graphing polynomial functions. finding the end behavior of a function degree leading coefficient...

GraphingPolynomial Functions

Finding the End Behavior of a functionDegree

LeadingCoefficient

GraphComparison

EndBehavior

As x –,

Rise right

Rise left

Fall right

Fall left

Rise right

Fall left

Fall right

Rise left

y = x2

y = –x2

y = x3

y = –x3

Positive

Negative

Positive

Negative

Even

Even

Odd

Odd

f(x) +

As x +, f(x)

As x +, f(x)

As x –, f(x)

As x +,

As x +,

As x –,

As x –,

f(x)

f(x)

f(x)

f(x)

+

–

–

+

–

–

+

Determine the end behavior of the function.

f(x) = 4x7 + 5x4 + 2

Degree:________

LeadingCoefficient: ___________

Graph Comparison:________

End Behavior

As x –, f(x) –

Odd

Positive

y = x3

As x +, f(x) +

Rises rightFalls left

1


f(x) = –7x6 + 2x2 – 3x

Degree:________


End Behavior

As x –, f(x) –

Even

Negative

y = –x2

As x +, f(x) –

Falls rightFalls left

Determine the end behavior of the function.2


f(x) = –2x5 – x3 + 6x

Degree:________


End Behavior

As x –, f(x) +

Odd

Negative

y = –x3

As x +, f(x) –

Falls rightRises left

Determine the end behavior of the function.3

Left Side

GraphComparison

4 Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

Right SideFalls Rises

y = x3

Degree:________Odd

Leading Coefficient: ___________Positive

Left Side

GraphComparison


Right SideRises Rises

Degree:________Even

Leading Coefficient: ___________Positive

y = x2

Left Side

GraphComparison


Right SideRises Falls

Degree:________Odd

Leading Coefficient: ___________Negative

y = –x3

The multiplicity of root r is the number of times that x – r is a factor of P(x).

Multiplicity

The graph of P(x) touchesthe x-axis, but does notcross it. (i.e. U-turn)

Odd MultiplicityThe graph of P(x)crosses the x-axis.

Even Multiplicity

Root a hasodd multiplicity

Root b haseven multiplicity

Find the following on chart for problems #5, #6, #7• Zeros (Solutions)• x-intercepts• Multiplicity of zeros• Explain what occurs on graph at

each x-intercept

Find the zeros and x-intercepts for the function.

Let P(x) = 0

2( ) ( 1) ( 3)P x x x 5

0 = (x – 1)2 (x – 3)

x – 1 = 0 , x – 3 = 0Let eachfactor = 0

Solve for x x = 1 x = 3

x-intercepts (1,0)

zeros

(3,0)

Find the multiplicity for each zero, and explain

what occurs at each x-intercept.2( ) ( 1) ( 3)P x x x

5

x = 1 x = 3Zeros

Multiplicity

At x = 1 2The graph touchesthe x-axis and turnsaround at the x-intercept

Multiplicity

At x = 3 1The graph crossesthe x-axis at the x-intercept.


Let P(x) = 0

6

x + 2 = 0 , x – 3 = 0Let eachfactor = 0

Solve for x x = –2 x = 3

x-intercepts (–2,0)

zeros

(3,0)

2 21( ) ( 2) ( 3)

12P x x x

2 210 ( 2) ( 3)

12x x


what occurs at each x-intercept.

6

x = –2 x = 3Zeros

Multiplicity

At x = –2 2The graph touchesthe x-axis and turnsaround at the x-intercept

Multiplicity

At x = 3 2

2 21( ) ( 2) ( 3)

12P x x x

The graph touchesthe x-axis and turnsaround at the x-intercept


Let P(x) = 0

7

0 = x(x – 3)(x + 2)

, x – 3 = 0 , x + 2 = 0Let eachfactor = 0

Solve for x x = 3 x = –2

x-intercepts (3,0)

zeros

(–2,0)

( ) ( 3)( 2)P x x x x

x = 0

x = 0

(0,0)


what occurs at each x-intercept.

7

Zeros

Multiplicity


Multiplicity


( ) ( 3)( 2)P x x x x

x = 3 x = –2x = 0

Multiplicity

At x = –2 1The graph crossesthe x-axis at the x-intercept.

• Degree• Leading Coefficient• Graph Comparison• End Behavior• Maximum Number of Turning Points

Find the following on chart for problems #5, #6, #7

Find the following. 2( ) ( 1) ( 3)P x x x

5

Degree:________


Graph Comparison:______

End Behavior

As x –, f(x) –

Odd

Positive

y = x3

As x +, f(x) +


2 + 1 = 3P(x) = 1(x – 1)2 (x – 3)1

1

Maximum NumberOf Turning Points = Degree – 1 = 3 – 1 = 2

Find the following.

6


Degree:________


End Behavior

As x –, f(x) +

Even

Positive

y = x2

As x +, f(x) +

Rises rightRises left

( ) ( 2) ( 3)P x x x 221

12

2 + 2 = 4

1

12


Find the following.

7

Degree:________



End Behavior

As x –, f(x) –

Odd

Positive

y = x3

As x +, f(x) +


1 + 1 + 1 = 3P(x) = 1x1(x – 3)1 (x + 2)1

1

P(x) = x(x – 3)(x + 2)


• Find y-intercept• Find Additional Points• Plot Points• Use Multiplicity• Draw Graph

Find the following on chart for problems #5, #6, #7

Find the y-intercept for the function.

Substitute x = 0.

y-intercept: (0,–3)

5

y = (x – 1)2 (x – 3)

y = (0 – 1)2 (0 – 3)

y = (–1)2 (–3) = (1)(–3) = –3Solve for y.

Find additional points. Substitute x-value between zeros x = 1 and x = 3.

Let x = 2: y = (2 – 1)2 (2 – 3)y = (1)2 (–1)y = (1)(–1) = –1

Point: (2,–1)

Draw graph.

y-intercept: (0,–3)

5y = (x – 1)2 (x – 3)

Point: (2,–1)

x-intercepts(1,0)(3,0)

Multiplicity21

Falls left / Rises rightEnd Behavior


Substitute x = 0.

y-intercept: (0,3)

6

= 3

Solve for y.

2 21( 2) ( 3)

12y x x

2 21( 2) (0 0 3)

12y

2 21(2) ( 3)

12y

1(4)(9)

12y

1(36)

12

6

2 21( 2) ( 3)

12y x x

Find additional points. Substitute x-value between zeros x = –2 and x = 3.

Let x = 1:

Point: (1,3)

2 21( 2) (1 1 3)

12y

2 21(3) ( 2)

12y

1(9)(4)

12y = 3

1(36)

12

Draw graph.

y-intercept: (0,3)

6

Point: (1,3)

x-intercepts(–2,0)(3,0)

Multiplicity22

Rises left / Rises rightEnd Behavior

2 21( 2) ( 3)

12y x x


Substitute x = 0.

y-intercept: (0,0)

7

Solve for y.

y = x(x – 3)(x + 2)

y = 0(0 – 3)(0 + 2)

y = 0(–3)(2)

y = 0

7

y = x(x – 3)(x + 2)

Find additional points. Substitute x-values between zeros x = –2, x = 0, and x = 3.

Let x = –1:

Point: (–1,4)

y = –1(–1 – 3)(–1 + 2)

y = –1(–4)(1) = 4

Let x = 1:

Point: (1,–6)

y = 1(1 – 3)(1 + 2)

y = 1(–2)(3) = –6

Let x = 2:

Point: (2,–8)

y = 2(2 – 3)(2 + 2)

y = 2(–1)(4) = –8

Draw graph.

y-intercept: (0,0)

7

Points(–1,4)(1,–6)(2,–8)

x-intercepts(0,0)(3,0)

(–2,0)

Multiplicity111

Falls left / Rises rightEnd Behavior

y = x(x – 3)(x + 2)

graphing polynomial functions. finding the end behavior of a function degree leading coefficient...

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