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