Analyzing Graphs of Polynomial Functions

A Turning point of a graph of a polynomial functions is a point on the graph at which the functions changes from

• Increasing to decreasing

or

• Decreasing to increasing​

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Essential Question: How Many turning points can a graph of a polynomial functions have?

Turning Points

So far this chapter we have seen that zeros, factors, solutions and x-intercpets are closely related. Here is a summary of these relationships

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Step 1: Plot the x - intercepts, Because -3 and 2 are zeros of f, we will plot (-3,0) and (2,0).

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Step 2: Create a table and plot points beyond the x - intercpets as well as between

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Step 3: Determine the end behavior.

​ Because f(x) has 3 factors in the form (x-k) AND a constant factor of 1/6, f has a degree of 3 and is cubic, and has a positive LC. That means that the ends will approch opposite infinities. So, f(x) → −∞ as x → −∞ and f(x) → +∞ as x → +∞.

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Step 4: Draw the graph so that it passes all of the plotted points and has the appropriate end behavior.

If f is a polynomial function, and a and b are two real numbers such that f(a) < 0 and f(b) > 0, then f has at least one real zero between a and b.

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To use this principle to locate real zeros of a polynomial function, fi nd a value a at which the polynomial function is negative and another value b at which the function is positive. You can conclude that the function has at least one real zero between a and b​

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Location Principle

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Find all REAL zeros of

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Step 1) Use a graphing calculator to make a table​

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Find all REAL zeros of

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Step 2) Use the Location Principle. From the table shown, you can see that f(1) < 0 and f(2) > 0. So, by the Location Principle, f has a zero between 1 and 2. Because f is a polynomial function of degree 3, it has three zeros. The only possible rational zero between 1 and 2 is (3/2) . Using synthetic division, you can confirm that (3/2) is a zero.

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Find all REAL zeros of

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Find all REAL zeros of

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​​Functions can have "hills and valleys": places where they reach a minimum or maximum value. These are the turning points

It may not be the minimum or maximum for the whole function, but locally it is.

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Every polynomial with the degree of in has AT MOST n - 1 turning points.

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Local Maximum and Local Minimum

​1) Choose an interval

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2) The top of the hill or bottom of the vallley in that interval is your local maximum or minimum. Meaning for that function given that specific domain no point is ​high if it is a maximum or lower if it is a minimum.

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Write a maximum as it as f(a) ≥ f(x) for all x in the interval​

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Write a minimum as ​f(a) ≤ f(x) for all x in the interval

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Finding Local Maximum and Local Minimum