AP PreCalculus Flashcards: The Tangent Function
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 14 cards to help you master important concepts.
Why is the period of the tangent function $\\pi$ instead of $2\\pi$?
The slope values of the terminal ray on the unit circle repeat every one-half revolution ($\\pi$ radians), which is faster than the sine or cosine values.
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Why is the period of the tangent function $\\pi$ instead of $2\\pi$?
The slope values of the terminal ray on the unit circle repeat every one-half revolution ($\\pi$ radians), which is faster than the sine or cosine values.
How is the period of a transformed tangent function, $y = a \\tan(b(\\theta+c)) + d$, calculated?
The period of a transformed tangent function is calculated using the formula $|\\frac{\\pi}{b}|$.
What is meant by the tangent function's 'periodic asymptotic behavior'?
This describes how the function's graph approaches vertical asymptotes at regular, repeating intervals (the period).
What is the mathematical reason for the vertical asymptotes in the tangent function?
Vertical asymptotes occur where the function is undefined, which is when its denominator, $\\cos \\theta$, is equal to 0.
What is the definition of the tangent function as a ratio of other trigonometric functions?
The tangent function is the ratio $\\frac{\\sin \\theta}{\\cos \\theta}$, for all values where $\\cos \\theta \ne 0$.
What is the period of the parent tangent function, $y = \\tan \\theta$?
The period of the tangent function is $\\pi$.
In the general form $y = a \\tan(b(\\theta+c)) + d$, what transformation is caused by the parameter 'c'?
The parameter 'c' causes a horizontal phase shift of the graph by $-c$ units.
What is the period of the function $y = 5 \\tan(2\\theta)$?
Using the formula $|\\frac{\\pi}{b}|$, the period is $|\\frac{\\pi}{2}| = \\frac{\\pi}{2}$ units.
Identify the vertical shift and vertical dilation for the function $y = -3 \\tan(\\theta) + 4$.
The function has a vertical dilation (stretch) by a factor of 3 and a vertical shift up by 4 units.
How is the tangent function, $f(\\theta) = \\tan \\theta$, defined in relation to the unit circle?
The tangent function gives the slope of the terminal ray for a given angle $\\theta$ on the unit circle.
Describe the phase shift for the function $y = \\tan(\\theta + \\frac{\\pi}{6})$.
The phase shift is $-c$, so it is $-\\frac{\\pi}{6}$ units, which represents a shift to the left.
In the general form $y = a \\tan(b(\\theta+c)) + d$, what transformation is caused by the parameter 'd'?
The parameter 'd' causes a vertical shift of the graph by $d$ units, moving the line that contains the points of inflection.
At which values of $\\theta$ does the tangent function have vertical asymptotes?
The tangent function has vertical asymptotes at input values of $\\theta = \\frac{\\pi}{2} + k\\pi$, for all integer values of k.
In the general form $y = a \\tan(b(\\theta+c)) + d$, what transformation is caused by the parameter 'a'?
The parameter 'a' causes a vertical dilation (stretch or compression) of the graph by a factor of $|a|$.