Main Graphs (Page 1)
Graphs First Introduced in (by Color): Intermediate Algebra or before, College Algebra, Precalculus Algebra, or Trigonometry
Web links can be opened by right-clicking on link, and selecting "Open Hyperlink".
Type
Graph
Absolute Value Functions
Vee
Polynomial Functions
Constant (0 degree) Function
Equation
Main Characteristics
y  a | x  h | k
Vertex = (h,k)
Axis of Symmetry: x = h
Opens up if a>0
Opens down if a<0
Continuous and smooth curve
y-int = b
Horizontal Line
yb
Line
y  mx  b
Quadratic (2 degree)
Function
Parabola
y  ax 2  bx  c
Transformations of a Perfect
Cube
Cubic and higher degree
Vertical S
y  a(x  h)3  k
(Varies)
y  a n x n  a n 1x n 1 +a n 2 x n 2  . . .  a 2 x 2  a1x  a 0
y-int = a0
x-intercepts when y = 0
x c
x-int = c
slope = -A/B
y-int = C/B
Linear (1st degree) Function
nd
y  a(x  h)2  k
1st Degree Polynomial
Equations in 2 variables
Linear Equation in Just x
Linear Equation
Line
2nd Degree Polynomial
Equations in 2 variables
Parabolic Equation
[A=0 or C=0, but not both;
and B=0]
Conic Sections,
usually
Parabola
Vertical Line |
General Line
Ax  By  C
4p(y  k)  (x  h) 2 (vertical orientation, VO)
or
4p(x  h)  (y  k) 2 (horizontal orientation, HO)
Circle
Ellipse
For a>b:
(x  h)2 (y  h)2

 1 (horizontal orientation, HO)
a2
b2
or
(y  k)2 (x  h)2

 1 (vertical orientation, VO)
a2
b2
Hyperbolic Equation
[AC<0, and B=0]
Hyperbola
(x  h)2 (y  h)2

 1 (horizontal orientation, HO)
a2
b2
or
(y  k)2 (x  h)2

 1 (vertical orientation, VO)
a2
b2
(x  h) 2  (y  k) 2 = r 2
Hyperbola at 45º
Orientation
Reciprocal Power Functions
y
y
or
General Rational Functions
Radical (Root) Functions
Square Root Function
(Varies)
b
2a
k  ah 2  bh  c
h
Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0
Circular Equation
[A=C≠0, and B=0]
Elliptic Equation
[AC>0, A≠C, and B=0]
Rational Functions
Reciprocal Functions
slope = m
y-int = b
y-int = c
Vertex = (h,k)
Axis of Symmetry: x = h
Opens up if a>0
Opens down if a<0
Inflection Point = (h,k)
y
a
k
xh
a
x  h
n
k
An x n  An1x n1  An2 x n2  ...  A2 x 2  A1x  A0
Bm x m  Bm1x m1  Bm2 x m2  ...  B2 x 2  B1x  B0
Half parabola
y  a b(x  h)  k
Cube Root Functions
Horizontal S
y  a 3 b(x  h)  k ;make b  0
Even Root Functions
Half U-shaped
curve
y  a n b(x  h)  k ;n is even
Odd Root Functions
Horizontal S
y  a n b(x  h)  k ;n is odd, make b  0
Vertex = (h,k)
Axis of Symmetry either x = h for VO or y = k for HO
Focus either (h,k+p) for VO or (h+p,k) for HO
Directrix either y = k-p for VO or x = h-p for HO
Center = (h,k)
Radius = r
Center = (h,k)
Vertices either (h±a, k) for HO or (h, k±a) for VO
Endpoints of minor axis either (h, k±b) for HO or (h±b, k) for VO
Foci either (h±c, k) for HO or (h, k±c) for VO where c2 = a2-b2
Center = (h,k)
Vertices either (h±a, k) for HO or (h, k±a) for VO
Foci either (h±c, k) for HO or (h, k±c) for VO where c2 = a2+b2
b
a
Asymptotes either y =   x  h   k for HO or y =   x  h   k for VO
a
b
Vertical Asymptote: x = h
Horizontal Asymptote: y = k
Vertical Asymptote: x = h
Horizontal Asymptote: y = k
Right-hand graph when n is odd and left-hand graph when n is even
Holes (possibly)
Vertical Asymptotes, VA (possibly)
Horizontal Asymptotes, HA, when n < m
Oblique (Slant) Asymptotes, OA, when n = m+1
Intercepts with either HA or OA (possibly)
x-intercepts when y = 0
A
y-intercept = 0 ; if B0  0
B0
Endpoint = (h,k)
Top half if a > 0, or bottom half if a < 0
Opens to the right if b > 0, or opens to the left if b < 0
Inflection point = (h,k)
Increasing if a>0, or decreasing if a<0
Endpoint = (h,k)
Top half if a > 0, or bottom half if a < 0
Opens to the right if b > 0
Opens to the left if b < 0
Inflection point = (h,k)
Increasing if a>0
Decreasing if a<0
Main Graphs (Page 2)
Graphs First Introduced in (by Color): Intermediate Algebra or before, College Algebra, Precalculus Algebra, or Trigonometry
Web links can be opened by right-clicking on link, and selecting "Open Hyperlink".
Type
Graph
Exponential Functions
Natural Exponential Functions
Equation
y  a  exh   k , where e  2.71828... ;( case)
or
y  a  e(xh)   k , where e  2.71828... ;( case)
y  a  b x h   k
General Exponential
Functions
Logarithmic Functions
Natural Logarithmic Functions
y  a Ln c  x  h   k
y  a Logb c  x  h   k
General Logarithmic
Functions
Trigonometric Functions
Cosine Functions
or
Sine Functions
Sinusoidal Wave
y  a cos(bx  c)  d ; make b  0
or
y  a sin(bx  c)  d ; make b  0
Main Characteristics
Horizontal Asymptote, HA: y = k
Above HA if a>0
Below HA if a<0
Approaches HA to the left if + case
Approaches HA to the right if - case
Horizontal Asymptote, HA: y = k
Above HA if a>0
Below HA if a<0
Approaches HA to the left if b>1
Approaches HA to the right if 0<b<1
Vertical Asymptote, VA: x = h
Right of VA if c>0
Left of VA if c<0
Approaches VA going down if a>0
Approaches VA going up if a<0
Vertical Asymptote, VA: x = h
Right of VA if c>0
Left of VA if c<0
Approaches VA going down if b>1 and a>0, or if 0<b<1 and a<0
Approaches VA going up if 0<b<1 and a>0, or if b>1 and a<0
Periodic
2
b
Amplitude = |a|
Period 
c
b
Vertical Shift = d
Original Basic Period =  0,2 
Phase Shift  
y  a tan(bx  c)  d ; make b  0
Tangent Function
Period 

b
c
b
Vertical Shift = d
Phase Shift  
  
Original Basic Period =   , 
 2 2
Vertical Asymptotes
y  a sec(bx  c)  d ; make b  0
or
y  a csc(bx  c)  d ; make b  0
Secant Functions
or
Cosecant Functions
Period 
2
b
c
b
Vertical Shift = d
Original Basic Period =  0,2 
Phase Shift  
Vertical Asymptotes
y  a cot(bx  c)  d ; make b  0
Cotangent Functions
Period 

b
c
b
Vertical Shift = d
Original Basic Period =  0,  
Phase Shift  
Vertical Asymptotes
y  Arccos(x)
or
Inverse Cosine Function
y  cos 1 (x)
y  Arcsin(x)
or
Inverse Sine Function
y  sin 1 (x)
Inverse Tangent Function
Bounded
Horizontal S
y  Arctan(x)
or
y  tan 1 (x)
Domain   1,1
Range   0,  
Domain   1,1
  
Range    , 
 2 2
Domain   ,  
  
Range    , 
 2 2
Horizontal Asymptotes: y  

2
and y 

2