Exercise 3 Page 205

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.

• The factor $a$ causes a reflection in the $x\text{-axis}$ 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 $n\ge 3$ 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 $y=x^4-5x^2+4.$ 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, $x^4-5x^2+4$ cannot be written in the form $a(x-h{)}^4+k.$