1 6 Graphs of Other Trigonometric Functions Part

1. 6 Graphs of Other Trigonometric Functions Part 1

Graph of the Tangent Function Recall that the tangent function is odd, that is tan(-x) = -tan x Therefore, the graph of y = tan x is symmetric with respect to the origin.

Graph of the Tangent Function You also know that tan x = sin x/cos x. Therefore it is undefined when? ? When cos x = 0. This is when x = π/2 and x = - π/2

Lets make a chart and find points for the graph: x -Π -1. 57 -1. 5 -1 0 1 1. 57 2 Π 2 tan x undef. -1255. 8 -14. 1 -1. 56 0 1. 56 14. 1 1255. 8 undef. tan x approaches - ∞ as x approaches –π/2 from the right tan x approaches ∞ as x approaches π/2 from the left

Analyzing the chart: x -Π -1. 57 -1. 5 -1 0 1 1. 5 -1. 57 2 Π 2 tan x undef. -1255. 8 -14. 1 -1. 56 0 1. 56 14. 1 1255. 8 undef. tan x increases forever as it approaches π/2 from the left ∞ ∞ tan x decreases forever as it approaches -π/2 from the right Therefore, the graph has vertical asymptotes at x = π/2 and x = - π/2

Graph of the Tangent Function: Because the period of tangent is π, vertical asymptotes also occur when x = π/2 + nπ

Characteristics of the Tangent Function Vertical asymptotes are where cos x = 0 (odd multiples of π/2) Period π Domain: all x ≠ π/2 + nπ Range (-∞, ∞)

Transformations of y = tan x l Sketching the graph of the form y = a tan(bx – c) is similar to sketching the graph of y = a sin(bx – c). l You will locate key points and identify the intercepts and asymptotes. l Two consecutive asymptotes can be found by solving the equations: bx – c = - π/2 l bx – c = π/2 The midpoint between these two asymptotes is an x-intercept of the graph.

Transformations of y = tan x l After plotting the asymptotes and the x-intercept, plot a few additional points between the two asymptotes. l Sketch one cycle. l Finally, sketch one or two additional cycles to the left and right.

Transformations of y = tan x Remember: l Shifts (vertical & horizontal) are done as the shifts to y = sin x. l Period change (same as to y = sin x, except the original period of tan x is π, not 2π). l Tangent has no defined amplitude, since the graph increases (or decreases) without bound.

Sketch the graph of = tan(x/2) l y First, find the asymptotes: Solve the equations x = -π 2 2 and x = π 2 2 x = -π and x = π Therefore, two consecutive asymptotes occur at x = -π and x = π

Sketch the graph of y = tan(x/2) Between the two asymptotes x = -π and x = π, plot a few points, including the x-intercept. Make a table of x and y values: x -π/2 0 π/2 y = tan(x/2) -1 0 1

x -π/2 y = tan(x/2) -1 0 0 π/2 1 Graph of y = tan(x/2) Asymptotes at x = - π and (π/2, 1) (0, 0) (-π/2, -1) x=π

Graph y = -3 tan (2 x) l Solve the equations: 2 x = -π/2 and 2 x = π/2 x = - π/4 and l l l x = π/4 You can see that two consecutive asymptotes occur at x = - π/4 and x = π/4 Between these asymptotes, plot a few points, including the x-intercept. Make a table of these values: x y = -3 tan(2 x) -π/8 3 0 0 π/8 -3

x -π/8 0 y = -3 tan(2 x) 3 0 π/8 -3 Graph of y = -3 tan(2 x) Asymptotes at x = - π/4 and (-π/8, 3) (0, 0) π 2 -π 2 (π/8, -3) x = π/4

Compare If a > 0, then the graph is increasing between two consecutive asymptotes. l If a < 0, then the graph is decreasing between two consecutive asymptotes. l There is a reflection in the x-axis. y =tan(x/2) y = -3 tan(2 x) l

Classwork: Section 1. 6 Practice Worksheet The Graph of the Tangent Function

Homework: