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. 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≥ 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.