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