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

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:
 Translations Reflection Vertical Stretch/Shrink

Symmetric 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:

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:

 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 = 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):

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:

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

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

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