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quadraticfeaturesShape: Parabola
10Symmetric about the y-axis.
Turning point (minimum) at (0,0).
Equation Forms:
• Vertex Form:
y = a(x - h)2 + k
with vertex (h,k)
easy access to finding vertex, max/min, inc/dec

• Point-Slope Form:
y = ax2 + bx + c
shows general form of graph
and roots (zeros)

• Intercept Form:
y = a(x - p)(x - q)
p and q are x-intercepts.
shows roots, pos/neg

Axis of Symmetry:
LQFsym
locates "turning point"
(vertex)

Average rate of change
NOT constant

x-intercept(s):
determine roots/zeros

y-intercept:
(0, y)

End Behavior: Parent Function: f (x) = x²
as  x → + ∞,  f (x) → + ∞
as  x → - ∞,  f (x) → + ∞
parabola opens upward 		Negative Coefficient: f (x) = -x²
as  x → + ∞,  f (x) → - ∞
as  x → - ∞,  f (x) → - ∞
parabola opens downward

The quadratic function y = x2 is an even function: f (-x) = f (x)

Quadratic Function - Possible Real Roots and Complex Roots:
lqfroot1
y = (x + 2)(x + 2)
x = -2;   x = -2
Repeated root
Multiplicity 2
lqfroot2
y = (x - 2)(x + 2)
x = 2;   x = -2
Each root is
multiplicity 1
lqfroo3
y = x² + 2

roots are complex (imaginary)

Maximum/Minimum: Finding the "turning point" (vertex) will locate the maximum or minimum point. The intervals of increasing/decreasing are also determined by the vertex.

Quadratic Function - Transformation Examples:

Translations

Translations:

Vertical Shift: f (x) + k

Horizontal Shift: f (x + k)

Reflections:

-f (x) over x-axis

f (-x) over y-axis

Reflection

Vertical Stretch/Compress

Vertical Stretch/Compress

k • f (x) stretch (k > 1)

k • f (x) compress (0 < k < 1)

Horizontal Stretch/Compress

f (k • x) stretch (0 < k < 1)

f (k • x) compress ( k > 1)
Horizontal Stretch/Compress


Remember: for y = ax2 + bx + c, negative "a" opens down.

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