01 · See the movement
A translation is a slide. Every point moves the same distance in the same direction; the line does not turn, stretch, or bend.
1Move one point.
2Move every point by (h,k).
3Keep the slope.
4Rewrite the equation.
02 · Build the rule
Moving up by k adds k to every output. Moving right by h means the old output at x=0 must now appear at x=h, so replace x with x−h.
g(x) = f(x − h) + k
| Movement | New rule |
|---|---|
| right h | f(x−h) |
| left h | f(x+h) |
| up k | f(x)+k |
| down k | f(x)−k |
For y=mx+b, the translated equation is (g(x)=m(x-h)+b+k=mx+(b-mh+k)). The coefficient of x is still m, proving the slope stays unchanged.
03 · Contest moves
Warm-up
Move y=x+2 up 3: (y=x+5).
Core pattern
Move y=−2x+5 right 4: (y=−2(x−4)+5=−2x+13).
AMC 8
A parallel line to y=½x+4 through (6,1) has equation (y=½x−2).
AMC 10
For y=3x+2 translated by (h,k), the new intercept is (2−3h+k). If it equals 11, then (k−3h=9).
04 · Your turn
Translate y=2x+3 up 5.
y=2x+8.
Translate y=−x+4 right 3.
y=−(x−3)+4=−x+7.
Translate 3x−4y=12 by (2,−3).
Replace x with x−2 and y with y+3: 3x−4y=30.