Function:

Value of A: 1

Value of B: 1

Value of C: 0

Value of D: 0

The graphs of the six basic trigonometric functions can be transformed by adjusting their height or period, or translating them horizontally or vertically.

The amplitude of a sinusoidal trig function (sine or cosine) $$f(x)$$ is it's 'height,' the distance from the average value of the curve to its maximum (or minimum) value. We can change the amplitude of these functions by multiplying the function by a constant $$A$$. The other trig functions (tangent, cotangent, secant, and cosecant) do not have an amplitude, but multiplication will affect their steepness. Note that a negative value of $$A$$ will flip the graph of any function across the $$x$$-axis.

The period of any trig function is the length of one cycle. Sine, cosine, secant, and cosecant all have periods of $$2\pi$$, while tangent and cotangent have periods of $$\pi$$. The period of the trig function $$f(x)$$ can be changed by multiplying $$x$$ by a constant $$B$$ before applying the function.

Subtracting a value $$C$$ from $$x$$ before applying a trig function will translate its graph horizontally. If $$C$$ is positive, the graph will shift to the right by a factor of $$C$$; if $$C$$ is negative, the graph will shift to the left.

Adding a value $$D$$ to a trig function will translate its graph vertically. If $$D$$ is positive, the graph will shift up by a factor of $$D$$; if $$D$$ is negative, the graph will shift down.

Of course, all of these transformations can be applied to a trig function simultaneously. The graph of $A f(Bx - C) + D$ is obtained by stretching the graph of $$f$$ vertically by a factor of $$A$$ and horizontally by a factor of $$B$$, while translating it vertically by a factor of $$D$$ and horizontally by a factor of $$C$$.

## Using the Applet

Choose a trig function to transform from the drop down menu. Then, use the sliders to adjust the values of $$A$$, $$B$$, $$C$$, and $$D$$ to see how the graph changes. The transformed graph will be displayed in black. The blue dots show the original trig function.