments of 1.
8. Continue dragging vertically until you
have filled the column with numbers
from 0 to 10 as shown at right.
Fill handle
9. Click on cell B1. We need to enter the
formula for the first y value. So type
=100*EXP(LN(.5)*
10. Now we need to click on the
appropriate x value. Click on cell A1.
Now finish the formula by typing
another parentheses.You’ll see that
the formula now contains A1 as you
can see in the function bar on the
right.
11. Press Enter on your keyboard to
calculate the cell.
12. Click on cell B1 to select. Move your
mouse over the fill handle.
13. Hold down the left mouse button and
fill the cells as shown to the right.
Notice that every 1 million years, the
percent of the substance drops by half.
14. Now that we have the points for the
exponential function, we need to
select the data. Click on cell A1 and
hold down the left mouse button to
select both columns.
15. From the Insert tab, select Scatter and
then Scatter with Smooth lines as
shown to the right.
Function bar 16. You should see a graph similar to the
one on the right.
17. The series label is not really needed.
Left mouse click on the label to select
it.
18. Press the Delete button on your
keyboard to remove the series label.
19. You can make further modifications
to the graph by clicking on the Design
or Layout tabs that are in the red box
shown to the right.
20. You can copy and paste your final
graph into a Word document.