College Algebra Quick Reference Sheet
Set Notation
Interval Notation
Set-Builder Notation
(a, b)
{ x | a < x < b }
[a, b]
{ x | a ≤ x ≤ b }
[a, b)
{ x | a ≤ x < b }
(a, b]
{ x | a < x ≤ b }
(a, ∞)
{ x | a < x }
[a, ∞)
{ x | a ≤ x }
(-∞, b)
{ x | x < b }
(-∞, b]
{ x | x ≤ b }
Set Operations
Operation
Elements
Logic
Union
All
OR
Intersection
Common
AND
Coordinate Plane Quadrants
II
I
III
IV
Distance and Midpoint Formulas
If P1=(x1,y1) and P2=(x2,y2) are two points,
the distance between them is
and the midpoint coordinates are
Intercepts of an Equation
x-intercepts
Set y = 0; solve for x
y-intercepts
Set x = 0; solve for y
Symmetry of the Graph of an Equation
Type
Mathematical
Geometrical
x-axis
Unchanged when
y replaced by -y
Unchanged when
reflected about
x-axis
y-axis
Unchanged when
x replaced by -x
Unchanged when
reflected about
y-axis
origin
Unchanged when
y replaced by –y &
x replaced by -x
Unchanged when
rotated 180°
about origin
Function Notation y = f(x)
Domain
Set of all valid x
Range
Set of all valid y
Function Arithmetic
Transformations of Graphs of Functions
Horizontal
Vertical
Shift
(left)
(right)
(up)
(down)
Reflect
(y-axis)
(x-axis)
Scale
(compress)
(expand)
1. Subtract h from each of the x-coordinates of
the points on the graph of f. This results in a
horizontal shift to the left if h > 0 (positive h) or
right if h < 0 (negative h).
2. Divide the x-coordinates of the points on the
graph obtained in Step 1 by b. This results in a
horizontal scaling, but may also include a
reflection about the y-axis if b < 0 (negative b).
3. Multiply the y-coordinates of the points on
the graph obtained in Step 2 by a. This results in
a vertical scaling, but may also include a
reflection about the x-axis if a < 0 (negative a).
4. Add k to each of the y-coordinates of the
points on the graph obtained in Step 3. This
results in a vertical shift up if k > 0 (positive k)
or down if k < 0 (negative k).
Properties of Equality
Properties of Inequalities
Lines or Linear Functions
Slope of Line through points (x1, y1) & (x2, y2)
Slope-Intercept Form - slope m and point (0, b)
Point-Slope Form - slope m and point (x1, y1)
or
Horizontal Line through point (0, b)
Vertical Line through point (a, 0)
Average Rate of Change
The average rate of change m for function y=f(x)
between x=a and x=b is
Absolute Value Properties
Absolute Value Function as a
Piecewise-Defined Function
Absolute Value Equations and Inequalities
If c is a positive number:
Parabolas or Quadratic Functions
General Form
The graph has a smile if a is positive and a frown
if a is negative, and has a vertex at coordinates:
Vertex Form
The graph has a smile if a is positive and a frown
if a is negative, and has a vertex at (h, k).
Special Factoring Formulas
Special Product Formulas
Quadratic Formula
Solve
If , then 2 real unequal solutions
If , then 2 real duplicate solutions
If , then no real solutions
Factored Form for real factors:
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