Algebra Cheat Sheet (Reduced) - Pauls Online Math Notes

Functions and Graphs
Constant Function
Parabola/Quadratic Function
y= a or f ( x)
= a
2
x= ay + by+ c
2
g( y)
= ay + by+
c
Graph is a horizontal line passing
through the point ( 0,a ) .
The graph is a parabola that opens right
if a > 0 or left if a < 0 and has a vertex
Line/Linear Function
æ æ b ö b ö
at ç gç - ,
y= mx+ b or f ( x)
= mx+
b
2a
÷ -
2a
÷
è è ø ø .
Graph is a line with point ( 0,b ) and
Circle
slope m.
( x- h) 2 + ( y- k)
2 = r
2
Slope
Slope of the line containing the two
Graph is a circle with radius r and center
( hk , ) .
points ( x1,
y
1)
and ( x2,
y
2)
is
y2 - y1
rise
Ellipse
m= = x
2
- x
1
run
( x-h) 2 ( y-k)
2
+ = 1
2 2
Slope – intercept form
a b
The equation of the line with slope m Graph is an ellipse with center ( hk , )
and y-intercept ( 0,b ) is
with vertices a units right/left from the
y = mx+
b
Point – Slope form
center and vertices b units up/down from
the center.
The equation of the line with slope m
and passing through the point ( x1,
y
1)
is Hyperbola
y= y1+ m( x-
x1)
( x-h) 2 ( y-k)
2
- = 1
2 2
a b
Graph is a hyperbola that opens left and
Parabola/Quadratic Function
2 2 right, has a center at
y= a( x- h) + k f ( x) = a( x- h)
( hk , ) , vertices a
+ k
units left/right of center and asymptotes
b
The graph is a parabola that opens up if that pass through center with slope ± .
a > 0 or down if a < 0 and has a vertex
a
Hyperbola
at ( hk , ) .
( y-k) 2 ( x-h)
2
- = 1
2 2
Parabola/Quadratic Function
b a
2 2
y = ax + bx+ c f ( x)
= ax + bx+
c
Graph is a hyperbola that opens up and
hk , , vertices b
The graph is a parabola that opens up if
a > 0 or down if a < 0 and has a vertex
æ b æ b öö
at ç - , f ç - ÷
2a
2a
÷ .
è è øø
down, has a center at ( )
units up/down from the center and
asymptotes that pass through center with
b
slope ± .
a
Common Algebraic Errors
Error
Reason/Correct/Justification/Example
2
0
0 ¹ and 2 ¹ 2
Division by zero is undefined!
0
2
- 3 ¹ 9
( x
2 ) 3
¹ x
5
( ) 3
a a a
¹ +
b+
c b c
1
¹ x + x
2 3
x + x
-2 -3
a + bx
¹ 1+
bx
a
2
- 3 =- 9, (- 3) 2
= 9 Watch parenthesis!
x = xxx = x
a( x 1)
( )
- - ¹-ax- a
2 2 2 2 6
1 1 1 1
= ¹ + = 2
2 1+
1 1 1
A more complex version of the previous
error.
a+
bx a bx bx
= + = 1+
a a a a
Beware of incorrect canceling!
-a x- 1 =- ax+
a
Make sure you distribute the “-“!
( ) 2 2 2
x+ a ¹ x + a
( ) ( )( )
2 2
x + a ¹ x+
a
2 2 2
x+ a = x+ a x+ a = x + 2ax+
a
2 2 2 2
5= 25 = 3 + 4 ¹ 3 + 4 = 3+ 4 = 7
x+ a ¹ x+ a
See previous error.
+ ¹ + and n x a n x n More general versions of previous three
+ ¹ + a
errors.
( ) n n n
x a x a
( x+ ) ¹ ( x+
)
2 2
2 1 2 2
( 2x+ 2) ¹ 2( x+
1)
2 2
( ) ( )
2 2 2
2 x+ 1 = 2 x + 2x+ 1 = 2x + 4x+
2
( ) 2 2
2x+ 2 = 4x + 8x+
4
Square first then distribute!
See the previous example. You can not
factor out a constant if there is a power on
the parethesis!
2 2 2 2
- x + a ¹- x + a
( ) 1
- x 2 + a 2 = - x 2 + a
2 2
Now see the previous error.
æaö
a ab
¹
a
ç
1
÷
æaöæcö
ac
æb
ö c
=
è ø
=
ç
c
÷
b b
ç ÷ç ÷ =
æ ö æ ö è1øèbø
b
è ø
ç ÷ ç ÷
ècø ècø
æaö æa
ö
æaö
ç
ç ÷
b
÷ ç
b
÷
è ø æaöæ1ö
a
èb
ø ac
=
è ø
= ¹ c c
ç
b
֍
c
÷ =
æ ö è øè ø bc
c b
ç ÷
è1ø
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu.
© 2005 Paul Dawkins