Answer :
To determine which transformation was not done to convert the linear parent function \( f(x) = x \) to the function \( g(x) = -5(x + 3) - 8 \), let's analyze each component of the transformation step-by-step.
1. Inside the Parentheses \((x+3)\):
- The expression \( (x+3) \) indicates a horizontal shift. When we add 3 to \( x \), it translates the graph to the left by 3 units.
- Shift to the left by 3 units.
2. Multiplication by a Negative Coefficient and a Factor \(-5\):
- The multiplication by -5 outside the parentheses performs two operations on the graph:
- The negative sign indicates a reflection over the \( x \)-axis.
- The factor 5 represent a vertical stretch by a factor of 5.
- Reflection over the \( x \)-axis.
- Vertical stretch by a factor of 5.
3. Subtraction Outside the Parentheses (−8):
- Subtraction of 8 outside the parentheses indicates a vertical shift downward by 8 units.
- Shift down by 8 units.
Given the transformations, let's match them with the provided options:
A. Shift right 3 units:
- From our analysis, the function \( (x + 3) \) shifts the graph left by 3 units, not right. So, shift right by 3 units is not performed.
B. Vertical stretch by a factor of 5:
- This transformation is indeed done, as indicated by the coefficient 5. So, vertical stretch by a factor of 5 is performed.
C. Reflection over the \( x \)-axis:
- This transformation is also performed, as indicated by the negative sign. So, reflection over the \( x \)-axis is performed.
D. Shift down 8 units:
- The subtraction of 8 indicates that this transformation is done. So, shift down by 8 units is performed.
Based on the analysis, the transformation that was not done is:
A. Shift right 3 units.
1. Inside the Parentheses \((x+3)\):
- The expression \( (x+3) \) indicates a horizontal shift. When we add 3 to \( x \), it translates the graph to the left by 3 units.
- Shift to the left by 3 units.
2. Multiplication by a Negative Coefficient and a Factor \(-5\):
- The multiplication by -5 outside the parentheses performs two operations on the graph:
- The negative sign indicates a reflection over the \( x \)-axis.
- The factor 5 represent a vertical stretch by a factor of 5.
- Reflection over the \( x \)-axis.
- Vertical stretch by a factor of 5.
3. Subtraction Outside the Parentheses (−8):
- Subtraction of 8 outside the parentheses indicates a vertical shift downward by 8 units.
- Shift down by 8 units.
Given the transformations, let's match them with the provided options:
A. Shift right 3 units:
- From our analysis, the function \( (x + 3) \) shifts the graph left by 3 units, not right. So, shift right by 3 units is not performed.
B. Vertical stretch by a factor of 5:
- This transformation is indeed done, as indicated by the coefficient 5. So, vertical stretch by a factor of 5 is performed.
C. Reflection over the \( x \)-axis:
- This transformation is also performed, as indicated by the negative sign. So, reflection over the \( x \)-axis is performed.
D. Shift down 8 units:
- The subtraction of 8 indicates that this transformation is done. So, shift down by 8 units is performed.
Based on the analysis, the transformation that was not done is:
A. Shift right 3 units.
