Chapter 5.9, Problem 41PPSE

Solution

The explanation for some quadratic polynomials cannot be written in the form $y=a{\left(x-h\right)}^4+k$ and give two examples also.

The required two examples are $y=x^4-x^2$ and $y=x^4+x^3-x^2$ .

Given Information:

The polynomial is defined as,

  $y=a{\left(x-h\right)}^4+k$

Calculation:

Consider the given polynomial,

  $y=a{\left(x-h\right)}^4+k$

The parent polynomial for the given function is $y=x^4$ . The given polynomial can be obtained by transformation of $y=x^4$ .

By using basic transformations, it is not possible to change the shape of the graph. It may at best reflect the graph across the axes to change its orientation. By transformations it is not possible to change the number of turning points.

First graph the function $y=x^4$ , which have only one turning point.

  

So, there exist a quadratic polynomial, which have three turning points. But the given polynomial $y=a{\left(x-h\right)}^4+k$ is not a quadratic polynomial with required turning points 3.

For example:

There are two polynomials $y=x^4-x^2$ and $y=x^4+x^3-x^2$ .

Now, draw these polynomials with parent polynomial.

The green graph is $y=x^4+x^3-x^2$ and the blue graph is $y=x^4-x^2$ .

  

Therefore, some quadratic polynomial cannot be written in the form of $y=a{\left(x-h\right)}^4+k$ .