Rotating Lines What is the general rule for rotating the line y=mx+b 90 degrees, 180 degrees, and 270 degrees?

Check out the reference chart below for a summary of each rotation equation. It is attached as an image.

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Explanation:

In each rotation, I'm assuming the center of rotation is the origin.

For 180 degree rotations, the line stays the same. This is because a line is a combination of two rays that point in opposite directions (eg: one ray points directly northwest and the other ray points directly southeast).

So effectively it's like starting to face the northwest direction, rotating 180 degrees to face the southeast direction, and you're on the same line when all is said and done.

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For 90 and 270 degree rotations, it's a bit trickier. I'll go over counterclockwise rotations of each.

Let's start with a 90 degree counterclockwise rotation.

The slope of y = mx+b is m

The negative reciprocal of this is -1/m. We flip the fraction and the sign from positive to negative.

When doing a 90 degree counterclockwise rotation, the point (0,b) moves to (-b,0). It doesn't matter if b is positive or negative or zero.

Notice the y intercept of the original equation (b) is now connected to the x intercept of the answer.

We'll use this to find the y intercept

y-y1 = m(x-x1)

y-0 = (-1/m)(x - (-b))

y = (-1/m)(x + b)

y = (-1/m)x - b/m

this is the equation for 90 degree counterclockwise rotations. It also applies to 270 degree clockwise rotations.

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Now onto 270 degree counterclockwise rotations

Once again, the center of rotation is assumed to be the origin.

Like before, the old slope m becomes -1/m so that the lines are perpendicular.

So far, everything appears to be the same as the last section. However, the y intercept will be different.

The point (0,b) rotates to (b,0) when doing a 270 degree counterclockwise rotation.

We'll use the point slope formula to find the equation we're after.

y - y1 = m(x - x1)

y - 0 = (-1/m)(x - b)

y = (-1/m)(x - b)

y = (-1/m)x + b/m

this is the equation for 270 degree counterclockwise rotations. It also works for 90 degree clockwise rotations.

So the only real difference is the -b/m changed to +b/m.

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The chart summarizing everything is posted below.

shift 4 units up, and 1 unit left.

it will translate f(x) = 3^x to g(x) = 3^(x+1) + 4