Graph the function:

2.2 Vertical and Horizontal Shifts of Graphs

Vertical Shift of graphs

• Discussion 1

x

y

f(x) = x²

f(x) = x²+1

f(x) = x²-2

f(x) = x²-5

↑ 1 unit

↓ 2 unit

↓ 5 unit

What about shift f(x) up by 10 unit?

shift f(x) down by 10 unit?

Vertical Shift of Graphs

• Discussion 2

x

y

f(x) = x³

f(x) = x³+2

f(x) = x³-3

↑ 2 unit

↓ 3 unit

Vertical Shift of Graphs

• If c>0, then the graph of y = f(x) + c is obtained by shifting the graph of y = f(x) upward a distance of c units. The graph of y = f(x) – c is obtained by shifting the graph of y = f(x) downward a distance of c units.

↑ f(x) + c

↓ f(x) - c

Horizontal Shift of graphs

• Discussion 1

x

y

f(x) = x²

f(x) = (x+1)²

f(x) = (x-2)²

f(x) = (x-5)²

← 1 unit

→ 2 unit

→ 5 unit

What about shift f(x) left by 10 unit?

shift f(x) right by 10 unit?

Horizontal Shift of Graphs

• Discussion 2

x

y

f(x) = |x|

f(x) = |x + 2|

f(x) = |x - 3|

← 2 unit

→ 3 unit

Horizontal Shift of Graphs

• If c > 0, the graph of y = f(x + c) is obtained by shifting the graph of y = f(x) to the left a distance of c units. The graph of y = f(x - c) is obtained by shifting the graph of y = f(x) to the right a distance of c units.

f(x + c) ←

→ f(x - c)

Conclusion

Combinations of vertical and horizontal shifts

• Equation → write a description

y₁ = |x - 4|+ 3. Describe the transformation of f(x) = |x|. Identify the domain / range for both.

answer: shifting f(x) up by 3 units, then shift f(x) right by 4 units. ( or shift f(x) right by 4 units, then shift f(x) up by 3 units.)

Combinations of vertical and horizontal shifts

• Description → equation

Write the function that shifts y = x² two units left and one unit up.

answer: y1 = (x+2)²+1

Combinations of vertical and horizontal shifts

• Graph → equation

Write the equation for the graph below. Assume each grid mark is a single unit.

​

Answer:

f(x) = (x-1)³-2

x

y

Combinations of vertical and horizontal shifts

• Equation → graph

Sketch the graph of

y = f(x) = √x-2 -1.

How does the transformation affect the domain and range?

​

x

y

Combinations of vertical and horizontal shifts

• Graph & symbolic transformation → new graph

Using the given graph of

f(x), sketch the graph of

f(x) +2

f(x+2)

f(x-1) - 3

​

x

y