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India
Class XI

Cube Root Functions, Transformations and Graphing

Lesson

 

Transformations on the basic curve 

Just like the square root curve, the transformed cubic root curve given as $y_2=a\sqrt[3]{x-h}+k$y2=a3xh+k is the basic function $y_1=\sqrt[3]{x}$y1=3x dilated by a factor $a$a and translated so that its centre point (at the origin) is relocated to the point $\left(h,k\right)$(h,k)

If $a$a is negative, the basic function is transformed by first reflecting in the $x$x axis, and then dilating and reflecting as before.

Dealing with the switched argument $\left(h-x\right)$(hx)

One very interesting property of the cubic root function is that the slightly changed form $y_3=a\sqrt[3]{h-x}+k$y3=a3hx+k , with the argument switched to $\left(h-x\right)$(hx), is simply a reflection (across the $x$x axis) of $y_2=a\sqrt[3]{x-h}+k$y2=a3xh+k.  

This is because:

$y_3$y3 $=$= $a\sqrt[3]{h-x}+k$a3hx+k
  $=$= $a\sqrt[3]{-\left(h-x\right)}+k$a3(hx)+k
  $=$= $a\times\sqrt[3]{-1}\sqrt[3]{\left(h-x\right)}+k$a×313(hx)+k
  $=$= $a\times\left(-1\right)\sqrt[3]{\left(h-x\right)}+k$a×(1)3(hx)+k
  $=$= $-a\sqrt[3]{\left(h-x\right)}+k$a3(hx)+k
  $=$= $-y_2$y2

Note that if $y_2$y2 has a negative dilation factor, then $y_3$y3 will be a reflection of an already reflected curve.

To emphasise this point, the curve $y=2\sqrt[3]{\left(3-x\right)}+5$y=23(3x)+5 is identical to the curve $y=-2\sqrt[3]{\left(x-3\right)}+5$y=23(x3)+5, and the curve $y=-2\sqrt[3]{\left(x-3\right)}+5$y=23(x3)+5 is identical to the curve $y=2\sqrt[3]{\left(3-x\right)}+5$y=23(3x)+5

That is, if you wish to switch the argument from $\left(h-x\right)$(hx) to $\left(x-h\right)$(xh), or vice versa, you must change the sign of the dilation factor.

A typical cube root function

Here is the graph of $y=-3\sqrt[3]{x-1}-2$y=33x12. The grey curve shows the basic cube root function  $y=\sqrt[3]{x}$y=3x . 

 

The Applets

The first applet allows you to vary the dilation factor and the translation constants $h$h and $k$k. Make sure you look at the full range of possibilities - in particular negative values of $a$a.

Applet 1

 

This second applet gives you the opportunity to compare two different transformed cube root functions - one with the argument under the cube root sign given as $\left(x-h\right)$(xh) and one with the argument as $\left(h-x\right)$(hx). Choose different values for the dilation factors and translation constants.

In particular, satisfy yourself that $y=2\sqrt[3]{\left(3-x\right)}+5$y=23(3x)+5 is identical to $y=-2\sqrt[3]{\left(x-3\right)}+5$y=23(x3)+5 and that $y=-2\sqrt[3]{\left(x-3\right)}+5$y=23(x3)+5 is identical to $y=2\sqrt[3]{\left(3-x\right)}+5$y=23(3x)+5

Applet 2

 

Worked Examples

QUESTION 1

Consider the function $f\left(x\right)=\sqrt[3]{x}$f(x)=3x.

  1. Complete the table of values below:

    $x$x $-8$8 $-1$1 $0$0 $1$1 $8$8
    $\sqrt[3]{x}$3x $-2$2 $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
  2. Now draw a graph of the function, making sure to include the point of symmetry.

    Loading Graph...

  3. What is the domain of the function?

    $x>0$x>0

    A

    All real numbers.

    B

    $x\le0$x0

    C

    $x\ge0$x0

    D
  4. What is the range of the function?

    All real numbers.

    A

    $y\le0$y0

    B

    $y>0$y>0

    C

    $y\ge0$y0

    D

QUESTION 2

Consider the function $f\left(x\right)=\sqrt[3]{x-1}+4$f(x)=3x1+4.

  1. Which of the following are points that lie on the graphed function? Select the two that apply.

    $\left(1,0\right)$(1,0)

    A

    $\left(0,4\right)$(0,4)

    B

    $\left(2,-6\right)$(2,6)

    C

    $\left(1,4\right)$(1,4)

    D

    $\left(2,5\right)$(2,5)

    E
  2. Graph the curve.

    Loading Graph...

  3. What is the domain?

    $x\ge4$x4

    A

    All real numbers.

    B

    $x\ge0$x0

    C

    $x\ge1$x1

    D
  4. What is the range?

    $y\ge0$y0

    A

    $y\ge1$y1

    B

    $y\ge4$y4

    C

    All real numbers.

    D

Outcomes

11.SF.RF.2

Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain and range of a function. Real valued function of the real variable, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum and greatest integer functions with their graphs. Sum, difference, product and quotients of functions.

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