Answer :
To solve this problem, we need to understand how the graph of [tex]\( y = x \)[/tex] changes when it becomes [tex]\( y = x - 82 \)[/tex].
1. Understand the Transformation: The equation [tex]\( y = x - 82 \)[/tex] indicates a transformation from [tex]\( y = x \)[/tex]. The expression [tex]\(- 82\)[/tex] suggests a vertical shift.
2. Vertical Translation: When you subtract a number from [tex]\( y = x \)[/tex], like in [tex]\( y = x - 82 \)[/tex], it means you are moving the graph down. Specifically, subtracting 82 from [tex]\( y = x \)[/tex] translates the graph 82 units downward. This is because each [tex]\( y \)[/tex]-value on the original line [tex]\( y = x \)[/tex] is reduced by 82, resulting in a downward shift.
3. Select the Correct Option: Now, let's look at the options:
- A. Translated 8 units to the left: This would involve a change to the [tex]\( x \)[/tex]-values, not [tex]\( y \)[/tex]-values. So this is incorrect.
- B. Translated 8 units down: This description involves vertically shifting, which matches our transformation type, but the units are incorrect.
- C. Translated 8 units up: This is also incorrect, as the graph is shifted down, not up.
- D. Slope decreased by 8: The slope of [tex]\( y = x \)[/tex] is 1. The slope isn't changing here; it's the position of the line that's shifted, not its steepness.
Given the provided answer, the graph is actually translated 82 units down. So the correct statement, based on the question's meaning, should be conceptualized correctly as a vertical translation:
Final Answer: The graph of [tex]\( y = x - 82 \)[/tex] is the graph of [tex]\( y = x \)[/tex] translated 82 units down.
1. Understand the Transformation: The equation [tex]\( y = x - 82 \)[/tex] indicates a transformation from [tex]\( y = x \)[/tex]. The expression [tex]\(- 82\)[/tex] suggests a vertical shift.
2. Vertical Translation: When you subtract a number from [tex]\( y = x \)[/tex], like in [tex]\( y = x - 82 \)[/tex], it means you are moving the graph down. Specifically, subtracting 82 from [tex]\( y = x \)[/tex] translates the graph 82 units downward. This is because each [tex]\( y \)[/tex]-value on the original line [tex]\( y = x \)[/tex] is reduced by 82, resulting in a downward shift.
3. Select the Correct Option: Now, let's look at the options:
- A. Translated 8 units to the left: This would involve a change to the [tex]\( x \)[/tex]-values, not [tex]\( y \)[/tex]-values. So this is incorrect.
- B. Translated 8 units down: This description involves vertically shifting, which matches our transformation type, but the units are incorrect.
- C. Translated 8 units up: This is also incorrect, as the graph is shifted down, not up.
- D. Slope decreased by 8: The slope of [tex]\( y = x \)[/tex] is 1. The slope isn't changing here; it's the position of the line that's shifted, not its steepness.
Given the provided answer, the graph is actually translated 82 units down. So the correct statement, based on the question's meaning, should be conceptualized correctly as a vertical translation:
Final Answer: The graph of [tex]\( y = x - 82 \)[/tex] is the graph of [tex]\( y = x \)[/tex] translated 82 units down.
