Exponential Functions
Use this page to explore how the parameters in the exponential equation
y = a * (1 + r / t) ^ [t * (x - h)] + k
affect the appearance of its graph.
Start by dragging the green dots to change the settings of each slider below, and watch the effect in both the equation and the graph.
Can you:
- Make the curve go through the origin?
- Make the curve go through the blue point displayed at (4,10)?
- Make the curve slope downwards as it goes to the right?
- Make the curve horizontal (parallel to the x-axis)?
- Make the curve shift vertically without changing its shape at all?
- Make the curve shift horizontally without changing its shape at all?
The equation being graphed above looks a bit intimidating with all of its parameters:
y = a(1 + r/t)^(t*(x-h)) + k
However, if we take out the h, k, and t it looks like this:
which should look a little better! To play around with this version, adjust the sliders as follows:
t = 1
h = 0
k = 0
Now adjust the a and r sliders to different values and see what affect they have on the curve. Can you explain why each has the effect that it does?
What happens when one of a or r is negative? Why?
What happens when both a and r are negative? Why?
The constant t is used when the rate of growth (r) applies to a smaller time period (weekly or monthly, instead of annually). For example, if I wish to graph the value of my bank account over a series of years, but the bank pays interest monthly , then by setting t = 12, r/t becomes the monthly interest rate, and t*x becomes the number of months for which interest will be paid (when x is measured in years).
y = a * (1 + r / t) ^ [ t * x ]
What happens to the graph when you change the value of t? How does this compare to changing the value of a or r? Why is there such a difference?
And now, finally, to h and k. What effect do they each have, and why?
This equation can model exponential growth or decay, at a variety of growth/decay rates, for a variety of initial balances/populations, and be easily translated horizontally and vertically to match initial conditions. While there are many letters in the generic form of the equation, once you have used the sliders to set values for a, r, t, h, and k the equation only has two variables left: x and y.
If you wish to use other applets similar to this, you may find an index of all my applets here: https://mathmaine.com/2010/04/27/geogebra/