Answer :
Let's start with the original function [tex]\( f(x) = x \)[/tex]. We will apply the transformations sequentially as described.
### Step 1: Horizontal Shift of 7 Units Left
A horizontal shift to the left by 7 units means that we replace [tex]\( x \)[/tex] with [tex]\( x + 7 \)[/tex] in our function. Thus, we transform:
[tex]\[ f(x) = x \quad \text{to} \quad f(x) = x + 7 \][/tex]
### Step 2: Reflection Across the x-axis
Reflecting the function across the x-axis means we take the negative of the function. Therefore, we transform:
[tex]\[ f(x) = x + 7 \quad \text{to} \quad f(x) = -(x + 7) = -x - 7 \][/tex]
### Step 3: Vertical Stretch of 3 Units Away from the x-axis
A vertical stretch by a factor of 3 means we multiply the entire function by 3. Therefore, we transform:
[tex]\[ f(x) = -x - 7 \quad \text{to} \quad f(x) = 3(-x - 7) = -3x - 21 \][/tex]
### Step 4: Vertical Shift of -11 Units
A vertical shift down by 11 units means we subtract 11 from our function. Therefore, we transform:
[tex]\[ f(x) = -3x - 21 \quad \text{to} \quad f(x) = -3x - 21 - 11 = -3x - 32 \][/tex]
### Final Result
After applying all the transformations, the resulting formula for the transformed function is:
[tex]\[ f(x) = -3x - 32 \][/tex]
### Step 1: Horizontal Shift of 7 Units Left
A horizontal shift to the left by 7 units means that we replace [tex]\( x \)[/tex] with [tex]\( x + 7 \)[/tex] in our function. Thus, we transform:
[tex]\[ f(x) = x \quad \text{to} \quad f(x) = x + 7 \][/tex]
### Step 2: Reflection Across the x-axis
Reflecting the function across the x-axis means we take the negative of the function. Therefore, we transform:
[tex]\[ f(x) = x + 7 \quad \text{to} \quad f(x) = -(x + 7) = -x - 7 \][/tex]
### Step 3: Vertical Stretch of 3 Units Away from the x-axis
A vertical stretch by a factor of 3 means we multiply the entire function by 3. Therefore, we transform:
[tex]\[ f(x) = -x - 7 \quad \text{to} \quad f(x) = 3(-x - 7) = -3x - 21 \][/tex]
### Step 4: Vertical Shift of -11 Units
A vertical shift down by 11 units means we subtract 11 from our function. Therefore, we transform:
[tex]\[ f(x) = -3x - 21 \quad \text{to} \quad f(x) = -3x - 21 - 11 = -3x - 32 \][/tex]
### Final Result
After applying all the transformations, the resulting formula for the transformed function is:
[tex]\[ f(x) = -3x - 32 \][/tex]
