Question

Graph each function. Label x-axis.​

Answer 1

Answer:

Here's what I get.

Step-by-step 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).

$\boxed{a = 3; b = 4; h = \frac{\pi}{2}; k = -2}}$

$\text{amplitude = 3; period = } \dfrac{\pi}{2}}$

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

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)

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