NZ Level 7 (NZC) Level 2 (NCEA)

Transformations of Square Root Functions

Lesson

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`=`a`√`x`−`h`+`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`).

Strictly speaking, we really should write that if the original function $y_1=\sqrt{x}$`y`1=√`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 $y_2=a\sqrt{x-h}+k$`y`2=`a`√`x`−`h`+`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`(`x`−5)−3=2√`x`−5−3

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`=`a`√`x`−`h`+`k`, note that we need $x-h\ge0$`x`−`h`≥0 and thus $x\ge h$`x`≥`h`.

The new range, because of the lift (or fall) caused by $k$`k`, has also changed to the interval $k\le y<\infty$`k`≤`y`<∞ for values of $a>0$`a`>0. If however $a$`a` is negative the new range becomes $-\infty`y`≤`k`. Experiment with the applet, taking careful note of the natural domains and ranges.

We can extend our idea of root functions to include $n$`n`th roots rather than just square roots.

The nth root of x can be written as $x^{\frac{1}{n}}$`x`1`n`, where n is a positive integer greater than $1$1. The function $f\left(x\right)=\sqrt[n]{x}=x^{\frac{1}{n}}$`f`(`x`)=^{n}√`x`=`x`1`n` is defined for positive $x$`x` irrespective of the parity (oddness or evenness) of $n$`n` . However it is defined for negative $x$`x` only if $n$`n` is odd.

As an example, the domain for $f\left(x\right)=\sqrt[3]{x}$`f`(`x`)=^{3}√`x` includes all real numbers, but the domain for $f\left(x\right)=\sqrt[4]{x}=x^{\frac{1}{4}}$`f`(`x`)=^{4}√`x`=`x`14 is only defined for $x\ge0$`x`≥0. The reason that values of $x$`x` cannot be negative when $n$`n` is $4$4 (or indeed any even number), is because numbers like $\sqrt[4]{-1}$^{4}√−1 , $\sqrt[4]{-3.5}$^{4}√−3.5 etc. are not real.

This second applet, illustrating the general root function $y=a\sqrt[n]{x-h}+k$`y`=`a`^{n}√`x`−`h`+`k`, demonstrates this very well. It has the option of dilating and translating the basic root function just like the first applet, but also has a slider to increase $n$`n`. As you play with the sliders in combination, it is important that you record your new learnings somewhere.

Use the graph of $y=f\left(x\right)$`y`=`f`(`x`) to graph $y=f\left(x-3\right)+4$`y`=`f`(`x`−3)+4.

- Loading Graph...

Consider the function $y=2\sqrt{x}+3$`y`=2√`x`+3.

Is the function increasing or decreasing from left to right?

Decreasing

AIncreasing

BDecreasing

AIncreasing

BIs the function more or less steep than $y=\sqrt{x}$

`y`=√`x`?More steep

ALess steep

BMore steep

ALess steep

BWhat are the coordinates of the vertex?

Hence graph $y=2\sqrt{x}+3$

`y`=2√`x`+3Loading Graph...

Consider the function $y=-2\sqrt{x-2}$`y`=−2√`x`−2.

Is the function increasing or decreasing from left to right?

Decreasing

AIncreasing

BDecreasing

AIncreasing

BIs the function more or less steep than $y=\sqrt{x}$

`y`=√`x`?More steep

ALess steep

BMore steep

ALess steep

BWhat are the coordinates of the vertex?

Plot the graph $y=-2\sqrt{x-2}$

`y`=−2√`x`−2Loading Graph...

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

Apply graphical methods in solving problems