Big Ideas Math Algebra 2, 2014
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Big Ideas Math Algebra 2, 2014 View details
8. Analyzing Graphs of Polynomial Functions
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Exercise 36 Page 217

A turning point is either a local maximum or a local minimum of the function.

Turning Points:
Local maximum: None.
Local minimum:
Zeros: and
Smallest Possible Degree:

Practice makes perfect

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 coordinate of the point at which the graph intercepts the axis. The smallest possible degree is given by the number of zeros of the function.

Turning Points

We want to estimate the coordinate 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.


The graph has none relative maximum point and one relative minimum point.

coordinate of Relative Maximum Point coordinate of Relative Minimum Point
None About

Zeros

We want to estimate the zeros of the polynomial. A zero is the coordinate of the point at which the graph intercepts the axis. Let's consider the given graph.

The zeros occur at approximately and

Smallest Possible Degree

To find the least possible degree of a polynomial function, there are two things we should consider.

  1. The number of turning points plus one.
  2. The number of real zeros.
Whichever of these two numbers is larger is the smallest possible degree. Let be the number of turning points and be the degree of the given function. We know, that the polynomial function has at most turning points.
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 is turning point, giving us the following.
Now, let's consider the number of real zeros. Again from our work above we know the following.
We know that the function has two real zeros. Therefore, we may think its smallest possible degree is However, the graph is not a parabola. This means its degree is not Moreover, the end behavior is up and up. This means that the function has an even degree. Therefore, the smallest possible degree is