Certainly! Let's go through the problem step by step to find the formula for the transformed function.
### Step 1: Vertical Stretch
The original function is [tex]\( y = \cos(x) \)[/tex]. When a function is vertically stretched by a factor of 2, you multiply the function by 2. Thus, the new function becomes:
[tex]\[ y = 2 \cos(x) \][/tex]
### Step 2: Horizontal Shift
Next, we need to shift the graph 11 units to the left. Horizontal shifts are achieved by adding or subtracting a constant to the input variable [tex]\(x\)[/tex]. To shift the graph left by 11 units, we replace [tex]\(x\)[/tex] with [tex]\(x + 11\)[/tex]. Thus, the function now becomes:
[tex]\[ y = 2 \cos(x + 11) \][/tex]
### Step 3: Vertical Translation
Lastly, we translate the graph 15 units upward. A vertical translation is done by adding a constant to the entire function. Adding 15 to [tex]\(y\)[/tex] results in:
[tex]\[ y = 2 \cos(x + 11) + 15 \][/tex]
Thus, the formula for the function whose graph undergoes these transformations is:
[tex]\[ f(x) = 2 \cos(x + 11) + 15 \][/tex]
So, the resultant function is:
[tex]\[ f(x) = 2 \cos(x + 11) + 15 \][/tex]
This is the final formula describing the graph after it has been vertically stretched by a factor of 2, shifted 11 units to the left, and translated 15 units upward.
