
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:

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 ∞.
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.)

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 
NOTE: The reposting of materials (in part or whole) from this site to the Internet
is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use". 
