Examples

y = \sqrt{x - 3} + 6

$y = \sqrt{x - 3} + 6$

Step 1

The parent function is the simplest form of the type of function given.

y = \sqrt{x}

$y = \sqrt{x}$

Step 2

Assume that

y = \sqrt{x}

$y = \sqrt{x}$ is

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$ and

y = \sqrt{x - 3} + 6

$y = \sqrt{x - 3} + 6$ is

g (x) = \sqrt{x - 3} + 6

$g (x) = \sqrt{x - 3} + 6$ .

f (x) = \sqrt{x}

$f (x) = \sqrt{x}$

g (x) = \sqrt{x - 3} + 6

$g (x) = \sqrt{x - 3} + 6$

Step 3

The transformation from the first equation to the second one can be found by finding

a

$a$ ,

h

$h$ , and

k

$k$ for each equation.

y = a \sqrt{x - h} + k

$y = a \sqrt{x - h} + k$

Step 4

Factor a

1

$1$ out of the absolute value to make the coefficient of

x

$x$ equal to

1

$1$ .

y = \sqrt{x}

$y = \sqrt{x}$

Step 5

Factor a

1

$1$ out of the absolute value to make the coefficient of

x

$x$ equal to

1

$1$ .

y = \sqrt{x - 3} + 6

$y = \sqrt{x - 3} + 6$

Step 6

Find

a

$a$ ,

h

$h$ , and

k

$k$ for

y = \sqrt{x - 3} + 6

$y = \sqrt{x - 3} + 6$ .

a = 1

$a = 1$

h = 3

$h = 3$

k = 6

$k = 6$

Step 7

The horizontal shift depends on the value of

h

$h$ . When

h > 0

$h > 0$ , the horizontal shift is described as:

g (x) = f (x + h)

$g (x) = f (x + h)$ - The graph is shifted to the left

h

$h$ units.

g (x) = f (x - h)

$g (x) = f (x - h)$ - The graph is shifted to the right

h

$h$ units.

Horizontal Shift: Right

3

$3$ Units

Step 8

The vertical shift depends on the value of

k

$k$ . When

k > 0

$k > 0$ , the vertical shift is described as:

g (x) = f (x) + k

$g (x) = f (x) + k$ - The graph is shifted up

k

$k$ units.

g (x) = f (x) - k

$g (x) = f (x) - k$ - The graph is shifted down

k

$k$ units.

Vertical Shift: Up

6

$6$ Units

Step 9

The sign of

a

$a$ describes the reflection across the x-axis.

- a

$- a$ means the graph is reflected across the x-axis.

Reflection about the x-axis: None

Step 10

The value of

a

$a$ describes the vertical stretch or compression of the graph.

a > 1

$a > 1$ is a vertical stretch (makes it narrower)

0 < a < 1

$0 < a < 1$ is a vertical compression (makes it wider)

Vertical Compression or Stretch: None

Step 11

To find the transformation, compare the two functions and check to see if there is a horizontal or vertical shift, reflection about the x-axis, and if there is a vertical stretch.

Parent Function:

y = \sqrt{x}

$y = \sqrt{x}$

Horizontal Shift: Right

3

$3$ Units

Vertical Shift: Up

6

$6$ Units

Reflection about the x-axis: None

Vertical Compression or Stretch: None

Step 12

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