Vertical and Horizontal Shifts of Graphs

Identify the basic function with a graph as below:

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

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 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)

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.

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

Step 1: f(x) = √x

Step 2: f(x) = √x-2

Step 3: f(x) = √x-2 -1