3.3 graphs of polynomial functions - dr. travers page of...

3.3 Graphs of Polynomial Functions

Roots

In this section, we want to look at how to take a function andbreak it into the pieces we need to be able to sketch the graph.

DefinitionA root of a polynomial f is a value x such that f (x) = 0

Does anyone know the Fundamental Theorem of Algebra? Forour purposes, it says ...

TheoremFundamental Theorem of AlgebraEvery real values polynomial of degree n has n roots that are notnecessarily distinct.

Roots



Does anyone know the Fundamental Theorem of Algebra?

Forour purposes, it says ...


Roots



Does anyone know the Fundamental Theorem of Algebra? Forour purposes, it says ...


Finding Roots

How can we find the roots of a polynomial?

Factoring

If we can write a polynomial in factored form, we can find theroots of the polynomial.

Example

Find the roots of f (x) = x2 − 4x + 3

x2 − 4x + 3 = 0(x− 3)(x− 1) = 0

x = 1, 3

Finding Roots


Factoring


Example


x2 − 4x + 3 = 0

(x− 3)(x− 1) = 0x = 1, 3

Finding Roots


Factoring


Example


x2 − 4x + 3 = 0(x− 3)(x− 1) = 0

x = 1, 3

Finding Roots

Example

Find the roots of f (x) = x3 − 3x2 − 4x + 12.

x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0

(x− 3)(x2 − 4) = 0(x− 3)(x− 2)(x + 2) = 0

x = ±2, 3

Finding Roots

Example


x3 − 3x2 − 4x + 12 = 0

x2(x− 3)− 4(x− 3) = 0(x− 3)(x2 − 4) = 0

(x− 3)(x− 2)(x + 2) = 0x = ±2, 3

Finding Roots

Example


x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0

(x− 3)(x2 − 4) = 0(x− 3)(x− 2)(x + 2) = 0

x = ±2, 3

Finding Roots

Example


x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0

(x− 3)(x2 − 4) = 0

(x− 3)(x− 2)(x + 2) = 0x = ±2, 3

Finding Roots

Example


x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0

(x− 3)(x2 − 4) = 0(x− 3)(x− 2)(x + 2) = 0

x = ±2, 3

An Example

Example

Find the roots of f (x) = x2 − 4x + 4.

x2 − 4x + 4 = 0(x− 2)(x− 2) = 0

x = 2

Anyone remember what a root with multiplicity 2 is called? Adouble root.

An Example

Example


x2 − 4x + 4 = 0

(x− 2)(x− 2) = 0x = 2


An Example

Example


x2 − 4x + 4 = 0(x− 2)(x− 2) = 0

x = 2


An Example

Example


x2 − 4x + 4 = 0(x− 2)(x− 2) = 0

x = 2

Anyone remember what a root with multiplicity 2 is called?

Adouble root.

An Example

Example


x2 − 4x + 4 = 0(x− 2)(x− 2) = 0

x = 2


Finding Roots

Example

Find the roots of f (x) = x5 + 3x3 − 4x.

x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0

x(x2 + 4)(x2 − 1) = 0x(x2 + 4)(x + 1)(x− 1) = 0

x = 0,±1

What about x2 + 4?

Finding Roots

Example


x5 + 3x3 − 4x = 0

x(x4 + 3x2 − 4) = 0x(x2 + 4)(x2 − 1) = 0

x(x2 + 4)(x + 1)(x− 1) = 0x = 0,±1

What about x2 + 4?

Finding Roots

Example


x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0

x(x2 + 4)(x2 − 1) = 0x(x2 + 4)(x + 1)(x− 1) = 0

x = 0,±1

What about x2 + 4?

Finding Roots

Example


x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0

x(x2 + 4)(x2 − 1) = 0

x(x2 + 4)(x + 1)(x− 1) = 0x = 0,±1

What about x2 + 4?

Finding Roots

Example


x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0

x(x2 + 4)(x2 − 1) = 0x(x2 + 4)(x + 1)(x− 1) = 0

x = 0,±1

What about x2 + 4?

Roots

Why are roots important?

The roots are the horizontal intercepts (x-intercepts) of thepolynomial. We need them to accurately plot the polynomials.

We also want to be able to find the vertical intercept - how dowe do this?

Example

Find the vertical intercept of f (x) = 3x2 − 3x + 1.

f (0) = 1, so the vertical intercept is (0, 1).

More Roots

What other methods do we know for finding roots?

Quadratic formula (only for quadratics)Completing the square (only for quadratics)Technology

Pictures and Roots

Single Root

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

Pictures and Roots

Double Root

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

Pictures and Roots

Triple Root

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

End Behavior

Our concern here is what happens as x→ ∞ and as x→ −∞.

Example

What is the end behavior of f (x) = x2 − 2x + 1.

What happens to f as x→ ∞? f (x)→ ∞.

What happens to f as x→ −∞? f (x)→ ∞.

End Behavior


Example


What happens to f as x→ ∞?

f (x)→ ∞.


End Behavior


Example




End Behavior


Example



What happens to f as x→ −∞?

f (x)→ ∞.

End Behavior


Example




End Behavior


Example

What is the end behavior of f (x) = −x3 + 3x2.

What happens to f as x→ ∞? f (x)→ −∞.


End Behavior


Example


What happens to f as x→ ∞?

f (x)→ −∞.


End Behavior


Example




End Behavior


Example



What happens to f as x→ −∞?

f (x)→ ∞.

End Behavior


Example




Putting it all together

Example

Plot f (x) = x2 − 5x + 6.

Roots? x = 2, 3, so (2, 0) and (3, 0)

y-intercept? (0, 6)

End behavior?


Example

Plot f (x) = x2 − 5x + 6.

Roots?

x = 2, 3, so (2, 0) and (3, 0)

y-intercept? (0, 6)

End behavior?


Example

Plot f (x) = x2 − 5x + 6.

Roots? x = 2, 3, so (2, 0) and (3, 0)

y-intercept? (0, 6)

End behavior?


Example

Plot f (x) = x2 − 5x + 6.

Roots? x = 2, 3, so (2, 0) and (3, 0)

y-intercept?

(0, 6)

End behavior?


Example

Plot f (x) = x2 − 5x + 6.

Roots? x = 2, 3, so (2, 0) and (3, 0)

y-intercept? (0, 6)

End behavior?


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

6

5

4

3

2

1

-4

-3

-2

-1

••

•

•

Graphing Polynomials

Example

Plot f (x) = (x− 2)(x− 1)(x + 2).

Roots? (2, 0), (1, 0), (−2, 0).

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

•••


Example

Plot f (x) = (x− 2)(x− 1)(x + 2).

Roots?

(2, 0), (1, 0), (−2, 0).

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

•••


Example

Plot f (x) = (x− 2)(x− 1)(x + 2).

Roots? (2, 0), (1, 0), (−2, 0).

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

•••


Example

Plot f (x) = (x− 2)(x− 1)(x + 2).

y-intercept? f (0) = (0− 2)(0− 1)(0 + 2) = 4, so we have (0, 4)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

•••

•


Example

Plot f (x) = (x− 2)(x− 1)(x + 2).

y-intercept?

f (0) = (0− 2)(0− 1)(0 + 2) = 4, so we have (0, 4)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

•••

•


Example

Plot f (x) = (x− 2)(x− 1)(x + 2).

y-intercept? f (0) = (0− 2)(0− 1)(0 + 2) = 4, so we have (0, 4)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

•••

•


Example

Plot f (x) = (x− 2)(x− 1)(x + 2).

End behavior?

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

•••

•


Example

Graph g(x) = (x− 1)(x + 2)2.

Roots? (1, 0) and (−2, 0) with multiplicity 2.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• •


Example

Graph g(x) = (x− 1)(x + 2)2.

Roots?

(1, 0) and (−2, 0) with multiplicity 2.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• •


Example

Graph g(x) = (x− 1)(x + 2)2.

Roots? (1, 0) and (−2, 0)

with multiplicity 2.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• •


Example

Graph g(x) = (x− 1)(x + 2)2.

Roots? (1, 0) and (−2, 0) with multiplicity 2.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• •


Example

Graph g(x) = (x− 1)(x + 2)2.

y-intercept? g(0) = (0− 1)(0 + 2)2 = −4 so (0,−4).

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• •

•


Example

Graph g(x) = (x− 1)(x + 2)2.

y-intercept?

g(0) = (0− 1)(0 + 2)2 = −4 so (0,−4).

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• •

•


Example

Graph g(x) = (x− 1)(x + 2)2.

y-intercept? g(0) = (0− 1)(0 + 2)2 = −4 so (0,−4).

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• •

•


Example

Graph g(x) = (x− 1)(x + 2)2.

End behavior?

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• •

•

Finding the Polynomial

Example

Find the equation of a polynomial with roots at x = ±1, 3 andwith y-intercept at (0, 2).

We know we have

f (x) = a(x− 1)(x + 1)(x− 3)

but we need to find the parameter a. How can we do this?

a(0− 1)(0 + 1)(0− 3) = 23a = 2

a =23

So, our equation is

f (x) =23(x− 1)(x + 1)(x− 3)


Example


We know we have

f (x) = a(x− 1)(x + 1)(x− 3)


a(0− 1)(0 + 1)(0− 3) = 2

3a = 2

a =23

So, our equation is

f (x) =23(x− 1)(x + 1)(x− 3)


Example


We know we have

f (x) = a(x− 1)(x + 1)(x− 3)


a(0− 1)(0 + 1)(0− 3) = 23a = 2

a =23

So, our equation is

f (x) =23(x− 1)(x + 1)(x− 3)


We can do the same thing from a picture ...

Example

Find an equation for the the polynomial given below.

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

4

3

2

1

-4

-3

-2

-1

• • •

•


Roots?

(3, 0), (1, 0), (−3, 0) with multiplicity 2.

So we have ...

f (x) = a(x− 3)(x− 1)(x + 3)2

Now ...

a(0− 3)(0− 1)(0 + 3)2 = 327a = 3

a =19

So,

f (x) =19(x− 3)(x− 1)(x + 3)2


Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.

So we have ...

f (x) = a(x− 3)(x− 1)(x + 3)2

Now ...

a(0− 3)(0− 1)(0 + 3)2 = 327a = 3

a =19

So,

f (x) =19(x− 3)(x− 1)(x + 3)2



So we have ...

f (x) = a(x− 3)(x− 1)(x + 3)2

Now ...

a(0− 3)(0− 1)(0 + 3)2 = 3

27a = 3

a =19

So,

f (x) =19(x− 3)(x− 1)(x + 3)2



So we have ...

f (x) = a(x− 3)(x− 1)(x + 3)2

Now ...

a(0− 3)(0− 1)(0 + 3)2 = 327a = 3

a =19

So,

f (x) =19(x− 3)(x− 1)(x + 3)2

Solving Inequalities

One more skill we need from here ...

Example

Solve (x− 4)(x− 1)(x + 3) > 0

What are the roots? x = 4, 1,−3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •



Example

Solve (x− 4)(x− 1)(x + 3) > 0

What are the roots?

x = 4, 1,−3

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •



Example

Solve (x− 4)(x− 1)(x + 3) > 0


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •



Example

Solve (x− 4)(x− 1)(x + 3) > 0


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

• • • •



Example

Solve (x− 4)(x− 1)(x + 3) > 0


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •


f (−4) = (−4− 4)(−4− 1)(−4 + 3)

f (−4) = (−)(−)(−) < 0f (−1) = (−1− 4)(−1− 1)(−1 + 3)

f (−1) = (−)(−)(+) > 0f (2) = (2− 4)(2− 1)(2 + 3)

f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)

f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−)

< 0f (−1) = (−1− 4)(−1− 1)(−1 + 3)

f (−1) = (−)(−)(+) > 0f (2) = (2− 4)(2− 1)(2 + 3)

f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)

f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0

f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0

f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)

f (−1) = (−)(−)(+) > 0f (2) = (2− 4)(2− 1)(2 + 3)

f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)

f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+)

> 0f (2) = (2− 4)(2− 1)(2 + 3)

f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)

f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0

f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0

f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0

f (2) = (2− 4)(2− 1)(2 + 3)

f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)

f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0

f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+)

< 0f (5) = (5− 4)(5− 1)(5 + 3)

f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0

f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0

f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0

f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0

f (5) = (5− 4)(5− 1)(5 + 3)

f (5) = (+)(+)(+) > 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0

f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0

f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+)

> 0


f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0

f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0

f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0

f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+) > 0


-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •

f (x)

f (x)

f (x)

f (x)

So, (x− 4)(x− 1)(x + 3) > 0 on (−3, 1) ∪ (4, ∞)

Last Example

Example

Solve (x + 2)(x− 1)2 > 0

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •

f (x)

f (x) f (x)

Last Example

Example

Solve (x + 2)(x− 1)2 > 0

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

• • •

f (x)

f (x) f (x)

Last Example

Example

Solve (x + 2)(x− 1)2 > 0

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •

f (x)

f (x) f (x)

Last Example

Example

Solve (x + 2)(x− 1)2 > 0

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •

f (x)

f (x)

f (x)

Last Example

Example

Solve (x + 2)(x− 1)2 > 0

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •

f (x)

f (x) f (x)

Last Example

Example

Solve (x + 2)(x− 1)2 > 0

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •

f (x)

f (x) f (x)

(x + 2)(x− 1)2 > 0 on the interval

(−2, 1) ∪ (1, ∞)

Last Example

Example

Solve (x + 2)(x− 1)2 > 0

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •

f (x)

f (x) f (x)

(x + 2)(x− 1)2 > 0 on the interval (−2, 1) ∪ (1, ∞)

3.3 graphs of polynomial functions - dr. travers page of...

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