What happens when you evaluate polynomial functions for very large positive numbers and very large negative numbers? How does that manifest on the graph? Let’s investigate!
Without evaluating, determine if each expression will be positive or negative.
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Question 1
1.
100^{5}
1
1
1
1
Consider f(x)=x^{3}+5.
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Question 6
6.
What happens to the y-values as the x-values get bigger and bigger? Try a few values to investigate.
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Question 7
7.
What happens to the y-values as the x-values decrease without bound? Try a few values to investigate.
Consider g(x)=3x^{4}-x^{3}+5.
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Question 8
8.
What happens to the y-values as the x-values get bigger and bigger? Try a few values to investigate.
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Question 9
9.
What happens to the y-values as the x-values decrease without bound? Try a few values to investigate.
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Question 10
10.
How would your answers to numbers 8 and 9 change if the function was g(x)=-3x^{4}-x^{3}+5?
The graphs of the functions g(x)=-2x^{5} and h(x)=-2x^{5}+5x^{4}-10 are shown.
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Question 11
11.
Which graph is g? __________
Which graph is h? __________
Use the Show Your Work tool to explain how you know.
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Question 12
12.
Compare the end behavior of the two graphs.
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Question 13
13.
Which term in the polynomial seems to have the biggest impact on the end behavior of the graph?
Why do you think this is?
For very large positive and very large negative input values (which are at the end of graphs), the polynomial behavior is dominated by the leading term.
Polynomials have unbounded end behavior.
Right end behavior comes from whether the function is positive or negative (determined by the leading term.)
Positive: ____________ ↑
Negative: ___________ ↓
Then the degree fills in the left end behavior
Even degree's left end behavior has the same as the right end behavior.
Odd degree's left end behavior has the opposite of the right end behavior.
In Algebra, we mainly used ↑↑, ↓↑, ↑↓, and ↓↓.
But for Precalculus, we will discuss the limits.
\lim_{x\to\infty}f(x)=\infty reads as "the limit as x approaches infinity of f(x) is infinity."
Question 14
14.
and
and
and
and
↑↑
↓↑
↑↓
↓↓
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Question 15
15.
Use limit notation to describe the end behavior of the graph shown.
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Question 16
16.
Is it possible for this graph to have a degree of 5? Why or why not?
Question 17
17.
Which of the following terms, when added to the given polynomial, will change the end behavior? Check all that apply.
y=-2x^{7}+5x^{6}-24
Question 18
18.
Use limit notation to describe the end behavior of y=\frac{1}{6}(x-9)(x+4)^{2}.
Question 19
19.
The graph of f(x)=8x^{3}-5x^{6}+2x^{2}-24 has the same end behavior as...
