Question

How do you reflect a line y = 2x + 1 about y = 1/2x?

a) Rotate it 90 degrees counterclockwise
b) Reflect it over the x-axis
c) Reflect it over the y-axis
d) Rotate it 180 degrees counterclockwise

Answer 1

To reflect a line across another line, adjust the slopes and y-intercepts; the given options (a-d) don't directly apply to the question as it involves a complex reflection not covered by these choices.

The question asks how to reflect the line y = 2x + 1 about the line y = 1/2x. To reflect a line across another line that is not one of the coordinate axes, you should examine the slopes and y-intercepts of both lines. The main answer in 2 lines: the line is neither rotated nor reflected over the x-axis or y-axis; instead it is transformed such that the slopes and y-intercepts are adjusted according to reflection rules. An explanation in 160 words: When reflecting a line over another line, you should try to visualize or draw how the reflection will look like. Since we are not reflecting over the x-axis or y-axis, we can't simply apply options b or c. Also, rotation (options a or d) does not apply here because rotation changes the orientation of the entire plane and does not result in a reflection over a line. When reflecting y=2x+1 over y=1/2x, the slope of the reflected line will be the negative reciprocal of the original slope if the line of reflection is y=x, but since the line of reflection here has a different slope, it's a more complex reflection that involves finding an equation that would make the original and reflected lines equidistant from y=1/2x at every point. Such a transformation maintaining the slope is not given in the options a, b, c or d, thus the reflection cannot be achieved through the given choices.

Answer 2

To reflect the line y = 2x + 1 about y = ½x, we can use the formula for reflecting a point about a line. Using this formula, we can find the reflection of a given point on the line y = 2x + 1 about the line y = ½x and then use these reflections to determine the equation of the reflected line.

Step-by-step explanation:

To reflect the line y = 2x + 1 about y = ½x, we can use the fact that the reflection of a point P(x, y) about a line ax + by + c = 0 is given by:

P'(x', y') = Q + (Q - P), where Q is the foot of the perpendicular from P to the line.

In this case, the line y = 2x + 1 has a slope of 2 and a y-intercept of 1. The line y = ½x has a slope of ½ and passes through the origin.

First, we find the foot of the perpendicular from a point on the line y = 2x + 1 to the line y = ½x. Let's take a point on the line y = 2x + 1, for example, (1, 3):

So, the foot of the perpendicular from (1, 3) to the line y = ½x is (0.8, 0.4).

Now, we can find the reflection of point (1, 3) about the line y = ½x using the formula:

P'(x', y') = Q + (Q - P)

P'(1', 3') = (0.8, 0.4) + ((0.8, 0.4) - (1, 3))

= (0.8, 0.4) + (-0.2, -2.6)

= (0.6, -2.2)

Therefore, the reflection of the line y = 2x + 1 about y = ½x is the line y = -2x + 8.