To analyze the change from the function \( f(x) = x \) to the equation \( f(x) = 3x - 4 \), we can break down the transformations involving slope and vertical shifts.
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Change in Slope:
- The original function \( f(x) = x \) has a slope of 1 (since it can be expressed as \( f(x) = 1x + 0 \)).
- In Equation A, \( f(x) = 3x - 4 \), the slope is 3.
- Therefore, the slope increases from 1 to 3, which is an increase by a factor of 3.
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Change in Vertical Position:
- The original function \( f(x) = x \) has a y-intercept of 0 (the graph intersects the y-axis at the point (0,0)).
- In Equation A, the y-intercept is -4 (the graph intersects the y-axis at the point (0,-4)).
- This means the entire graph of the function moves down the y-axis by 4 units.
Combining these observations, we can conclude that:
The correct response is: The slope increases by a factor of 3 and the graph of the function moves down the y-axis 4 units.