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LQfgraph1
Unless otherwise stated:
   Domain: lqfmath1 (All Reals)
   Range: lqfmath2 (All Reals)
Equation Forms:
• Slope-Intercept Form:
y = mx + b

m = slope;   b = y-intercept

• Point-Slope Form:

y - y1 = m(x - x1)
uses point (x1,y1) and m

• Standard Form:
Ax + By = C

A, B and C are integers.
A is positive.
Finding Slope:
LQFmath3

Average rate of change (slope) is constant.
No relative or absolute maxima or minima unless domain is altered.
x-intercept (for y = x):
crosses x-axis
(x, 0)
Set y = 0, solve for x.

y-intercept (for y = x):
crosses y-axis
(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 = y-intercept)
• 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:
graph
Translations
lqfgraph9
Reflection
LQRgraph7
Vertical Stretch/Shrink

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quadraticfeaturesclickquad
10Symmetric about the y-axis.
Equation Forms:
• Vertex Form:
y = a
(x - h)2 + k
with vertex (h,k)

shows vertex, max/min, inc/dec

• Point-Slope Form:

y = ax2 + bx + c
negative "a" opens down

• 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: Both ends approach +∞ (or both ends approaches -∞ when a's negative).
The quadratic function y = x2 is an even function: f (-x) = f (x)

Quadratic Function - Possible Real Roots:

lqfroot1
y = (x + 2)(x + 2)
x = -2;   x = -2
lqfroot2
y = (x - 2)(x + 2)
x = 2;   x = -2
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:
transgraph11
Translation
transgraph3
Reflection
transgraph5a
Vertical Stretch/Shrink

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cubicfeatures
CUBE1
Cubic functions are of degree 3.
Example Equation Forms:
y = x3
(1 real root - repeated)

y = x3- 3x2= x2(x - 3)
(two real roots - 1 repeated)

y = x3+2x2+x = x(x + 1)2
(three visible terms)

y = x3+3x2+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

x-intercept(s):
determine roots/zeros

y-intercept:
(0, y)

End Behavior:

One end approaches +∞,
other end approaches -∞.

(Unless domain is altered.)
The cubic function y = x3 is an odd function: f (-x) = -f (x)

Cubic Function - Possible Real Roots:

cube4
y = x³
1 Real Root (repeated)
cube3
y = x³ - 3x²
2 Real roots (1 repeated)
cube2
y = x³ - 3x² + 2
3 Real roots

Cubic Function - Transformation Examples:
cube6
Translations
cube5
Reflection
cube7
Vertical Stretch/Shrink

 

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