o Combined Rule
-> Use g(x) = a*f(b*(x-h)) + k with a,b != 0.
-> Map parent (u,v) to (u/b+h,a*v+k).
-> Divide by b; do not multiply the input by b.
-> Dilation and reflection come before the shifts.
-> a controls vertical scale and x-axis reflection.
-> b controls reciprocal horizontal scale and y-reflection.
-> h shifts horizontally; k shifts vertically.
-> Test the rule with the parent vertex or any known point.
-> Domain and range come from the parent; k moves the range.
o Factor Before Reading
-> Factor the entire inside to find b and h.
-> 0.4*x+2 = 0.4*(x+5), so b = 0.4, h = -5.
-> The shift is left 5, not left 2.
-> g(x) = -(0.4*x+2) uses identity f(x) = x.
-> Here a = -1, b = 0.4, h = -5, k = 0.
-> Stretch horizontally by 2.5; reflect x; shift left 5.
-> Equivalent: g(x) = -0.4*x-2; (0,0) maps to (-5,0).
-> Always factor out b first, then read the leftover constant.
-> Confirm with a value: g(0) = -2 matches the expanded form.
o Absolute Value Combined
-> g(x) = -(2/3)*|x+3| + 1 uses a = -2/3, b = 1.
-> Reflect x; compress vertically 2/3; left 3; up 1.
-> h = -3, k = 1, so the vertex is (-3,1).
-> (3,3) maps to (0,-1); check: g(0) = -2+1 = -1.
-> The V opens down; its maximum output is 1.
-> Domain: R. Range: (-inf,1].
-> With b = 1 only, the mapping is (u+h,a*v+k).
-> Its ray slopes are 2/3 on the left and -2/3 on the right.
o Quadratic Combined
-> Start with f(x) = x^2.
-> Open down, stretch vertically 4, right 2, down 5.
-> Choose a = -4, b = 1, h = 2, k = -5.
-> The equation is g(x) = -4*(x-2)^2 - 5.
-> Vertex: (2,-5); axis: x = 2; maximum: -5.
-> (1,1) maps to (3,-9); check: g(3) = -4-5 = -9.
-> Domain: R. Range: (-inf,-5]; there are no real zeros.
-> Expand to confirm: g(x) = -4*x^2 + 16*x - 21.
-> The y-intercept (0,-21) lies 16 units below the maximum.
o Dolphin Model
-> g(x) = -0.125*(x-12)^2 + 18 models height.
-> Vertex: (12,18); maximum height: 18 distance units.
-> Zero height: 0 = -0.125*(x-12)^2 + 18.
-> Thus (x-12)^2 = 144; x-12 = +/-12.
-> Zeros: x = 0 and x = 24; symmetry axis: x = 12.
-> Context domain: [0,24]; context range: [0,18].
-> The packet does not specify distance units.
-> Spot check the middle: g(12) = 18, the peak of the arc.
-> At x = 6 the height is 13.5, three quarters of 18.
o Reflection and Point Check
-> g(x) = -2*|-(x-5)| + 4 has a = -2, b = -1.
-> Reflect y then x; stretch vertically 2; right 5; up 4.
-> h = 5, k = 4; vertex: (5,4); range: (-inf,4].
-> (0,0) maps to (5,4); domain: R.
-> (-1,1) maps to (6,2); (1,1) maps to (4,2).
-> Absolute value hides the y-axis reflection.
-> Check each mapped point by substituting into g(x).
-> Verify the far ray: g(9) = -2*4 + 4 = -4.