8. Analyzing Graphs of Polynomial Functions
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Consider a polynomial function with a specific number of turning points. How many times can it intersect the x-axis?
One less than its degree.
Let's start by reviewing what a turning point is.
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A turning point is a point on the graph at which the function changes from increasing to decreasing or decreasing to increasing. |
In the graph below, we can see the example function f(x) = 2x^3-2x, which has two turning points.
As we can see, a function with two turning points can intersect the x-axis up to three times, with three different real zeros. In general a function with n different real zeros has n-1 turning points. Now that we know the relation between the number of zeros and turning points, let's recall the Fundamental Theorem of Algebra.
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If f(x) is a polynomial of degree n where n>0, then the equation f(x) = 0 has exactly n solutions provided that each repeated solution is counted as many times as it repeats. |
According to the theorem, a polynomial function of degree n, where n>0, can have at most n different real solutions. Therefore, it can have at most n-1 turning points.