The line $y = x$ is transformed into the line $y=\frac{x}{4}+2$. Fill in the blanks to describe the slope and $y$-intercept of the new line.
The slope is $\square$, and the line is shifted $\square$.
flatter
steeper
Answer (2)
The original line y = x has a slope of 1 and a y-intercept of 0.
The transformed line y = 4 x + 2 has a slope of 4 1 and a y-intercept of 2.
Since 4 1 < 1 , the slope of the transformed line is flatter.
The line is shifted up by 2, so the final answer is flatter.
Explanation
Analyze the Original Line The original line is given by the equation y = x . This is a linear equation in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. For the original line, the slope is m = 1 and the y-intercept is b = 0 .
Analyze the Transformed Line The transformed line is given by the equation y = 4 x + 2 . This is also a linear equation in slope-intercept form, y = m x + b . For the transformed line, the slope is m = 4 1 and the y-intercept is b = 2 .
Compare the Slopes Now we compare the slope of the transformed line to the slope of the original line. The original slope is 1, and the transformed slope is 4 1 . Since 4 1 < 1 , the transformed line is flatter than the original line.
Compare the Y-Intercepts Next, we compare the y-intercept of the transformed line to the y-intercept of the original line. The original y-intercept is 0, and the transformed y-intercept is 2. The line is shifted up by 2.
Final Answer Therefore, the slope is flatter, and the line is shifted up.
Examples
Understanding how lines transform is crucial in various fields. For instance, in computer graphics, transformations like scaling and translation are used to manipulate objects on the screen. If you have a line representing an object's edge, knowing how its slope and position change after a transformation helps you accurately render the object. Similarly, in physics, understanding linear transformations can help analyze motion and forces. For example, if you are analyzing the trajectory of a projectile, understanding how transformations affect the line representing the projectile's path can help you predict its landing point.
The slope of the new line is flatter, and the line is shifted up by 2. Specifically, the slope is 4 1 compared to the original slope of 1, and the y-intercept changes from 0 to 2. Thus, the statement can be filled in as: 'The slope is flatter, and the line is shifted up.'
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