Graphing Polynomials

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Section 4.2

• Graphing Polynomials

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May seem obsolete

• After all, you have a graphing calculator
• You must be smarter than your calculator

Graph and copy the graph of

• You must be able to know when the answer makes
  sense
• Calculators are tools, they do not think

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If P(x) is a polynomial function of degree n, the
graph of the function has

• At most n real zeros

How many x-intercepts?
y-axis

• At most n-1 turning points

x-axis
Relative or local maximum
How many x intercepts?
How many turning points?
What is the minimum degree?
What is the velocity at a turning point?
TI-84
Relative or local minimum
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Beginning the graph
After writing in standard polynomial form the
best place to begin is at the beginning
Ask what is the degree and leading coefficient
Even degree, positive coefficient
Even degree, negative coefficient
Odd degree, positive coefficient
Odd degree, negative coefficient
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Next, look for zeros (x-intercepts)
How? Factoring and other methods!
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When you have zeros (x-intercepts), they will
divide the x-axis into regions
Plot them on a number line to see the regions
0
-1/2
We now have three intervals of interest
g(1)-6
g(-.4).0256
g(-1)-2
We will test a value in each region to see if the
function is positive (above the line) or negative
( below the line
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So far we know

-
-
0
-1/2
Now find the y-intercept by setting x 0
To be accurate, select more points in each region
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Graph
Did the graph you copied earlier look like this?
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In Calculus

• You will learn about first derivatives
  (velocity), second derivatives (acceleration) and
  more.
• You should be able to graph the first 2
  derivatives, using this technique, when just
  given the graph of a function.

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The Intermediate Value Theorem

• Remember, polynomials are continuous
• A graph of a polynomial will cover values without
  skipping over some

Suppose we have a graph
and we have a negative y value along with a
positive y value
To get from negative to positive, can we get
there without going through zero?
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• For any polynomial function P(x) with real
  coefficients, suppose that

and
are opposite signs.

• Then, the function has a real zero between a and b

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Using the intermediate value theorem, determine,
if possible, whether the function has a real zero
between a and b
For the polynomial f(x), there is a real zero
between -5 and -4 because f(-5) and f(-4) have
opposite signs.
Unless there is a statement like this youve done
nothing except plug in numbers