Select the correct answer.
Each statement describes a transformation of the graph of $y=x$. Which statement correctly describes the graph of $y=x-13$?
A. It is the graph of $y=x$ translated 13 units up.
B. It is the graph of $y=x$ translated 13 units to the left.
C. It is the graph of $y=x$ where the slope is decreased by 13.
D. It is the graph of $y=x$ translated 13 units to the right.
Answer (2)
The graph of y = x − 13 is a transformation of the graph y = x indicating a downward shift of 13 units. This means every point on the original graph is lowered by 13 units. None of the provided options accurately describe this transformation.
;
The equation y = x − 13 represents a transformation of the graph y = x .
Subtracting 13 from x translates the graph vertically downwards by 13 units.
This is equivalent to translating the graph horizontally to the right by 13 units.
Therefore, the correct answer is D: It is the graph of y = x translated 13 units to the right. D
Explanation
Understanding the Problem We are given the equation y = x − 13 and asked to identify the correct transformation of the graph y = x .
Identifying the Transformation The equation y = x − 13 represents a vertical translation of the graph y = x . Subtracting a constant from the function shifts the graph downwards.
Determining the Direction and Magnitude In this case, we are subtracting 13 from x , which means the graph of y = x is translated 13 units downwards. A downward translation is the opposite of an upward translation.
Relating to the Options However, the options do not have 'downwards' as an option. Note that y = x − 13 can also be interpreted as a translation of 13 units to the right. To see why, consider a point ( a , a ) on the graph of y = x . The corresponding point on the graph of y = x − 13 would be ( a + 13 , a ) , since when x = a + 13 , y = ( a + 13 ) − 13 = a . Thus, the point ( a , a ) has been shifted to the right by 13 units to ( a + 13 , a ) .
Conclusion Therefore, the correct answer is that the graph of y = x − 13 is the graph of y = x translated 13 units to the right.
Examples
Understanding transformations of graphs is crucial in many real-world applications. For instance, in physics, if you're tracking the position of an object over time, a transformation might represent a change in the starting point or a shift in the time scale. Similarly, in economics, transformations can model shifts in supply and demand curves due to external factors like taxes or subsidies. Knowing how to interpret these transformations allows you to quickly understand and analyze the underlying relationships.
