Let's analyze the transformation given in the options step-by-step:
1. Understanding the Original Graph [tex]\( y = x \)[/tex]:
- This is a straight line with a slope of 1, passing through the origin (0, 0).
2. Analyzing the Transformation [tex]\( y = x - 8 \)[/tex]:
- This equation can be interpreted as taking the graph of [tex]\( y = x \)[/tex] and applying a vertical translation.
- Here, the subtraction of 8 affects the [tex]\( y \)[/tex]-value at every point on the line [tex]\( y = x \)[/tex].
Let's go through each option to see which one correctly describes this transformation:
A. It is the graph of [tex]\( y = x \)[/tex] translated 8 units up:
- If we translate [tex]\( y = x \)[/tex] 8 units up, the equation would be [tex]\( y = x + 8 \)[/tex].
- This is not correct, as our transformation involves subtracting 8, not adding.
B. It is the graph of [tex]\( y = x \)[/tex] where the slope is decreased by 8:
- The equation [tex]\( y = x - 8 \)[/tex] does not change the slope of the line.
- The slope remains 1; only the y-intercept is affected.
- Hence, this option is incorrect.
C. It is the graph of [tex]\( y = x \)[/tex] translated 8 units down:
- Translating a graph downwards means subtracting a value from the [tex]\( y \)[/tex]-values.
- The equation [tex]\( y = x - 8 \)[/tex] represents the graph of [tex]\( y = x \)[/tex] translated 8 units down.
- This matches our transformation perfectly.
D. It is the graph of [tex]\( y = x \)[/tex] translated 8 units to the left:
- Translating the graph to the left involves shifting the [tex]\( x \)[/tex]-values which modifies the equation to [tex]\( y = (x + 8) \)[/tex].
- This does not match our given equation [tex]\( y = x - 8 \)[/tex].
Based on this analysis:
The correct answer is C. It is the graph of [tex]\( y = x \)[/tex] translated 8 units down.
