Question

Graph each function. Label x-axis.​

Answer 1

Here's what I get.

Explanation:

Question 4

The general equation for a sine function is

y = a sin[b(x - h)] + k

where a, b, h, and k are the parameters.

Your sine wave is

y = 3sin[4(x + π/4)] - 2

Let's examine each of these parameters.

Case 1. a = 1; b = 1; h = 0; k = 0

y = sin x

This is a normal sine curve (the red line in Fig. 1).

(Sorry. I forgot to label the x-axis, but it's always the horizontal axes)

Case 2. a = 3; b = 1; h = 0; k = 0

y = 3sin x

The amplitude changes from 1 to 3.

The parameter a controls the amplitude of the wave (the blue line in Fig. 1).

Case 3. a = 3; b = 1; h = 0; k = 2

y = 3sin x - 2

The graph shifts down two units.

The parameter k controls the vertical shift of the wave (the green line

in Fig. 1).

Case 4. a = 3; b = 4; h = 0; k = 2

y = 3sin(4x) - 2

The period decreases by a factor of four, from 2π to π/2.

The parameter b controls the period of the wave (the purple line in Fig. 2).

Case 5. a = 3; b = 4; h = -π/4; k = 2

y = 3sin[4(x + π/4)] - 2

The graph shifts π/4 units to the left.

The parameter h controls the horizontal shift of the wave (the black dotted line in Fig. 2).

`\boxed{a = 3; b = 4; h = (\pi)/(2); k = -2}}`

`\text{amplitude = 3; period = } (\pi)/(2)}`

`\textbf{Transformations:}\\\text{1. Dilate across x-axis by a scale factor of 3}\\\text{2. Translate down two units}\\\text{3. Dilate across y-axis by a scale factor of } (1)/(4)\\\text{4. Translate left by } (\pi)/(4)`

Question 6

y = -1cos[1(x – π)] + 3

`\boxed{a = -1, b = 1, h = \pi, k = 3}`

`\boxed{\text{amplitude = 1; period = } \pi}`

Effect of parameters

Refer to Fig. 3.

Original cosine: Solid red line

m = -1: Dashed blue line (reflected across x-axis)

k = 3: Dashed green line (shifted up three units)

b = 1: No change

h = π: Orange line (shifted right by π units)

`\textbf{Transformations:}\\\text{1. Reflect across x-axis}\\\text{2. Translate up three units}\\\text{3. Translate right by } \pi`