Question

Sketch the transformed function y = =3cos [1/2 (x + π/4)] - 4

Answer 1

To sketch the transformed function y = 3cos [1/2 (x + π/4)] - 4, start by identifying the parent function as y = cos(x). Then, apply each transformation step by changing the amplitude, period, and shifting the graph horizontally and vertically.

Step-by-step explanation:

To sketch the transformed function y = 3cos [1/2 (x + π/4)] - 4, we can start by identifying the parent function, which is y = cos(x). This function represents the basic cosine curve. The transformation involves multiplying the amplitude (A) by 3, changing the period (T) of the function by dividing the coefficient of x by 1/2, shifting the graph horizontally by π/4 units to the left, and shifting the graph vertically by 4 units downward.

Let's go through each step:

1. The amplitude is multiplied by 3, so the new amplitude is 3.
2. The coefficient of x in the argument of cos(x) is 1/2, which means the period is 2π divided by 1/2, or 4π. So, the new period is 4π.
3. The graph is shifted horizontally by π/4 units to the left. This means we need to subtract π/4 from x in the argument of cos(x).
4. The graph is shifted vertically by 4 units downward, so we subtract 4 from the function.

Combining all these transformations, the transformed function becomes y = 3cos [1/2 (x + π/4)] - 4.

Answer 2

The graph of the transformed function y = 3cos[1/2(x + π/4)] - 4 is added as an attachment

Sketching the graph of the transformed function

From the question, we have the following parameters that can be used in our computation:

y = 3cos[1/2(x + π/4)] - 4

The above function is a cosine function

The parent of the above function is

y = cos(x)

First, the function is horizontally stretched by a factor of 2

So, we have

y = cos(1/2x)

Next, it is translated to the left by π/4

So, we have

y = cos(1/2(x + π/4))

First, the function is vertically stretched by a factor of 3

So, we have

y = 3cos(1/2(x + π/4))

Lastly, it is shifted down by 4 units

So, we have

y = 3cos(1/2(x + π/4)) - 4

Next, we plot the graph using a graphing tool by taking not of the above transformations rules

The graph of the function is added as an attachment