Write an equation for the following transformation of y =x; a vertical compression by a factor of 1/4
a. y=-4x
b. y= -1/4x
c. y = 4x
d. y = 1/4x
The Correct Answer and Explanation is :
The correct answer is d. y = 1/4x.
Explanation:
The transformation described in the problem is a vertical compression by a factor of 1/4. Let’s break down the components to understand how this transformation affects the graph of the function.
- The Original Function:
The function given is y = x, which represents a straight line with a slope of 1. In this case, for every increase of 1 unit in x, y increases by 1 unit. - Vertical Compression:
A vertical compression by a factor of 1/4 means that the graph of the function is “squished” vertically, making it flatter. For each value of x, the corresponding y value is multiplied by 1/4, meaning the values of y become one-quarter of their original values for any given x. - Mathematical Effect:
To apply a vertical compression by a factor of 1/4, we multiply the entire function y = x by 1/4:
[
y = \frac{1}{4}x
]
This equation represents a line with the same slope as the original line, but the y-values are scaled down by a factor of 1/4. The line will still pass through the origin, but it will be closer to the x-axis compared to the original line y = x. - Comparison with Other Options:
- a. y = -4x: This is a vertical stretch by a factor of 4, and it also has a negative slope, which is not the transformation described.
- b. y = -1/4x: This represents a vertical compression by a factor of 1/4, but with a negative slope. The negative sign is not part of the described transformation.
- c. y = 4x: This represents a vertical stretch by a factor of 4, which is the opposite of the transformation described.
- d. y = 1/4x: This is the correct transformation, as it represents a vertical compression by a factor of 1/4 with a positive slope.
Thus, the correct transformation of y = x with a vertical compression by a factor of 1/4 is y = 1/4x.