1.4 Polynomials

1.4
Polynomials

​The degree and leading coefficient of a function can tell us about multiple features of its graph.

Even degree polynomials have ends that either both go up or both go down.

Odd degree polynomials have ends that go in opposite directions.

A graph faces up on the right if it has a positive leading coefficient.

A graph faces down on the right if it has a negative leading coefficient.

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The function in the graph has an even degree and positive leading coefficient. We know this because both ends of the graph are going up forever.

End behavior in words:
As x is approaching negative infinity, y is approaching positive infinity. As x is approaching positive infinity, y is approaching positive infinity.

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Multiple Choice

What are the degree and leading coefficient of $$f\left(x\right)=-2x^4+8x^3+5x-1$$ ?

• Degree = 4
  LC = -2

• Degree = 7
  LC = 2

• Degree = 8
  LC = -2

• Degree = 4
  LC = -1

Multiple Choice

Which of the following could be the graph of $$y=-2x^4+8x^3+5x-1$$ ? Do not use a calculator!

Multiple Select

Select the interval(s) on which the function f(x) is increasing.

• $$\left(-\infty,-1.732\right)$$

• $$\left(-1.732,1.732\right)$$

• $$\left(-1.732,\ \infty\right)$$

• $$\left(1.732,\ \infty\right)$$

Multiple Select

Select the interval(s) on which the function f(x) is decreasing.

• $$\left(-\infty,-1.732\right)$$

• $$\left(-1.732,1.732\right)$$

• $$\left(-1.732,\ \infty\right)$$

• $$\left(1.732,\ \infty\right)$$

The point where a graph switches from increasing to decreasing is called a local, or relative, maximum.



The point where a graph switches from decreasing to increasing is called a local, or relative, minimum.

The function f(x) has a local, or relative, minimum at (1.732, -6.392).

The function f(x) has a local, or relative, maximum at (-1.732, 14.392).

There are no absolute, or global, extrema in this graph.

Multiple Choice

The point (0, 5) in the graph is a local

• maximum

• minimum

The point (0, 5) is considered an absolute, or global, minimum.


This is because 5 is the smallest output value of the entire function.

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Multiple Select

Select all local, or relative, minima.

• (-1.208, 8.314)

• (-3.191, -6.504)

• (0.649, -4.103)

Multiple Select

Select all local, or relative, maxima.

• (-1.208, 8.314)

• (-3.191, -6.504)

• (0.649, -4.103)

Multiple Choice

Does this graph have an absolute, or global, minimum?

• Yes, the point (-3.191, -6.504) is the absolute minimum because -6.504 is the smallest output value of the function.

• No, the function does not have a smallest output value. The y-values of the graph approach negative infinity.

• Yes, the point (0.649, -4.103) is the absolute minimum because -4.103 is the smallest output value of the function.

Multiple Choice

Does this graph have an absolute, or global, maximum?

• Yes, the point (-1.208, 8.314) is the absolute maximum because 8.314 is the largest output value of the function.

• No, the function does not have a greatest output value. The y-values of the graph approach positive infinity.

• Yes, the point (0.649, -4.103) is the absolute maximum because -4.103 is the largest output value of the function.

Match

Match the following graphs to their degree and leading coefficient.

Positive LC, Even Degree

Negative LC, Odd Degree

Negative LC, Even Degree

Positive LC, Odd Degree

• Positive LC, Even Degree

• Negative LC, Odd Degree

• Negative LC, Even Degree

• Positive LC, Odd Degree