Intermediate Algebra (11th edition)

Graphing a Parabola (Horizontal Shift)

Graph

If then giving the vertex The graph of

has the same shape as that of but is shifted 2 units to the right. Plotting

points on one side of the vertex, and using symmetry about the axis to find cor-

respondingpoints on the other side, gives the graph in FIGURE 7.

x= 2

ƒ 1 x 2 =x^2

x= 2, F 1 x 2 = 0, 1 2, 0 2. F 1 x 2 = 1 x- 222

F 1 x 2 = 1 x- 222.

EXAMPLE 2

SECTION 9.5 Graphs of Quadratic Functions 533

Vertical Shift

The graph of is a parabola.

• The graph has the same shape as the graph of


• The parabola is shifted kunits up if and units down if


• The vertex of the parabola is^1 0, k^2.

k 7 0, |k| k 6 0.

ƒ 1 x 2 =x^2.

F 1 x 2 x^2 k

0

1

2

3

4

4

1

0

1

4

x

(^02)
4
x
y

f (x) x^2

F (x) (x – 2 )^2

F (x) (x – 2 )^2

x 2

FIGURE 7

Vertex:

Axis:

Domain:

Range: 3 0, q 2

1 - q, q 2

x= 2

1 2, 0 2

F 1 x 2 = 1 x- 222

NOW TRY

CAUTION Errors frequently occur when horizontal shifts are involved.To

determine the direction and magnitude of a horizontal shift, find the value that causes

the expression x-hto equal 0, as shown below.

Shift the graph of 5 units to the right,

because causes to equal 0.

Shift the graph of 5 units to the left,

because - 5 causes x+ 5 to equal 0.

F 1 x 2

F 1 x 2 = 1 x+ 522

+ 5 x- 5

F 1 x 2

F 1 x 2 = 1 x- 522

Graphing a Parabola (Horizontal and Vertical Shifts)

Graph

This graph has the same shape as that of but is shifted 3 units to the

left (since if ) and 2 units down (because of the ). See FIGURE 8

on the next page.

x+ 3 = 0 x=- 3 - 2

ƒ 1 x 2 = x^2 ,

F 1 x 2 = 1 x+ 322 - 2.

EXAMPLE 3

NOW TRY
EXERCISE 2
Graph Give
the vertex, axis, domain,
and range.

ƒ 1 x 2 = 1 x+ 122.

NOW TRY ANSWER
2.

vertex: ; axis: ;

domain: ; range: 1 - q, q 2 3 0, q 2

1 - 1, 0 2 x=- 1

x

y

–1 0

4

1

f(x) = (x + 1)^2

Horizontal Shift

The graph of is a parabola.

• The graph has the same shape as the graph of


• The parabola is shifted hunits to the right if and units to the left

if

• The vertex of the parabola is^1 h, 0^2.

h 6 0.

h 7 0, |h|

ƒ 1 x 2 =x^2.

F 1 x 2  1 xh 22