Core Transformation Rules for Graphs: Shifts, Reflections, Stretches, and Coordinates

f(x)+k

Vertical Shift UP k units (k is outside)

f(x)−k

Vertical Shift DOWN k units (k is outside)

f(x−h)

Horizontal Shift RIGHT h units (−h is inside - OPPOSITE sign!)

f(x+h)

Horizontal Shift LEFT h units (+h is inside - OPPOSITE sign!)

−f(x)

Reflection over the x-axis (Flips UP/DOWN)

f(−x)

Reflection over the y-axis (Flips LEFT/RIGHT)

a⋅f(x), where a>1

Vertical Stretch by factor of a (Graph gets skinnier)

a⋅f(x), where 0

Vertical Compression by factor of a (Graph gets wider)

g(x)=f(x)+7

Vertical Shift Up 7

g(x)=f(x)−12

Vertical Shift Down 12

g(x)=f(x−1)

Horizontal Shift Right 1 (Opposite Sign!)

g(x)=f(x+9)

Horizontal Shift Left 9 (Opposite Sign!)

g(x)=f(x−3)+5

Shift Right 3, Shift Up 5

g(x)=f(x+1)−10

Shift Left 1, Shift Down 10

g(x)=−f(x)

Reflection over the x-axis

g(x)=f(−x)

Reflection over the y-axis

g(x)=−f(x)+2

Reflection over x-axis, Shift Up 2

g(x)=f(−x)−6

Reflection over y-axis, Shift Down 6

g(x)=−f(x−4)

Reflection over x-axis, Shift Right 4

g(x)=f(−x+1)

Reflection over y-axis, Shift Left 1

g(x)=4f(x)

Vertical Stretch by a factor of 4

g(x)=5/1 f(x)

Vertical Compression by a factor of 5/1

g(x)=2f(x)−1

Vertical Stretch by 2, Shift Down 1

g(x)=4/3 f(x)+5

Vertical Compression by 4/3, Shift Up 5

g(x)=−3f(x−1)+8

Reflection over x-axis, Vertical Stretch by 3, Shift Right 1, Shift Up 8

g(x)=−2/1 f(x+2)−4

Reflection over x-axis, Vertical Compression by 2/1, Shift Left 2, Shift Down 4

g(x)=f(x)+5

(2,10+5)=(2,15)

g(x)=f(x−3)

(2+3,10)=(5,10)

g(x)=−f(x)+1

(2,−10+1)=(2,−9)

g(x)=2f(x+6)−1

(2−6,2⋅10−1)=(−4,19)

g(x)=−2/1 f(x−1)+3

(2+1,−2/1 ⋅10+3)=(3,−2)