3.8

Topic

The Tangent Function

Practice by unit: Trigonometric and Polar Functions

Sub-standards

• 3.8.A.1 — Construct representations of the tangent function using the unit circle. Given an angle of measure in standard position and a unit circle centered at the origin, there is a point, P, where the terminal ray intersects the circle. The tangent function, f ( ) = tan , gives the slope of the terminal ray.
• 3.8.A.2 — Because the slope of the terminal ray is the ratio of the change in the y-values to the change in the x-values between any two points on the ray, the tangent function is also the ratio of the sine function to the cosine function. Therefore, t sinan = cos , where cos 0.
• 3.8.B.1 — Describe key characteristics of the tangent function. Because the slope values of the terminal ray repeat every one-half revolution of the circle, the tangent function has a period of .
• 3.8.B.2 — The tangent function demonstrates periodic asymptotic behavior at input values , for integer values of k, because cos = 0 at those values.
• 3.8.B.3 — The tangent function increases and its graph changes from concave down to concave up between consecutive asymptotes. UNIT 3 The graph of the additive transformation g ( ) = tan + d of the tangent function f ( ) = tan is a vertical translation of the graph of f and the line containing its points of inflection by d units.
• 3.8.C.2 — The graph of the additive transformation g ( ) = tan( + c) of the tangent function f ( ) = tan is a horizontal translation, or phase shift, of the graph of f by −c units.
• 3.8.C.4 — The graph of the multiplicative transformation g ( ) = tan(b ) of the tangent function f ( ) = tan is a horizontal dilation of the graph of f and differs in period by a factor of . If b < 0, the transformation involves a reflection over the y-axis.
• 3.8.C.5 — The graph of y = f ( ) = atan(b( + c)) + d is a vertical dilation of the graph of y = tan by a factor of , has a period of units, is a vertical shift of the line containing the points of inflection of the graph of y = tan by d units, and is a phase shift of −c units. a 1 b UNIT 3 Required Course Content − , 2 2 − , . ( ) 2 2