A polynomial is any combination of power functions where n is equal to a positive integer or zero.
When a function has an even amount of turning points at the same point, we have a point of inflection in our graph.
A point of inflection is characterised by a gradient which approaches zero and is 0 at its point, just like a turning point. However unlike the turning point it then continues in the same direction instead of reversing
For the blue cubic function there is a point of inflection at (-2,0), since both the turning points are located at the same point. Note how the function flattens out and then continues in the same direction. A cubic function is also able to have two turning points at different co-ordinates instead of a point of inflection.
For the red quadratic function, the turning point we can see at (2,0). Note how the function flattens out and then reverses direction.
Sometimes a polynomial equation may appear in a form similar to the turning point form of a quadratic, but its not possible to display all polynomials in this form unlike a quadratic:
This form lets us know where the turning point or the point of inflection is straight away.
If n is a positive odd integer greater or equal to 3 then we know that it will be a point of inflection at (-h,k). In this form all the turning points are located in the one spot.
If n is a positive even integer greater or equal to 2 then we know that it will be a turning point at (-h,k). This is the same form as our turning point form for quadratics.
To draw the rest of the graph we go through the same process as the quadratic of setting x to zero to calculate the y intercept and then y to zero to calculate the x intercept.
For example if we were required to sketch:
Step 1: Locate the point of Inflection
As the function is in turning point form and to the power of an odd number the x co-ordinate of the point of inflection is easy to find. In this case it is (-1,2).
Step 2: Find the y intercept, by setting x to zero
Step 3: Find the x intercept by setting y to zero
Step 4: Connect the dots in a polynomial shape like so:
Receiving a polynomial in turning point form is easiest to sketch as it will always be either the S shape of the cubic above when is an odd number or the U shape when n is an even number. However, since not all polynomials can be put in this form, they are not all this easy to sketch.
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