How to Draw Absolute Valued Functions Step by Step (involving Linear and Quadratic terms)
Author: Lew W. S.

Step1

Step2

Step3

Step4/Final

Sketching an Absolute Value Function

constant1 = 0 (slider from -5 to 5, step 1)
c = -3 (slider from -5 to 5, step 1)
b = 2 (slider from -5 to 5, step 1)
a = 0 (slider from -5 to 5, step 1)
fn = Simplify(a * x^(2.0) + b * x + c)
f = Simplify(abs(fn(x)))
coefficient1 = 1 (slider from -5 to 5, step 0.5)
g = Simplify(coefficient1 * f(x))
h = Expand(g(x) + constant1)
Step1 = true
Step2 = false
Step3 = false
Step4 = false
FinalStep = false
ShowAllSteps = false
absBasicFnText: "|" + (Simplify[fn(x)]) + "|"
LinearAbsoluteTerm: If[coefficient1 > 1.0, coefficient1 + absBasicFnText, If[coefficient1 ≟ 1.0, absBasicFnText, If[coefficient1 > 0.0 ∧ coefficient1 < 1.0, coefficient1 + absBasicFnText, If[coefficient1 ≟ 0.0, "No Absolute Valued Function !", If[coefficient1 < 0.0 ∧ coefficient1 > -1.0 * 1.0, " " + coefficient1 + absBasicFnText, If[coefficient1 ≟ -1.0 * 1.0, " - " + absBasicFnText, " " + coefficient1 + absBasicFnText]]]]]] + ""
AbsFunctionText: "y = " + (If[coefficient1 < 0.0 ∧ constant1 > 0.0, constant1 + LinearAbsoluteTerm, If[constant1 > 0.0, LinearAbsoluteTerm + " +" + constant1, If[constant1 ≟ 0.0, LinearAbsoluteTerm + "", LinearAbsoluteTerm + " -" + (abs(constant1))]]])
text1: " Step 1 : Draw basic function y = " + (Simplify[fn])
text4: " Step 4/Final : Shift  Step 3 function vertically " + constant1 + " units (constant term) 
   to get the final function " + AbsFunctionText
Title1: "Sketch  " + AbsFunctionText
text2: "Step 2 : Draw Absoluted Value Function y = " + absBasicFnText + "
            (reflect in the x-axis the part of the graph that
             is below the x-axis)"
text3: "Step 3 : Apply the vertical stretch factor 
       ie    " + LinearAbsoluteTerm

Tool: LineOver
Input: C, a
Output: b

b = Line(C, a)

Tool: EasyTable
f(x), start x, end x, interval
Input: f, a, b, c
Output: T1

f(x) = x^(6.0)
T1: "\begin{tabular}{cc} x & " + f + " \\ " + (Sequence["\\" + (i) + "&" + (f(i)) + "&&", i, a, b, c]) + "\end{tabular}"

Tool: Polynomial 3 point -Quadratic Curve (command Polynomial3point)
Input: A, B, C
Output: p_2

p_2 = Polynomial((x - x(B)) * (x - x(C)) / (x(A) - x(B)) / (x(A) - x(C)) * y(A) + (x - x(A)) * (x - x(C)) / (x(B) - x(A)) / (x(B) - x(C)) * y(B) + (x - x(A)) * (x - x(B)) / (x(C) - x(A)) / (x(C) - x(B)) * y(C))

Tool: Polynomial 4 Point Cubic Curve (command Polynomial4PointCubicCurve)
Input: A, B, C, D
Output: p_3

p_3 = Polynomial((x - x(B)) * (x - x(C)) * (x - x(D)) / ((x(A) - x(B)) * (x(A) - x(C)) * (x(A) - x(D))) * y(A) + (x - x(A)) * (x - x(C)) * (x - x(D)) / ((x(B) - x(A)) * (x(B) - x(C)) * (x(B) - x(D))) * y(B) + (x - x(A)) * (x - x(B)) * (x - x(D)) / ((x(C) - x(A)) * (x(C) - x(B)) * (x(C) - x(D))) * y(C) + (x - x(A)) * (x - x(B)) * (x - x(C)) / ((x(D) - x(A)) * (x(D) - x(B)) * (x(D) - x(C))) * y(D))

Tool: Quadratic Bezier Curve (command Bezier2)
Input: A, B, C
Output: loc1

a = Segment(A, B)
b = Segment(B, C)
D = Point(a)
c = Line(D, b)
d = Segment(A, C)
E = Intersect(c, d)
e = Line(E, a)
F = Intersect(b, e)
f = Segment(D, F)
g = Line(A, f)
G = Intersect(c, g)
h = Segment(B, G)
I = Intersect(f, h)
loc1 = Locus(I, D)

Tool: Cubic Bezier Curve (command Bezier3)
Input: A, B, C, D
Output: c

a = Segment(A, B)
E = Point(a)
t: Distance[A, E] / a
F: B + t * (C - B)
H: E + t * (F - E)
G: C + t * (D - C)
I: F + t * (G - F)
J: H + t * (I - H)
c = Locus(J, E)