Use this page to explore how the parameters in the exponential equation[br][br]y = [b]a[/b] * (1 + [b]r[/b] / [b]t[/b]) ^ [[b]t[/b] * (x - [b]h[/b])] + [b]k[/b][br][br]affect the appearance of its graph.[br][br]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:[br]- Make the curve go through the origin?[br]- Make the curve go through the blue point displayed at (4,10)?[br]- Make the curve slope downwards as it goes to the right?[br]- Make the curve horizontal (parallel to the x-axis)?[br]- Make the curve shift vertically without changing its shape at all?[br]- Make the curve shift horizontally without changing its shape at all?[br] [br]The equation being graphed above looks a bit intimidating with all of its parameters:[br] y = [b]a[/b](1 + [b]r[/b]/[b]t[/b])^([b]t[/b]*(x-[b]h[/b])) + [b]k[/b][br][br]However, if we take out the h, k, and t it looks like this:[br] [math] y = a(1 + r)^x[/math][br]which should look a little better! To play around with this version, adjust the sliders as follows:[br] [b] t[/b] = 1[br] [b]h[/b] = 0[br] [b]k[/b] = 0[br]Now adjust the [b]a[/b] and [b]r[/b] sliders to different values and see what affect they have on the curve. Can you explain why each has the effect that it does? [br][br]What happens when one of [b]a[/b] or [b]r[/b] is negative? Why?[br][br]What happens when both [b]a[/b] and [b]r[/b] are negative? Why?[br][br]The constant [b]t[/b] is used when the rate of growth ([b]r[/b]) 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, [b]r[/b]/[b]t[/b] becomes the monthly interest rate, and [b]t[/b]*x becomes the number of months for which interest will be paid (when x is measured in years).[br] y = [b]a[/b] * (1 + [b]r[/b] / [b]t[/b]) ^ [ [b]t[/b] * x ][br][br]What happens to the graph when you change the value of [b]t[/b]? How does this compare to changing the value of [b]a[/b] or [b]r[/b]? Why is there such a difference?[br][br]And now, finally, to [b]h[/b] and [b]k[/b]. What effect do they each have, and why?[br][br]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 [b]a[/b], [b]r[/b], [b]t[/b], [b]h[/b], and [b]k[/b] the equation only has two variables left: x and y.[br][br]If you wish to use other applets similar to this, you may find an index of all my applets here: [url=https://mathmaine.wordpress.com/2010/04/27/geogebra/]https://mathmaine.com/2010/04/27/geogebra/[/url]

Use this page to explore how the parameters of a logarithmic equation of the form[br][br]y = [b]a[/b] * log[ [b]b[/b] * (x - [b]h[/b]) ^ [b]c[/b] ] + [b]k[/b][br][br]affect the appearance of its graph.[br][br]Start by clicking on the green dots below and dragging them to the left or right to vary the value of each parameter, and watch the effect that it has on both the equation and the graph.

Can you:[br]- Make the curve go through the origin?[br]- Make the curve go through the blue point displayed at (2,4)?[br]- Make the curve slope downwards as it goes to the right?[br]- Make the curve horizontal (parallel to the x-axis)?[br]- Make the curve shift vertically without changing its shape at all?[br]- Make the curve shift horizontally without changing its shape at all?[br][br]The function being graphed looks a bit complicated with all the constants:[br][br]y = [b]a[/b] * log[ [b]b[/b] * (x - [b]h[/b]) ^ [b]c[/b] ] + [b]k[/b][br][br]To simplify things a bit, set[br] [b]b[/b] = 1[br] [b]c[/b] = 1[br] [b]h[/b] = 0[br] [b]k[/b] = 0[br][br]You are now looking at a graph of[br]y = [b]a[/b] * log(x)[br]What effect does changing the value of [b]a[/b] have on the graph?[br][br]Now consider the value of [b]b[/b] (please reset [b]a[/b] to 1). If you have been introduced to the laws of logarithms, you will hopefully be able to figure out a way to rewrite[br]y = log([b]b[/b]*x)[br]in a manner that separates [b]b[/b] from x. Once you have rewritten this equation (the log of a product equals...), what effect should the value of [b]b[/b] have on the graph? Check it out and see if you are right.[br][br]Turning to [b]c[/b] (please reset [b]b[/b] to 1), consider the equation[br]y = log (x ^ [b]c[/b])[br]Once again, this can be rewritten using the laws of logarithms (the log of x to a power is equal to...). Once you have rewritten this without [b]c[/b] inside the logarithm, what effect should changing [b]c[/b] have on the graph? Check it out and see if you are right.[br][br]Now that you have determined the effects of [b]a[/b], [b]b[/b], and [b]c[/b], what effects do [b]h[/b] and [b]k[/b] have? Does the way [b]h[/b] and [b]k[/b] appear in the equation remind you of the way they have been used in other equations? Play around with them and see what effect they have.[br][br]Which parameters have very similar effects? You should find that two parameters are redundant... which explains why you don't normally see logarithmic equations written this way! So, which two parameters would you drop to simplify things without compromising your ability to make the resulting equation conform to any logarithmic graph you might encounter?[br][br]If you wish to use other applets similar to this, you may find an index of all my applets here: [url=https://mathmaine.wordpress.com/2010/04/27/geogebra/]https://mathmaine.com/2010/04/27/geogebra/[/url]