2.6 graphs of factorable polynomials

Graphs of Factorable Polynomials

http://www.lahc.edu/math/precalculus/math_260a.html

http://www.lahc.edu/math/precalculus/math_260a.html

Graphs of Factorable PolynomialsIn this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions.

Graphs of Factorable PolynomialsIn this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions. A basic fact about the graphs of polynomials is that their graphs are continuous lines:

x

x

Graphs of Polynomials


x

x


x x

Graphs of Non–polynomials


x

x


x x


To graph a function, we separate the job into two parts.


x

x


x x


To graph a function, we separate the job into two parts.I. What does the graph look like in the “middle”, i.e. the curvy portion that we draw on papers?


x

x


x x


To graph a function, we separate the job into two parts.I. What does the graph look like in the “middle”, i.e. the curvy portion that we draw on papers?II. What does the graph look like beyond the drawn area?


x

x


x x


To graph a function, we separate the job into two parts.I. What does the graph look like in the “middle”, i.e. the curvy portion that we draw on papers?II. What does the graph look like beyond the drawn area?We start with part II with factorable polynomials.

Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = ±xN.

Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = ±xN.The graphs y = xeven

y = x2


The graphs y = xeven

y = x2y = x4

(1, 1)(-1, 1)

We start with the graphs of the polynomials y = ±xN.



y = x2y = x4y = x6

(1, 1)(-1, 1)




y = x2y = x4y = x6

y = -x2

y = -x4

y = -x6

(1, 1)(-1, 1)

(-1,-1) (1,-1)




y = x2y = x4y = x6

y = -x2

y = -x4

y = -x6

y = ±xeven:

y = xeven y = –xeven

(1, 1)(-1, 1)

(-1,-1) (1,-1)



y = x3

The graphs y = xodd


y = x3

y = x5

(1, 1)

(-1, -1)

The graphs y = xodd


y = x3

y = x5

y = x7

(1, 1)

(-1, -1)

The graphs y = xodd


The graphs y = xodd

y = x3

y = x5

y = x7 y = -x3

y = -x5

y = -x7

(1, 1)

(-1, -1)

(-1, 1)

(1,-1)


The graphs y = xodd

y = x3

y = x5

y = x7 y = -x3

y = -x5

y = -x7

y = ±xodd

y = xodd y = –xodd

(1, 1)

(-1, -1)

(-1, 1)

(1,-1)

Graphs of Factorable PolynomialsFacts about the graphs of polynomials:

Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.

Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.• Polynomial curves are smooth (no corners).


• Let P(x) = anxn + lower degree terms.


• Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms.


• Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms. For x's such that | x | are large, the "lower degree terms" is negligible compare to anxn.


• Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms. For x's such that | x | are large, the "lower degree terms" is negligible compare to anxn. Hence, for x where |x| is "large", the graph of P(x) resembles the graph y = anxn.


• Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms. For x's such that | x | are large, the "lower degree terms" is negligible compare to anxn. Hence, for x where |x| is "large", the graph of P(x) resembles the graph y = anxn. This means there're four behaviors of polynomial-graphs to the far left or far right (as | x | becomes large).


• Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms. For x's such that | x | are large, the "lower degree terms" is negligible compare to anxn. Hence, for x where |x| is "large", the graph of P(x) resembles the graph y = anxn. This means there're four behaviors of polynomial-graphs to the far left or far right (as | x | becomes large). These behaviors are based on the sign the leading term anxn, and whether n is even or odd.

Graphs of Factorable PolynomialsThis behaviors of polynomial-graphs to the "sides":


y = +xeven + lower degree terms:

This behaviors of polynomial-graphs to the "sides":


y = +xeven + lower degree terms: y = –xeven + lower degree terms:





y = +xodd + lower degree terms:




y = +xodd + lower degree terms: y = –xodd + lower degree terms:

Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts to sketch the central portion of the graphs.

Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts to sketch the central portion of the graphs.Recall that given a polynomial P(x), it's sign-chart is constructed in the following manner:


Construction of the sign-chart of polynomial P(x):



I. Find the roots of P(x) and their order respectively.



I. Find the roots of P(x) and their order respectively.II. Draw the real line, mark off the answers from I.



I. Find the roots of P(x) and their order respectively.II. Draw the real line, mark off the answers from I.III. Sample a point for it's sign, use the orders of the roots to extend and fill in the signs.



I. Find the roots of P(x) and their order respectively.II. Draw the real line, mark off the answers from I.III. Sample a point for it's sign, use the orders of the roots to extend and fill in the signs.

Reminder: Across odd-ordered root, sign changesAcross even-ordered root, sign stays the same.

Example A. Make the sign-chart of f(x) = x2 – 3x – 4



Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0



Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered.



Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line.



Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line.

4-1



Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Test for sign using x = 0 and we get f(0) negative.

4-1



Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Test for sign using x = 0 and we get f(0) negative.

0 4-1Test x = 0, we get that f(0) negative.



Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Test for sign using x = 0 and we get f(0) negative. Since both roots are odd-ordered, the sign changes to "+" across the them.







+ +





The graph of y = x2 – 3x – 4 is shown below:

+ +


+ + + + + – – – – – + + + + + 4-1

y=(x – 4)(x+1)


Note the sign-chart reflects the graph:I. The graph touches or crosses the x-axis at the roots.

+ + + + + – – – – – + + + + + 4-1

y=(x – 4)(x+1)


Note the sign-chart reflects the graph:I. The graph touches or crosses the x-axis at the roots.II. The graph is above the x-axis where the sign is "+".

+ + + + + – – – – – + + + + + 4-1

y=(x – 4)(x+1)


Note the sign-chart reflects the graph:I. The graph touches or crosses the x-axis at the roots.II. The graph is above the x-axis where the sign is "+". III. The graph is below the x-axis where the sign is "–".

+ + + + + – – – – – + + + + + 4-1

y=(x – 4)(x+1)

Graphs of Factorable PolynomialsII. The “Mid-Portions” of Polynomial Graphs

Graphs of Factorable PolynomialsII. The “Mid-Portions” of Polynomial Graphs

Graphs of an odd ordered root (x – r)odd at x = r.


+ +

order = 1 order = 1

r r

II. The “Mid-Portions” of Polynomial Graphs


y = (x – r)1 y = –(x – r)1


+

+ +

order = 1 order = 1

order = 3, 5, 7..

r r

r

II. The “Mid-Portions” of Polynomial Graphs


+

order = 3, 5, 7..

r

y = (x – r)1 y = –(x – r)1

y = (x – r)3 or 5.. y = –(x – r)3 or 5..


order = 2, 4, 6 ..

++

x=r

r

Graphs of an even ordered root at (x – r)even at x = r.

order = 2, 4, 6 ..y = (x – r)2 or 4.. y = –(x – r)2 or 4..


order = 2, 4, 6 ..

++

If we know the roots of a factorable polynomial, then we may construct the central portion of the graph (the body) in the following manner using its sign chart.

x=r

r




order = 2, 4, 6 ..

++

If we know the roots of a factorable polynomial, then we may construct the central portion of the graph (the body) in the following manner using its sign chart.I. Draw the graph about each root using the information about the order of each root.II. Connect all the pieces together to form the graph.

x=r

r



Graphs of Factorable PolynomialsFor example, given two roots with their orders and the sign-chart of a polynomial,

Graphs of Factorable PolynomialsFor example, given two roots with their orders and the sign-chart of a polynomial,

+order=2 order=3

Graphs of Factorable PolynomialsFor example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown.

+order=2 order=3

Graphs of Factorable PolynomialsFor example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. Connect them to get the whole graph.

+order=2 order=3


Example B. Given P(x) = -x(x + 2)2(x – 3)2, identify the roots and their orders. Make the sign-chart. Sketch the graph about each root. Connect them to complete the graph.

+order=2 order=3



The roots are x = 0 of order 1,

+order=2 order=3



The roots are x = 0 of order 1, x = -2 of order 2,

+order=2 order=3



The roots are x = 0 of order 1, x = -2 of order 2, and x = 3 of order 2.

+order=2 order=3

Graphs of Factorable PolynomialsThe sign-chart of P(x) = -x(x + 2)2(x – 3)2 is

++x = 3order 2

x = 0order 1

x = -2order 2


++x = 3order 2

x = 0order 1

x = -2order 2

By the sign-chart and the order of each root, we draw the graph about each root.


++x = 3order 2

x = 0order 1

x = -2order 2

By the sign-chart and the order of each root, we draw the graph about each root. (Note for x = 0 of order 1, the graph approximates a line going through the point.)


++x = 3order 2

x = 0order 1

x = -2order 2

By the sign-chart and the order of each root, we draw the graph about each root. (Note for x = 0 of order 1, the graph approximates a line going through the point.)Connect all the pieces to get the graph of P(x).


x = -2order 2

++x = 0order 1

x = 3order 2

By the sign-chart and the order of each root, we draw the graph about each root. (Note for x = 0 of order 1, the graph approximates a line going through the point.)Connect all the pieces to get the graph of P(x).

Graphs of Factorable PolynomialsNote the graph resembles its leading term: y = –x5,

when viewed at a distance:

Graphs of Factorable PolynomialsNote the graph resembles its leading term: y = –x5,

when viewed at a distance:

-2

++0 3

2.6 graphs of factorable polynomials

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