Algebra Examples

$y = \sqrt{- 4 x - 36}$

Step 1

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

$y = \sqrt{x}$

Step 2

Simplify $\sqrt{- 4 x - 36}$ .

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Step 2.1

Factor $4$ out of $- 4 x - 36$ .

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Step 2.1.1

Factor $4$ out of $- 4 x$ .

$y = \sqrt{4 (- x) - 36}$

Step 2.1.2

Factor $4$ out of $- 36$ .

$y = \sqrt{4 (- x) + 4 (- 9)}$

Step 2.1.3

Factor $4$ out of $4 (- x) + 4 (- 9)$ .

$y = \sqrt{4 (- x - 9)}$

Step 2.2

Rewrite $4$ as $2^{2}$ .

$y = \sqrt{2^{2} (- x - 9)}$

Step 2.3

Pull terms out from under the radical.

$y = 2 \sqrt{- x - 9}$

Step 3

Assume that $y = \sqrt{x}$ is $f (x) = \sqrt{x}$ and $y = \sqrt{- 4 x - 36}$ is $g (x) = 2 \sqrt{- x - 9}$ .

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

$g (x) = 2 \sqrt{- x - 9}$

Step 4

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 5

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

$y = \sqrt{x}$

Step 6

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

$y = 2 \sqrt{x + 9}$

Step 7

Find $a$ , $h$ , and $k$ for $y = 2 \sqrt{x + 9}$ .

$a = 2$

$h = - 9$

$k = 0$

Step 8

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: Left $9$ Units

Step 9

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: None

Step 10

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 11

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 Stretch: Stretched

Step 12

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: Left $9$ Units

Vertical Shift: None

Reflection about the x-axis: None

Vertical Stretch: Stretched

Step 13