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Wize Grade 11 Mathematics Textbook > Polynomials Factoring and Graphs

Graphs of Polynomials

Factoring PolynomialsPrevious Section
Graphing Polynomials using TransformationsNext Section

Basic Graphs of Polynomials



Basic Graphs

We can learn a lot about the graph of polynomials from looking at basic polynomials.

If the degree is
  • Even - the graph is similar to a quadratic or u-shape
  • Odd - the graph is similar to a cubic or side ways s-shape



Example 1

Use arrows to describe the end behavior of the following basic polynomials.

1. f(x)=x6f(x) = x^6

The end behavior is ↑ ↑\uparrow \ \uparrow

2. g(x)=x13g(x) = x^{13}

The end behavior is ↓ ↑\downarrow \ \uparrow

3. h(x)=xh(x) = x

The end behavior is ↓ ↑\downarrow \ \uparrow


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Domain and Range

All polynomials have a domain that includes all real numbers. If the degree is
  • Odd - the range will again include all real numbers
  • Even - there may will be some restrictions to the if the degree is even.
Example 2

Determine the domain and range of the following basic polynomials

1. f(x)=x17f(x) = x^{17}

  • Domain: All real numbers
  • Range: All real numbers

2. g(x)=x10g(x)= x^{10}

  • Domain: All real numbers
  • Range: [0,∞)[0, \infty)





End Behavior and Zeros



End Behavior

The end behavior of a polynomial describes the functions values as x gets large in the positive and negative direction.
The end behavior can be determined by leading term.

Degree
  • Odd: End behavior will be in opposite directions
  • Even: End behavior will be in the same direction

Leading Coefficient
  • Positive: Similar to basic polynomial with same degree
  • Negative: Similar to basic polynomial with the same degree, and reflected over x-axis
Example 1

Use arrows to describe the end behavior of the following polynomial functions.

1. f(x)=3x5−2x4+11f(x) = 3x^5-2x^4 + 11

The leading term is 3x53x^5 so the degree is odd and the leading coefficient is positive.
End behavior ↓ ↑\downarrow \ \uparrow

2. g(x)=−3x6+x4−4x3−x2g(x) = -3x^6 + x^4 - 4x^3 - x^2

The leading term is −3x6-3x^6 so the degree is even and the leading coefficient is negative.
End behavior ↓ ↓\downarrow \ \downarrow

3. k(x)=−x7+13xk(x) = - x^7 + 13x

The leading term is −x7-x^7 so the degree is odd and the leading coefficient is negative.
End behavior ↑ ↓\uparrow \ \downarrow

Wize Tip
To remember end behavior think of the basic polynomial graphs of x2x^2, −x2-x^2, x3x^3 and −x3-x^3.


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Zeros

The zeros of a function are the values of x that make the function equal to zero. These can been seen on the graph as places where the function touches the x-axis.


We can read the zeros of a polynomial by looking at the factors of the polynomial.
The exponent on a factor can give us the local behavior of the polynomial near the zero.
  • Even power: Graph touches at the zero
  • Odd: Graph go through the zero

Example 2

Determine the zeros of the following polynomials

1. f(x)=(x+3)(x−1)(x−2)f(x) = (x + 3)(x-1)(x - 2)

In this form we can see the factors of the polynomial. From the factor theorem we know that the zeros are at
x=−3,1,2x = -3, 1, 2.

2. g(x)=x2(2x−3)g(x) = x^2(2x - 3)

In this form we can see the factors of the polynomial. From the factor theorem we know that the zeros are at
x=0,32x = 0, \frac{3}{2}



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Graphing

We can sketch a graph of a polynomial by knowing both its end behavior and zeros. How we write the polynomial will help us find this information quickly.
  • Expanded form - all the terms have been multiplied out.
  • Factored form - the polynomial is factored

With this information we can graph a polynomial
  1. Graph End Behavior - Use expanded form to identity the leading term and highest degree
  2. Graph Zeros - Use factored form to mark out the zeros on the x-axis
  3. Connect the information with a smooth curve
Example 3

Sketch a graph of the polynomial

f(x)=−13(x+1)2(x−1)(x−3)f(x) = -\frac{1}{3} (x+1)^2(x-1)(x-3)

  • The zeros are at x=−1,1,3x = -1, 1, 3
  • The graph will touch at −1-1 and go through at 1,31, 3
  • The leading term is −13x4-\frac{1}{3}x^4 so the end behavior will be ↓ ↓\downarrow \ \downarrow

This gives us a sketch of the graph:



Example: Graphs of Polynomials


The design of a roller coaster can be modeled using the shape of a polynomial.
Suppose the following graph represents the track of a given roller coaster



1. Describe the end behavior of this graph.

The end behavior is ↓ ↑\downarrow \ \uparrow

2. What are the values and behaviors of the zeros.

The zeros are at
  • x=−1x=-1 touches
  • x=1x = 1 goes through
  • x=3x=3 touches

3. Find the equation of the polynomial that models the coaster if f(0)=95f(0) = \frac{9}{5}

From the previous part we know the polynomial must be of the form
f(x)=a(x+1)2(x−1)(x−3)2f(x) = a(x+1)^2(x-1)(x-3)^2
Putting in information when x=0x=0 we have

95=a(0+1)2(0−1)(0−3)295=a⋅915=a\begin{aligned} \frac{9}{5} &= a(0+1)^2(0 - 1)(0 - 3)^2 \\ \frac{9}{5} &=a\cdot9 \\ \frac{1}{5} &= a \end{aligned}

So the polynomial function is f(x)=15(x+1)2(x−1)(x−3)2f(x) = \frac{1}{5}(x+1)^2(x-1)(x-3)^2

Practice: Graphs of Polynomials

Match each polynomial expression with the arrows that represent their end behavior.
A.
↓ ↓\downarrow \ \downarrow
B.
↓ ↑\downarrow \ \uparrow
C.
↑ ↓\uparrow \ \downarrow
D.
↑ ↑\uparrow \ \uparrow
y=3x5−2x2+9y = 3x^5 - 2x^2 + 9
y=−7x3+7x−1y = -7x^3 + 7x -1
y=3x6+5x5−3x4+1y = 3x^6 + 5x^5 - 3x^4 + 1
y=−x4+2x3−7y = -x^4 + 2x^3 - 7

Practice: Graphs of Polynomials

Match each polynomial with a sketch of its graph.
A.
B.
C.
12x2(x+3)\frac{1}{2}x^2(x+3)
14x2(x+1)(x−3)\frac{1}{4}x^2(x+1)(x - 3)
y=−(x+1)2(x+3)y = -(x+1)^2(x + 3)

Practice: Graphs of Polynomials

Graph a sketch of the given polynomial
f(x)=−x4−x3−2x2f(x) = -x^4 - x^3 - 2x^2