Question

Start with g(x) = sin(x).
Hint: Make sure you got the correct function by graphing it on some electronic gadget.
a) Define gA(x) to change the amplitude of g to 2.
b) Define gB(x) to change the amplitude of g to 1/2.
c) Define gC(x) to change the period to π.
d) Define gD(x) to change the period to 4π

Answer 1

Start with $g(x) = \sin(x)$ .

Hint: Make sure you got the correct function by graphing it on some electronic gadget.

a) Define $\mathrm{gA}(\mathbf{x})$ to change the amplitude of $\mathbf{g}$ to 2.

b) Define $\mathrm{gB}(\mathbf{x})$ to change the amplitude of $\mathbf{g}$ to $1 / 2$ .

c) Define $\mathrm{gC}(\mathbf{x})$ to change the period to $\pi$ .

d) Define $\mathrm{gD}(\mathbf{x})$ to change the period to $4\pi$

Solution:

Use formula $g(x) = A \cdot \sin(\omega x)$

$$\omega = 2 \pi / T$$

$T - \text{period}$

$\operatorname{Sin}(\mathbf{x})$ is $2\pi$ period function

1) Define $\mathrm{gA}(\mathbf{x})$ to change the amplitude of $\mathbf{g}$ to 2. $A = 2$

$$\mathrm {g} (\mathrm {x}) := 2 \sin (\mathrm {x})$$

2) Define $\mathrm{gB}(\mathbf{x})$ to change the amplitude of $\mathbf{g}$ to $1/2$ . $A = 1/2$

$$\mathrm {g} (\mathrm {x}) := \frac {1}{2} \sin (\mathrm {x})$$

3) c) Define $\mathrm{gC}(\mathbf{x})$ to change the period to $\pi$ .

$$\omega = 2 \pi / \pi = 2$$

$\mathrm{g(x)} \gg \sin (2\mathrm{x})$

4) Define $\mathrm{gD}(\mathbf{x})$ to change the period to $4\pi$

$$\omega = 2 \pi / 4 \pi = 1 / 2$$

$\mathrm{g(x)} \gg \sin \left(\frac{1}{2}\mathrm{x}\right)$