Operations on Functions: Combining Operations on Graphs

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 x-axis (over line y = 0)
y = f (- x)	reflect over y-axis (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) = 2x².

Stretch f vertically by a factor of 2, and then shift f up 3 units: 2f (x) + 3 = 2(2x²) + 3 = 4x² + 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()x² = x².
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.