I’m trying to reflect the point (3, -1) across the line y = 2x + 1, and my brain keeps short-circuiting. With the x-axis or the line y = x I know the little tricks, but as soon as the “mirror” is tilted and doesn’t go through the origin, I’m lost. Do I have to draw a perpendicular to the line and find the midpoint, or is there a standard coordinate method that gets me the reflected point directly? I feel silly because I’m mixing up perpendicular slopes and the intercept and second-guessing every step.
Is there a straightforward way to get the coordinates without graph paper-like a formula or a reliable procedure I can do on a test? Do I need to first shift/rotate the line to make it easier and then undo that, or is there a direct approach?
Quick follow-up: if I’m reflecting a whole triangle over y = 2x + 1, is it always safe to reflect each vertex and then connect them, or is there something about orientation I should watch out for?
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