The line $y = x$ is transformed into the line $y=\frac{x}{3}-4$. Fill in the blanks to describe the slope and $y$-intercept of the new line.
The slope is ______, and the line is shifted ______.
A. steeper
B. flatter
Answer (2)
The slope of the original line y = x is 1, and the slope of the transformed line y = 3 x − 4 is 3 1 . Since 3 1 < 1 , the line becomes flatter.
The y-intercept of the original line is 0, and the y-intercept of the transformed line is -4, indicating a downward shift.
The slope is flatter.
The line is shifted down. f l a tt er and s hi f t e d .
Explanation
Understanding the Transformation We are given the original line y = x and the transformed line y = 3 x − 4 . We need to describe how the slope and y-intercept of the transformed line have changed compared to the original line.
Identifying Slopes and Intercepts The original line y = x has a slope of 1 and a y-intercept of 0. The transformed line y = 3 x − 4 has a slope of 3 1 and a y-intercept of -4.
Comparing Slopes To determine if the transformed line is steeper or flatter, we compare its slope to the original line's slope. Since 3 1 < 1 , the transformed line is flatter than the original line.
Determining the Shift The y-intercept of the transformed line is -4. This means the line has been shifted down by 4 units.
Final Answer Therefore, the slope is flatter, and the line is shifted down.
Examples
Understanding slope and y-intercept transformations is crucial in various real-world scenarios. For instance, consider adjusting the settings on a treadmill. The slope setting affects the steepness of your workout, while the speed setting influences your pace. Similarly, in economics, understanding how changes in tax rates (slope) and government spending (y-intercept) affect the overall economy is essential for policymakers. By analyzing these transformations, we can make informed decisions and optimize outcomes in diverse fields.
The slope of the transformed line y = 3 x − 4 is flatter than the original line y = x . The line has also been shifted down by 4 units. Thus, the slope is flatter and the line is shifted down.
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