Glossary
Amplitude (a)
For tangent functions, 'a' controls the vertical dilation (stretch or compression) and reflection over the x-axis, affecting the steepness of the graph.
Example:
In y = 2 tan(x), the amplitude value of 2 makes the graph vertically stretched compared to y = tan(x).
General Equation (y = a tan(b(x - c)) + d)
The standard form used to describe transformations of the tangent function, where a, b, c, and d represent different types of shifts and dilations.
Example:
Analyzing y = 3 tan(2(x - π/4)) + 1 requires understanding each parameter in the general equation to accurately sketch its graph.
Period (T = π/b)
The formula used to calculate the horizontal length of one complete cycle of a transformed tangent function, where 'b' is the coefficient of x.
Example:
For y = tan(4x), the period (T = π/b) is π/4, meaning the graph completes a cycle much faster.
Period of π
The horizontal distance over which the tangent function's graph completes one full cycle before repeating, which is π radians.
Example:
The graph of y = tan(x) repeats its pattern every period of π, meaning tan(x) = tan(x + π).
Phase Shift (c)
The horizontal translation of the tangent function's graph, determined by the 'c' value in the general equation.
Example:
The function y = tan(x - π/2) has a phase shift of π/2 units to the right, moving its vertical asymptotes accordingly.
Range of Tangent
The set of all possible output values (y-values) for the tangent function, which spans from negative infinity to positive infinity, denoted as (-∞, ∞).
Example:
No matter how steep a slope is, the range of tangent can represent it, from infinitely negative to infinitely positive.
Tangent Function (tan(θ))
A trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle, or y/x on the unit circle, representing the slope of the terminal ray.
Example:
If a ramp has a vertical rise of 3 feet and a horizontal run of 4 feet, the tangent function of the angle of elevation is 3/4.
Unit Circle
A circle with a radius of 1 centered at the origin, used to define trigonometric functions for any angle based on the coordinates (x, y) corresponding to (cos(θ), sin(θ)).
Example:
To find sin(π/2), we locate π/2 on the unit circle and observe that its y-coordinate is 1.
Vertical Asymptotes
Vertical lines on the graph of a function where the function's value approaches positive or negative infinity, occurring for tangent when cos(θ) = 0 (at θ = π/2 + kπ).
Example:
The graph of y = tan(x) has vertical asymptotes at x = π/2, 3π/2, and so on, indicating where the function is undefined.
Vertical Shift (d)
The vertical translation of the tangent function's graph, determined by the 'd' value in the general equation, moving the entire graph up or down.
Example:
In y = tan(x) + 5, the graph is moved 5 units up due to the vertical shift, raising its center line.