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A turning point is either a local maximum or a local minimum of the function.
Turning Points: (- 1.2,3), (0.2,- 48), (2,8), and (2.8,- 6)
Local Maximums: (-1.2,3) and (2,8)
Local Minimums: (0.2,- 48) and (2.8,- 6)
Real Zeros: - 1.4, -1, 1.5, 2.5, and 3
Smallest Possible Degree: 5
To begin let's recall the following information. A turning point is either a local maximum or a local minimum of the function. A zero is the x-coordinate of the point at which the graph intercepts the x-axis. The smallest possible degree is given by the number of zeros of the function.
We want to estimate the coordinates of every turning point. A turning point is either a relative maximum or minimum of the function. Let's look at the given graph to see where the turning points are.
Local Maximum Point | Local Minimum Point |
---|---|
About (- 1.2,3) | About (0.2,- 48) |
About (2,8) | About (2.8, - 6) |
We want to estimate the zeros of the polynomial. A zero is the x-coordinate of the point at which the graph intercepts the x-axis. Let's consider the given graph.
The zeros occur at approximately - 1.35, -1, 1.6, 2.5, and 3.
To find the least possible degree of a polynomial function, there are two things we should consider.
Whichever of these two numbers is larger is the smallest possible degree. Let k be the number of turning points and n be the degree of the given function. We know that the polynomial function has at most n-1 turning points. k ≤ n - 1 ⇔ k + 1 ≤ n The smallest possible degree of a polynomial function is given by the number of turning points plus one. From our work above, we know that there are 4 turning points, giving us the following. k turning points ⇒ k+1 degrees 4 turning points ⇒ 4+1 degrees = 5 Now, let's consider the number of real zeros. Again from our work above we know the following. n zeros ⇒ n degrees 5 zeros ⇒ 5 degrees Both gave us the same possibility, that the smallest possible degree is 5.