Math / Trigonometry

Question. Use Technology a) Predict the form of the graph of $y=\tan x+2$ .
Verify your prediction using graphing technology. b) Predict the form of the graph of $y=3 \tan x$ .
Verify your prediction using graphing 24. technology. c) Predict the form of the graph of $y=\tan \left(x-\frac{\pi}{4}\right)$ . Verify your prediction using graphing technology. d) Predict the form of the graph of $y=\tan 3 x$ . Verify your prediction using graphing technology.

Studdy Solution

STEP 1

1. We are working with transformations of the basic tangent function $y = \tan x$ .
2. The tangent function has a period of $\pi$ and vertical asymptotes at $x = \frac{\pi}{2} + k\pi$ for integer $k$ .
3. Transformations include vertical shifts, vertical stretches/compressions, horizontal shifts, and changes in period.

STEP 2

1. Predict the form of the graph of $y = \tan x + 2$ .
2. Predict the form of the graph of $y = 3 \tan x$ .
3. Predict the form of the graph of $y = \tan \left(x - \frac{\pi}{4}\right)$ .
4. Predict the form of the graph of $y = \tan 3x$ .

STEP 3

a) Predict the form of the graph of $y = \tan x + 2$ :
- The graph of $y = \tan x$ is vertically shifted upwards by 2 units. - The vertical asymptotes remain unchanged at $x = \frac{\pi}{2} + k\pi$ .

STEP 4

b) Predict the form of the graph of $y = 3 \tan x$ :
- The graph of $y = \tan x$ is vertically stretched by a factor of 3. - The vertical asymptotes remain unchanged at $x = \frac{\pi}{2} + k\pi$ . - The amplitude of the tangent graph is not defined, but the steepness of the graph increases.

STEP 5

c) Predict the form of the graph of $y = \tan \left(x - \frac{\pi}{4}\right)$ :
- The graph of $y = \tan x$ is horizontally shifted to the right by $\frac{\pi}{4}$ units. - The vertical asymptotes shift to $x = \frac{\pi}{2} + \frac{\pi}{4} + k\pi$ .

STEP 6

d) Predict the form of the graph of $y = \tan 3x$ :
- The period of the tangent function changes from $\pi$ to $\frac{\pi}{3}$ . - The vertical asymptotes are now at $x = \frac{\pi}{6} + \frac{k\pi}{3}$ .
Verify each prediction using graphing technology to ensure the transformations are accurately represented.

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