# Square Root Function

## Power Functions

A power function has the form y = x^{n}.

- For n = 1, we get a linear graph,
- For n = 2 we get a parabola,
- For n = 3 we get a cubic.
- For n = –1, we get a hyperbola.
- For n = –2, we get a truncus

For $n = \dfrac{1}{2}$, we get the square root graph.

## Graph of the Standard Square Root Function

… … $y = x^{\frac{1}{2}} \; \text{ is more familiar as } y = \sqrt{x}$

… … Domain: $x \in R^+ \cup \{ 0 \}$

… … … or $\{ x: x \geqslant 0 \}$

… … Range: $y \in R^+ \cup \{ 0 \}$

… … … or $\{ y: y \geqslant 0 \}$

… … Intercepts: (0, 0) (at the origin)

## Comparing Square Root Graph to a Parabola

Notice that if we combine the positive and negative versions of the square root function, we get the graph of a relation that is a parabola drawn sideways.

… … This is because $y = \pm \sqrt{x}$

… … is equivalent to $x = y^2$,

… … which is the inverse of $y = x^2$.

This means that they are a reflection across the line $y = x$

OR

the x and y coordinates of individual points are swapped so (x, y) → (y, x)

## Transformations on the Square Root Function

We can apply the standard transformations such as dilations, translations and reflections when sketching a square root function.

#### Example 1

… … Sketch $y = \sqrt{-2x-2}-1$

**Solution:**

… … This can be written as: $y=\sqrt{-2 \left(x+1 \right) }-1$

This is the standard square root function with the following transformations.

- Reflected across the y-axis (in the x-direction)
- Dilated by a factor of ½ in the x-direction
- Translated to the left by 1 unit
- Translated down by 1 unit

Domain: $\big\{ x: \; x \leqslant -1 \big\}$

… … or $x \in \big( -\infty, \; -1 \big]$

Range: $\big\{ y: \; x \geqslant -1 \big\}$

… … or $y \in \big[ -1, \; \infty \big)$

… … There are no stationary points.

… … There is an x-intercept at x = –1.5

… … There are no y-intercepts

**Note:**

- a negative sign outside the sqrt reflects the graph in the y-direction (across the x-axis)
- a negative sign with the x inside the sqrt reflects the graph in the x-direction (across the y-axis)

- the domain can always be found by setting the insides of the sqrt to > 0 then solving for x.

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