Understanding the tangent function | AP®︎/College Precalculus | Khan Academy

Overview

Khan Academy explains the tangent function by first graphing y = tangent(theta), illustrating its definition as the slope of a terminal ray and its vertical asymptotes at pi/2 + n*pi. The explanation then details transformations: multiplying theta by a coefficient (e.g., 3*theta) compresses the period, multiplying the function by a coefficient (e.g., -3*tangent(theta)) scales it vertically and reflects it across the x-axis, and adding constants shifts it horizontally (e.g., theta - pi/4) and vertically (e.g., +2).

Key takeaways

The tangent function, tangent(theta), is fundamentally defined by the slope of the angle's terminal ray.
Vertical asymptotes for tangent(theta) occur at theta = pi/2 + n*pi, where n is an integer.
The period of the tangent function is pi.
Multiplying theta by a coefficient 'b' (tangent(b*theta)) compresses the period to pi/|b|.
Multiplying the tangent function by a coefficient 'a' (a*tangent(theta)) scales the graph vertically by a factor of |a| and reflects it across the x-axis if 'a' is negative.
Horizontal shifts are achieved by replacing theta with (theta - h), moving the graph 'h' units to the right.
Vertical shifts are achieved by adding a constant 'k' to the function, moving the graph 'k' units up.

Chapters

0:00 Graphing the Basic Tangent Function (y = tangent(theta))

Defines tangent(theta) as the slope of the terminal ray of an angle theta.
Plots key points: tangent(0)=0, tangent(pi/4)=1.
Identifies vertical asymptotes at theta = pi/2 and theta = -pi/2, approaching +/- infinity.

6:57 Transformations: Period Compression and Vertical Scaling/Reflection

Multiplying theta by a coefficient (e.g., 3*theta) compresses the period by dividing it by that coefficient.
A negative coefficient (e.g., -tangent(theta)) reflects the graph across the x-axis.
A vertical scalar (e.g., -3*tangent(theta)) scales the graph vertically, changing the magnitude of y-values.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Khan Academy.