Dilation and Symmetry

This interactive will allow you to explore polynomials with varying degrees.  Take particular note of the similarities that occur between polynomials with an odd highest degree and polynomials with an even highest degree.

• What happens when you change h?
• What happens when you change k?
• What happens when you change a?
• Does the behaviour or a, h or k change with different values of N?
• What is the effect of having an odd or even number when you change N? 

 

Now, the previous interactive only looked at $\left(h,k\right)$(h,k)-Form.  There are many different forms of polynomials, for example a polynomial of degree $5$5 could have up to $6$6 terms, $ax^5+bx^4+cx^3+dx^2+ex+f$ax5+bx4+cx3+dx2+ex+f.  Often its the existence of these middle terms that create the most interesting graphs.

We don't have to know about all functions to be able to graph them.  We can use our existing knowledge of dilation, reflection and translations that we have used with Quadratics and apply this to lots of various functions and curves.  

Dilation

The dilation of a graph is the value of the coefficient of the term with the highest power. This is also the term that identifies the degree of the function.  Remember that dilation is a measure of how spread out the function is in relation to the axes.

Here are some examples:

$y=3x^3+2x^5$y=3x3+2x5  has dilation factor of $2$2, because the highest power is $5$5 and the coefficient of $x^5$x5 is $2$2.

$y=-8x^6+9x^2$y=−8x6+9x2  has dilation factor of $8$8, because the highest power is $6$6 and the coefficient of $x^6$x6 is $-8$−8.

The function $y=-8x^6+9x^2$y=−8x6+9x2 has a greater dilation than the function $y=3x^3+2x^5$y=3x3+2x5 so it is more steep, less spread out.   

 

End Behaviour

We have already seen reflection parallel to the $x$x axis in quadratics, this resulted in quadratics that were either concave up or concave down.  Similar behaviour is exhibited by other polynomials when the value of $a$a becomes negative.  The reflection will mean that the end points of the function will point in opposite directions.

Positive quadratics - both end points point in the positive $y$y direction.

Negative quadratics -  both end points point in the negative $y$y direction. (opposite to the positive)

Have a look at the following polynomial graphs.  

These ones are all of odd degrees.

These ones are the same polynomials but with a negative '$a$a' value, causing a reflection.

These ones are all of even degrees

These ones are the same polynomials but with a negative '$a$a' value, causing a reflection.

Symmetry

Some polynomials exhibit reflective symmetry when they are sketched.  

Quadratics, for example, have a reflective symmetry across a vertical line that passes through the vertex.

If the quadratic has no horizontal translation, than this reflective symmetry occurs on the $y$y axis. Such as in these cases.

Generally speaking, if any polynomial is of the form $y=ax^n+k$y=axn+k, and $n$n is even, then it will have a reflective symmetry across the $y$y axis.  

Have a look at this interactive to see this in action.

If a polynomial is of the form $y=ax^n+k$y=axn+k and $n$n is odd then there is NO reflective symmetry as the end values of the functions go in opposite directions.  Have a look at this interactive to see that there just is not any symmetry on functions where the highest power is odd. 

Functions cannot have reflective symmetry across the $x$x axis because then they would fail the vertical line test and no longer be functions.  But there are curves (not functions) that exist that have this symmetry.  

Examples

Question 1

Question 2