Use this page to explore the effect that each parameter of the Cosine function
has using the green sliders below.
Note that the slider value you choose appears in the equation shown at the top right of the graph.

Once you have a sense of the effect each slider has, can you:
- Change the period from 2π to π?
- Change the amplitude from 1 to 0.5?
- Move the entire curve up by 0.5?
- Shift the entire curve left by π/2?
- How many different ways can you get the curve to pass through the point at (-3π/2, -2)?
Which parameters worked the way you expected them to, and which did not?
Of the parameters whose effect surprised you, can you figure out why they did not quite do what you expected them to?
Note that h and k work exactly the way they do with the Point-Slope form of the equation of a line, or the Vertex form of the equation of a parabola: h is a horizontal translation, and k is a vertical translation. Why does changing the value of k cause a vertical translation?
The constant a modifies the amplitude of the function. Why does it have this effect?
Why does changing the value of h cause a horizontal translation?
And lastly, and this is a little more subtle, can you figure out and explain why and how b changes the period of the function the way it does?
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/