Graph each function for one period, and show (or specify) the intercepts and asymptotes. (a) y=t

step 1 All right, we'll come back. So for this problem, we're going to be grapping the functions. Y is

Graph each function for one period, and show (or specify)...

Key Concepts

Period of Tangent Functions

The period of a trigonometric function is the length along the x-axis over which the function repeats itself. For the standard tangent function, this period is ?; however, when the input is scaled by a factor (as in tan(x/3)), the period becomes ? divided by the absolute value of that factor. Thus, for tan(x/3), the period is 3?, meaning the pattern of the graph repeats every 3? units along the x-axis.

Function Transformations

Transformations in functions involve operations such as horizontal scaling and reflections, which affect the graph's shape and orientation. In tan(x/3), the division of x by 3 results in a horizontal stretch of the graph by a factor of 3, thereby increasing the period. Additionally, when the function is multiplied by -1, as in -tan(x/3), it reflects the graph over the x-axis, inverting the values of the function while preserving the period and asymptotic structure.

Vertical Asymptotes in Trigonometric Functions

Vertical asymptotes are lines where the function is undefined and the graph tends to infinity. In the standard tangent function, these asymptotes occur at x = ±?/2 + k?, where k is an integer. With a horizontal scaling such as tan(x/3), the asymptotes shift accordingly to x = ±(3?/2) + 3k?. These asymptotes are key in graphing the function as they delineate regions over which the function completes its cycle.

Intercepts in Trigonometric Functions

The intercepts of a function are the points where it crosses the axes. For tangent functions, the x-intercepts occur where the output equals zero, notably at multiples of the period (e.g., for tan(x/3), at x = 0, ±3?, etc.). The y-intercept, which is the point where x equals zero, is also determined by evaluating the function at x = 0. Since tan(0) = 0, the graph of tan(x/3) and its reflection -tan(x/3) both pass through the origin.

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