Question

Use equation A and describe the difference to show the change from f(x) =xto the equation A
f(x)= 3x-4

Answer 1

To show the change from the function \( f(x) = x \) to the new function \( f(x) = 3x - 4 \) using a general equation format, we can analyze the transformation involved in changing from one equation to the other.

1. **Initial Function**: Start with the simple linear function:
\[
f(x) = x
\]

2. **Target Function**: The new function is:
\[
f(x) = 3x - 4
\]

### Steps to Identify Differences:

- **Scaling**: The coefficient of \( x \) increases from \( 1 \) in the original function to \( 3 \) in the new function. This means the new function stretches the graph vertically by a factor of \( 3 \). Every \( y \)-value of the original function will be multiplied by \( 3 \).

- **Vertical Translation**: The equation \( 3x - 4 \) also includes a constant term of \( -4 \). This means that after scaling the output by \( 3 \), we then shift the entire graph downward by \( 4 \) units.

### Summary of the Transformations:
To summarize, the transformation from \( f(x) = x \) to \( f(x) = 3x - 4 \) involves two key changes:

1. **Vertical Scaling**: The graph is vertically stretched by a factor of \( 3 \).
2. **Vertical Shift**: The graph is shifted downwards by \( 4 \) units.

### Final Interpretation:
Thus, the complete transformation can be expressed as:
\[
f(x) = 3(x) - 4 \quad \text{(stretch by 3 and shift down by 4)}
\]

This process clearly illustrates how the original function \( f(x) = x \) is modified to reach the new function \( f(x) = 3x - 4 \).