Summary of Operations on Graphs

Translations 
y = f (x) + k  up k units 
y = f (x)  k  down k units 
y = f (x + h)  left h units 
y = f (x  h)  right h units 

Stretches/Shrinks 
y = m·f (x)  stretch vertically by a factor of
m 
y = ·f (x)  shrink vertically by a factor of
m (stretch by 
y = f (x)  stretch horizonally by a factor of
n 
y = f (nx)  shrink horizontally by a factor of n (stretch by
) 

Reflections 
y =  f (x)  reflect over xaxis (over line y = 0) 
y = f ( x)  reflect over yaxis (over line x = 0) 
x = f (y)  reflect over line y = x 
Combining Operations
We can combine operations, as long as we pay attention to the order in which we
alter inputs and outputs. Operations on outputs follow the order of
operations, and operations on inputs
follow the reverse order of operations (since we have to "undo" them). Thus,
the equation of a function stretched vertically by a factor of 2 and then
shifted 3 units up is y = 2f (x) + 3, and the equation of a function stretched
horizontally by a factor of 2 and then shifted 3 units right is y = f ((x  3)) = f (x  ).
Example: f (x) = 2x^{2}.
 Stretch f vertically by a factor of 2, and then shift f up 3 units:
2f (x) + 3 = 2(2x^{2}) + 3 = 4x^{2} + 3.
 Shrink f horizontally by a factor of 5, and then shift f right 2 units:
f (5(x  2)) = 2(5(x  2))^{2} = 2(25)(x  2)^{2} = 50(x  2)^{2}.
 Stretch f vertically by a factor of 3, stretch f horizontally by a
factor of 6, and shift f down 2 units: 3f (x)  2 = 3(2(x)^{2}) = 6()x^{2} = x^{2}.
 Shrink f vertically by a factor of 4, shrink f horizontally by a factor
of 2, and shift f left 6 units: f (2(x + 6)) = (2(x + 6))^{2} = (4)(x + 6)^{2} = (x + 6)^{2}.