We can apply transformations to a polynomial parent function of the form

f (x) = x^{n}

in the same way we do with a quadratic function in vertex form.

f (x) = a (x - h)^{n} + k

The factor $a$ causes a reflection in the $x - axis$ if $a < 0 .$ Furthermore, if $∣ a ∣ < 0$ it also causes a vertical shrink. And if $∣ a ∣ > 0$ it causes a vertical stretch.

The constant $h$ indicates a horizontal translation. If $h < 0$ it causes a translation to the left by $∣ h ∣$ units. On the other hand, if $h > 0$ it will cause a translation to the right by $h$ units.

The constant $k$ indicates a vertical translation. If $k < 0$ it will cause a translation downwards by $∣ k ∣$ units. And if $k > 0$ it will cause a translation upwards by $k$ units.

It is important to keep in mind that by using these transformations for polynomials of degree $n$ with $n \geq 3$ we cannot generate all the different possible polynomials of degree $n .$ For example, consider the polynomial function $f (x) = x^{4} .$

Notice that by applying all the different transformations discussed above, the resulting offspring function, $g (x),$ will have the same general shape and behavior of the parent function. An example can be seen below.

Now let's consider the quartic function $y = x^{4} - 5 x^{2} + 4 .$ This function has a different shape and behavior than that of $f (x) = x^{4},$ which cannot be obtained by translating, stretching, compressing or reflecting $f (x) = x^{4} .$ In other words, $x^{4} - 5 x^{2} + 4$ cannot be written in the form $a (x - h)^{4} + k .$

Exercise

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