Introduction to End Behavior
Object: To observe
the functional values of a polynomial as the value of the independent variable
increases and decreases without bound.
Exercises
- Look
at the graph of the polynomial . If you
use the Graphing Calculator software program, does it look like the graph
below?
We are not interested in specific
values of the function, we are only interested in what happens when we take
functional values of really big positive numbers or really big negative
numbers.
- Right
now, what do you think will happen to the values of f(x)
when taking larger positive x
values, or keep moving to the right on this graph?
- Also,
right now, what do you think will happen to the values of f(x)
when taking larger negative x
values, or keep moving to the left on this graph?
- LetŐs
make sure that our guess is true.
Use the zoom out feature of the graphing software. Does your Graphing Calculator
picture look like this?
- Does
this picture confirm your guesses from earlier? If it does not, what is your new guess?
- Use
the zoom out feature again.
Does that new picture still confirm your guesses or has it changed
again?
- Try
the zoom out feature repeatedly.
Has your guessed been confirmed in your mind?
- Fill
in the blanks with your conjecture.
i. As
the x values increases forever (keeps
going to the right), then the y
values _________________
ii. As
the negative x values decreases forever
(keeps going to the right), then the y values __________________
- Try
the same thing for the polynomial . Is your
conjecture the same as it was for Exercise #1?
- What
are the degrees of the polynomial in Exercises #1 and #2? What do those numbers have in
common? What are the leading
coefficients? What do these
numbers have in common?
- Use
those answers from Exercise #3 to fill in the blanks of the statement
below.
If the degree of the polynomial is
____________, and the sign of the leading coefficient is __________, then the
end behavior is as follows: as x approaches infinity (has a larger positive x value), then y approaches _________, as x
approaches negative infinity (has a larger negative x values), then y approaches _____________.
- LetŐs
try something slightly different: letŐs graph the polynomial .
- Trace
your graph on the axes provided below.
- What
happens to the y values as the x values get larger?
- What
happens to the y values as the x values become larger in the negative
direction?
- Use
the Zoom Out feature to see larger x
values. Sketch the graph
now.
- Are
your guesses still support by looking at a graph that is zoomed out?
- Keep
zooming out on your graphing software. Has your guessed been confirmed in your mind?
- Fill
in the blanks with your conjecture.
i. As
the x values increases forever (keeps
going to the right), then the y
values _________________
ii. As
the negative x values decreases forever
(keeps going to the right), then the y values __________________
- Try
the same approach for . Is your
conjecture still supported from Exercise #5?
- Use
those answers from Exercise #6 to fill in the blanks of the statement
below.
If the degree of the polynomial is
____________, and the sign of the leading coefficient is __________, then the
end behavior is as follows: as x approaches infinity (has a larger positive x value), then y approaches _________, as x
approaches negative infinity (has a larger negative x values), then y approaches _____________.
- Now
let us try a different polynomial: .
- Sketch
your graph on the axes below.
- What
happens to the y values as the x values approach infinity? In other words, in what direction
is the graph going as you go further to the right?
- What
happens to the y values as the x values approach negative infinity? In other words, in what direction
is the graph going as you go further to the left?
- Use
the Zoom Out feature to see what happens when you have larger x values.
What do you think is happening to the y values for large positive and negative x values?
- Are
your answers from parts b and c the same even after zooming out?
- If
you repeatedly zoom out, do you still get the same answers?
- Fill
in the blanks with your conjecture.
i. As
the x values increases forever (keeps
going to the right), then the y
values _________________
ii. As
the negative x values decreases forever
(keeps going to the right), then the y values __________________
- Try
the same thing approach for the polynomial . Do you
get the same answers when you keep zooming out?
- Use
those answers from Exercise #8 to fill in the blanks of the statement
below.
If the degree of the polynomial is
____________, and the sign of the leading coefficient is __________, then the
end behavior is as follows: as x approaches infinity (has a larger positive x value), then y approaches _________, as x
approaches negative infinity (has a larger negative x values), then y approaches _____________.
- Now
that you have the hang out, fill in the blanks for
- As
the x values increases forever
(keeps going to the right), then the y values _________________
- As
the negative x values decreases
forever (keeps going to the right), then the y values __________________
- Use
those answers from Exercise #11 to fill in the blanks of the statement
below.
If the degree of the polynomial is
____________, and the sign of the leading coefficient is __________, then the
end behavior is as follows: as x approaches infinity (has a larger positive x value), then y approaches _________, as x
approaches negative infinity (has a larger negative x values), then y approaches _____________.
- Sometimes the results from end
behavior can be expressed in a table form. Fill in the entries of the table with your answers from
earlier in the worksheet.
|
Degree of
Polynomial
|
Sign
of Leading Coefficient
|
|
Odd
|
Even
|
Positive
|
|
|
Negative
|
|
|