Precalculus Examples

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

Step 1

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

$y = \sqrt{x}$

Step 2

Assume that $y = \sqrt{x}$ is $f (x) = \sqrt{x}$ and $y = \sqrt{x - 3} + 6$ is $g (x) = \sqrt{x - 3} + 6$ .

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

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

Step 3

The transformation from the first equation to the second one can be found by finding $a$ , $h$ , and $k$ for each equation.

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

Step 4

Factor a $1$ out of the absolute value to make the coefficient of $x$ equal to $1$ .

$y = \sqrt{x}$

Step 5

Factor a $1$ out of the absolute value to make the coefficient of $x$ equal to $1$ .

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

Step 6

Find $a$ , $h$ , and $k$ for $y = \sqrt{x - 3} + 6$ .

$a = 1$

$h = 3$

$k = 6$

Step 7

The horizontal shift depends on the value of $h$ . When $h > 0$ , the horizontal shift is described as:

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

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

Horizontal Shift: Right $3$ Units

Step 8

The vertical shift depends on the value of $k$ . When $k > 0$ , the vertical shift is described as:

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

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

Vertical Shift: Up $6$ Units

Step 9

The sign of $a$ describes the reflection across the x-axis. $- a$ means the graph is reflected across the x-axis.

Reflection about the x-axis: None

Step 10

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

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

$0 < a < 1$ 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: $y = \sqrt{x}$

Horizontal Shift: Right $3$ Units

Vertical Shift: Up $6$ Units

Reflection about the x-axis: None

Vertical Compression or Stretch: None

Step 12

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