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.
About this Applet