Analyzing Graphs of Polynomial Functions

Analyzing Graphs of Polynomial Functions

By Justin Groth

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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Graph

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Graph

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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).

Graph

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

Graph

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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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Graph

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

Multiple Choice

In the above graph complete the following end behavior:
As x --> -∞, f(x) --> ____
As x --> +∞, f(x) --> ____

• -∞
  -∞
• +∞
  -∞
• -∞
  +∞
• +∞
  +∞

Multiple Choice

In the above graph complete the following end behavior: (f(x)=y)
As x --> -∞, f(x) --> ____
As x --> +∞, f(x) --> ____

• -∞
  -∞
• +∞
  -∞
• -∞
  +∞
• +∞
  +∞

Multiple Choice

Which equation MOST LIKELY matches the graph?

• y = x⁴ - x³ + 3x² + 2
• y = -x⁴ + x³ + 5x + 2
• y = x³ - 2x² + 1x + 3
• y = -x³ - 2x² + 1x + 3

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.

​

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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Multiple Choice

How many zeros does the polynomial function have?

f(x) = x³ - 4x² -4x + 16

• 2

• 3

• 4

• 1

Multiple Choice

List the possible rational zeros of

f(x) = x³ - 4x² -4x + 16

• 1, 2, 4, 8, 16

• $$\pm1,\pm2,\pm4,\pm8,\pm16$$

• $$\pm1,\pm4,\pm\frac{1}{4}$$

• -2 and-4

Multiple Choice

List the all REAL zeros of

f(x) = x³ - 4x² -4x + 16

• $$\pm2,\ 4$$  

• $$\pm1,\pm2,\pm4,\pm8,\pm16$$

• $$2,\pm4$$

• $$\pm2,\ \pm4$$  

Multiple Choice

How could you determine    if x-2 is a factor of 2x³-5x²+x-2?

• Use synthetic division and see if the quotient is even
• Ask the person sitting next to me
• Use synthetic division and see if the remainder is zero
• Flip a coin

​​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

​Graph

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​Graph

Multiple Choice

The function shown in the graph is:

• even
• odd
• neither even nor odd
• both even and odd

Multiple Choice

Is the graph an even, odd, or neither function?

• Even
• Odd
• Neither

Multiple Choice

Which one of the following functions is even?

• f(x) = x⁴ + x³
• g(x) = x⁴ + x²
• h(x) = x⁵ + x³
• k(x) = x³ + x

Multiple Choice

Even, odd, or neither?

• Even

• Odd

• Neither

Multiple Choice

$$y=8x^{12}-8x^4+6x+22$$  

• Even Function

• Odd Function

• Function - neither even nor odd

• Not a function

Multiple Choice

• Even Function

• Odd Function

• Function - Neither Even Nor Odd

• Both Even and Odd Function

• Not a Function