
Unless otherwise stated:
Domain: (All Reals)
Range: (All Reals)

Equation Forms:
• SlopeIntercept Form:
y = mx + b
m = slope; b = yintercept
• PointSlope Form:
y  y_{1} = m(x  x_{1})
uses point (x_{1},y_{1}) and m
• Standard Form: Ax + By = C
A, B and C are integers.
A is positive.


Finding Slope:
Average rate of change (slope) is constant. 
No relative or absolute maxima or minima unless domain is altered. 
xintercept (for y = x):
crosses xaxis
(x, 0)
Set y = 0, solve for x.
yintercept (for y = x):
crosses yaxis
(0, y)
"b" value
Set x = 0, solve for y.
End Behavior:
One end approaches +∞,
other end approaches ∞.
(Unless domain is altered.)

Effects of Changes in y = mx + b: (m = slope; b = yintercept)
• if m = 0, then line is horizontal (y = b)
• if m = undefined, then line is vertical ("run" =0) (not a function)
• if m > 0, the slope is positive (line increases from left to right)
(the larger the slope the steeper the line)
• if m < 0, the slope is negative (line decreases from left to right)
• Lines with equal slopes are parallel.
•  m  > 1 implies a vertical stretch
• 1 < m < 0 or 0 < m < 1, implies a vertical shrink
• if b > 0, then there is a vertical shift up "b" units
• if b < 0, then there is a vertical shift down "b" units 

Linear Function  Transformation Examples:
Symmetric about the yaxis. 
Equation Forms:
• Vertex Form:
y = a(x  h)^{2} + k
with
vertex (h,k)
shows vertex, max/min, inc/dec
• PointSlope Form:
y = ax^{2} + bx + c
negative "a" opens down
• Intercept Form:
y = a(x  p)(x  q)
p and q are xintercepts.
shows roots, pos/neg


Axis of Symmetry:
locates "turning point"
(vertex)
Average rate of change
NOT constant
xintercept(s):
determine roots/zeros
yintercept:
(0, y)

End Behavior: Both ends approach +∞ (or both ends approaches ∞ when a's negative).
The quadratic function y = x^{2} is an even function: f (x) = f (x) 
Quadratic Function  Possible Real Roots:
y = (x + 2)(x + 2)
x = 2; x = 2 
y = (x  2)(x + 2)
x = 2; x = 2 
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:
Translation

Reflection

Vertical Stretch/Shrink 
Cubic functions are of degree 3. 
Example Equation Forms:
• y = x^{3}
(1 real root  repeated)
• y = x^{3} 3x^{2}= x^{2}(x  3)
(two real roots  1 repeated)
• y = x^{3}+2x^{2}+x = x(x + 1)^{2}
(three visible terms)
• y = x^{3}+3x^{2}+3x+1=(x+1)^{3 }
(1 real root  repeated)
• y = (x+1)(x  2)(x  3)
(factored form  3 real roots)


Symmetric (for y = x³):
about origin
Average rate of change:
NOT constant
xintercept(s):
determine roots/zeros
yintercept:
(0, y)
End Behavior:
One end approaches +∞,
other end approaches ∞.
(Unless domain is altered.)

The cubic function y = x^{3} is an odd function: f (x) = f (x) 
Cubic Function  Possible Real Roots:
y = x³
1 Real Root (repeated) 
y = x³  3x²
2 Real roots (1 repeated) 
y = x³  3 x² + 2
3 Real roots 
Cubic Function  Transformation Examples:
Translations 
Reflection 
Vertical Stretch/Shrink 
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