(pen)o (g)Translation Rule(pen)
(pen)-> A (g)translation(pen) moves a graph without changing its shape.
(pen)-> Write g(x) = f(x-h) + k for parent function f.
(pen)-> h is the horizontal shift; k is the vertical shift.
(pen)-> h > 0 moves (li)right(pen); h < 0 moves (li)left(pen).
(pen)-> k > 0 moves (pu)up(pen); k < 0 moves (pu)down(pen).
(pen)-> Inside signs look opposite; outside signs act the same.
(pen)-> A parent point (u,v) becomes (u+h,v+k).
(pen)-> Translations (g)preserve(pen) the size of the domain and range.
(pen)-> Every point shifts the same way; the graph never flips.
(pen)o (li)Reading Shifts(pen)
(pen)-> g(x) = (x+2)^2 - 4 uses parent f(x) = x^2.
(pen)-> x+2 = x-(-2), so h = -2 and k = -4.
(pen)-> Shift (li)left(pen) 2 and (pu)down(pen) 4; vertex: (-2,-4).
(pen)-> Parent (1,1) becomes (-1,-3).
(pen)-> g(x) = |x-7| + 1 shifts right 7 and up 1.
(pen)-> Its vertex is (7,1); (1,1) becomes (8,2).
(pen)-> Check: g(8) = |8-7| + 1 = 2.
(pen)-> For g(x) = |x-7| + 1, the (li)range(pen) is [1,inf).
(pen)-> The parent range [0,inf) shifts to [-4,inf).
(pen)o (pu)Writing Equations(pen)
(pen)-> Begin with the parent, then insert the requested shifts.
(pen)-> Quadratic right 10 and up 4: h = 10, k = 4.
(pen)-> The equation is g(x) = (x-10)^2 + 4.
(pen)-> The (pu)vertex(pen) moves from (0,0) to (10,4).
(pen)-> Parent (2,4) moves to (12,8).
(pen)-> Check: g(12) = (12-10)^2 + 4 = 8.
(pen)-> Shifts preserve the parabola's width and opening.
(pen)-> Left 10 and up 4 gives h = -10 and k = 4.
(pen)o (br)Quadratic Worked Points(pen)
(pen)-> For g(x) = (x+5)^2 - 3, h = -5 and k = -3.
(pen)-> Translate the parent (li)left(pen) 5 and (pu)down(pen) 3.
(pen)-> (0,0) becomes vertex (-5,-3).
(pen)-> (1,1) becomes (-4,-2).
(pen)-> (2,4) becomes (-3,1).
(pen)-> Check: g(-3) = 2^2 - 3 = 1.
(pen)-> Domain: R. Range: [-3,inf); axis: x = -5.
(pen)-> (br)Symmetry(pen) pairs: (-6,-2) and (-4,-2) share output -2.
(pen)-> The x-intercept: (x+5)^2 = 3, so x = -5 +/- sqrt(3).
(pen)o (hp)Absolute Value Worked Points(pen)
(pen)-> For g(x) = |x+8| - 5, h = -8 and k = -5.
(pen)-> Translate left 8 and down 5; (hp)vertex(pen): (-8,-5).
(pen)-> Parent (-1,1) becomes (-9,-4).
(pen)-> Parent (1,1) becomes (-7,-4).
(pen)-> Domain: R. Range: [-5,inf).
(pen)-> The axis of symmetry is x = -8.
(pen)-> Check: g(-7) = |-7+8| - 5 = -4.
(pen)-> Each pair (hp)equidistant(pen) from x = -8 gives one output.
(pen)-> x-intercepts: |x+8| = 5 gives x = -3 and x = -13.
(pen)o (y)Factoring and Checks(pen)
(pen)-> Read h from a (y)fully factored(pen) inside expression.
(pen)-> With a scale, use b(x-h), not the raw constant.
(pen)-> Example: 2x-6 = 2(x-3), so h = 3, not 6.
(pen)-> This combines horizontal scaling with a shift.
(pen)-> For pure translations, the inside coefficient is 1.
(pen)-> Shift (li)domain(pen) horizontally and (pu)range(pen) vertically.
(pen)-> Verify a vertex and a second point by substitution.
(pen)-> (y)Factored(pen) check: x^2 - 6x + 9 = (x-3)^2, so h = 3, k = 0.