The graph of the function 𝑦 = 𝑠𝑖𝑛π‘₯ is transformed by vertically compressing it by a factor of 1/3, reflecting it in the x-axis, horizontally translating it πœ‹/2 units to the left and vertically translating it 15 units down. Write the equation of the resulting graph.

vertically compressing it by a factor of 1/3

y = 1/3 sin(x)

reflecting it in the x-axis
y = -1/3 sin(x)

and so on
see what you can do with the rest.

Well, well, well, fancy transforming a sine function, huh? Don't worry, I got you covered, my friend!

Let's break this down step by step. First, we vertically compress it by a factor of 1/3. So, we multiply the original function by 1/3: 𝑦 = (1/3)𝑠𝑖𝑛π‘₯.

Next, we reflect it in the x-axis. That means we multiply the whole thing by -1. So, we have 𝑦 = -(1/3)𝑠𝑖𝑛π‘₯.

Now, time for the horizontal translation πœ‹/2 units to the left. To perform this translation, we replace π‘₯ with (π‘₯ + πœ‹/2): 𝑦 = -(1/3)𝑠𝑖𝑛(π‘₯ + πœ‹/2).

Lastly, we vertically translate it 15 units down. This means we subtract 15 from the whole thing. Voila! The final equation of the resulting graph is:

𝑦 = -(1/3)𝑠𝑖𝑛(π‘₯ + πœ‹/2) - 15.

There you have it, the gender-bending, horizontally traveling, vertically challenged sine function! Hope this brings a smile to your face!

To transform the graph of 𝑦 = 𝑠𝑖𝑛π‘₯, we can apply the given transformations step-by-step.

1. Vertically compressing by a factor of 1/3:
To vertically compress a graph, we multiply the function by the compression factor.
The new function becomes 𝑦 = (1/3) sinπ‘₯.

2. Reflecting in the x-axis:
To reflect a graph in the x-axis, we take the negative of the function.
The new function becomes 𝑦 = - (1/3) sinπ‘₯.

3. Horizontally translating πœ‹/2 units to the left:
To horizontally translate a graph, we subtract the translation value from the argument.
The new function becomes 𝑦 = - (1/3) sin(π‘₯ - πœ‹/2).

4. Vertically translating 15 units down:
To vertically translate a graph, we add/subtract the translation value to the function.
The new function becomes 𝑦 = - (1/3) sin(π‘₯ - πœ‹/2) - 15.

Therefore, the equation of the resulting graph after all the transformations is 𝑦 = - (1/3) sin(π‘₯ - πœ‹/2) - 15.

To find the equation of the resulting graph, let's break down the transformation step by step.

1. Vertically compressing by a factor of 1/3:
This means that the amplitude of the graph is reduced by a factor of 1/3. Since the amplitude of the sine function is 1, the new amplitude will be 1/3 times 1. Therefore, the amplitude of the transformed graph is 1/3.

2. Reflecting it in the x-axis:
Reflecting the graph in the x-axis changes the sign of the function. So, the transformed graph has a negative sign, which means it is flipped vertically.

3. Horizontally translating it πœ‹/2 units to the left:
A horizontal translation moves the entire graph left or right. Since we are moving πœ‹/2 units to the left, we subtract πœ‹/2 from the original function. Therefore, the transformed graph can be represented by sin(x - πœ‹/2).

4. Vertically translating it 15 units down:
A vertical translation moves the entire graph up or down. Since we are moving 15 units down, we subtract 15 from the original function. Therefore, the transformed graph can be represented by sin(x - πœ‹/2) - 15.

Combining all the transformations, the equation of the resulting graph is:
𝑦 = -1/3 * sin(x - πœ‹/2) - 15