NZ Level 7 (NZC) Level 2 (NCEA) Transformations of Square Root Functions
Lesson

## Transformations on $y=\sqrt{x}$y=√x

The basic root function can be dilated and translated in a similar way to other functions.

The root function $y=\sqrt{x}$y=x can be transformed to $y=a\sqrt{x-h}+k$y=axh+k by dilating it using the factor $a$a and translating it, first horizontally $h$h units to the right, and then $k$k units upward. If the factor $a$a is negative, the basic curve reflects across the $x$x axis.

The subsequent translation then takes this reflected curve and moves the point at the origin to the point $\left(h,k\right)$(h,k).

## A word on notation

Strictly speaking, we really should write that if the original function $y_1=\sqrt{x}$y1=x is dilated by a factor $a$a and then translated right by $h$h and up by $k$k, then the translated function becomes $y_2=a\sqrt{x-h}+k$y2=axh+k. Most times though we just label both functions $y$y, because we understand that the basic function is different to the transformed function.

Using function notation, if $f\left(x\right)=\sqrt{x}$f(x)=x then, for example, after a dilation of $2$2, and translations of  $5$5 units to the right and $3$3 units down, we create a new transformed function, say $g\left(x\right)$g(x) ,where:

$g\left(x\right)=2\times f\left(x-5\right)-3=2\sqrt{x-5}-3$g(x)=2×f(x5)3=2x53

## Applet play

The best thing to do is to experiment with the first applet below showing how the variables $a,h$a,h and $k$k change the basic curve. Try both negative and positive values of $a$a.

Focussing on the domain and range of the transformed function $y=a\sqrt{x-h}+k$y=axh+k, note that we need $x-h\ge0$xh0 and thus $x\ge h$xh.

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems