Polynomial Function: Definition, Formulas & Graph Explained

Lesson Overview

• What Is a Polynomial Function?
• Types of Polynomial Functions with Graphs
• Polynomial Functions Graphs
• Turning Points of Polynomial Functions
• Roots of Polynomial Functions
• Solved Examples of Polynomial Function
• Polynomial Function Assessment

Polynomial functions are like curveballs. They twist and turn, rising and falling, just like a perfectly pitched ball. Imagine you're tracking how the height of a plant changes as it grows over time, where the growth follows a curved path. A polynomial equation can describe this growth perfectly.

What Is a Polynomial Function?

A polynomial function is a type of mathematical expression that involves a sum of terms, each consisting of a constant multiplied by a variable raised to a non-negative integer power. The key components of a polynomial are the coefficients (the constants) and the exponents (the powers of the variable).

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Types of Polynomial Functions with Graphs

Polynomial functions come in different "flavors" depending on their degree (the highest power of x). 

1. Constant Polynomial (P(x)=c)

A polynomial of degree 0, where P(x) is a constant value, represents a horizontal line.

• Example: If a ball is at rest on the ground and doesn't move, its height stays constant at 5 cm.
  P (x) = 5
  

2. Linear Polynomial (P(x) = ax + b)

A polynomial of degree 1 that forms a straight line. 

• Example: Imagine the ball is rolling on a flat surface. The height P(x) decreases linearly over time:

P (x) = -2x + 10

Here, the height decreases by 2 cm for every second.

3. Quadratic Polynomial (P(x) = ax² + bx + c)

A polynomial of degree 2 is a U-shaped or inverted U-shaped parabola.

• Example: When the ball is tossed into the air, it follows a parabolic trajectory. The height P(x) might look like:

P (x) = -3x² + 6x + 10

This equation describes how the height increases and then decreases due to gravity.

4. Cubic Polynomial (P(x) = ax³ + bx² + cx+ d)

A polynomial of degree 3 that can change direction up to two times forms an S-shaped curve.

• Example: Suppose a strong wind affects the ball's motion, causing an irregular bounce pattern. The height might be modeled as:

P(x) = x³ - 4x² + 3x + 5

5. Higher-Degree Polynomials (n ≥ 4)

Polynomials with degrees 4 or higher, showing complex patterns; wavy curves with more complex patterns.

Example: If the ball interacts with different surfaces like stairs or slopes, the motion might need a higher-degree polynomial to describe it.

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Polynomial Functions Graphs

The graph of a polynomial function is a smooth curve that can have multiple turning points, depending on the degree. 

Constant Polynomial: A horizontal line, showing no change in value.

Behavior: The value of the function remains constant for all xxx.

Linear Polynomial: A straight diagonal line, representing a consistent rate of change.
Behavior: It represents a consistent rate of change. The slope determines how steep the line is.

Quadratic Polynomial: A U-shaped (or inverted) parabola that illustrates a turning point.

Behavior: Contains one turning point, called the vertex.

Higher-Degree Polynomial: A wavy curve with more complex variations.
Behavior: The graph is influenced by the degree of the polynomial and the leading coefficient.

Turning Points of Polynomial Functions

Those points where you switch from going up to going down or vice versa are called turning points.

Types of Turning Points:

• Local Maximum: The highest point in a small section of the graph.
• Local Minimum: The lowest point in a small section of the graph.

Let's analyze the polynomial f(x)=x3−6x2+9x+2.

Steps to Find Turning Points:

Take the derivative to find the critical points where the slope is zero (or the graph is flat).

The derivative of f(x) is f′(x)=3x²−12x+9

Solve for critical points by setting f′(x)=0:

3x²−12x+9=0

Divide by 3: x²−4x+3=0

Factorize: (x−3)(x−1)=0

So, x=3 and x=1.

Analyze the turning points by plugging: x=3  and x=1 into f(x):

For x=3: f(3)=(3)³−6(3)²+9(3)+2=−4. 

For x=1: f(1)=(1)³−6(1)²+9(1)+2=6.

So, the turning points are at : (3,−4) and (1,6).

The graph above shows the polynomial function f(x)= x³− 6x²+ 9x + 2. The red dots highlight the turning points:

• Turning Point 1: (1,6)
• Turning Point 2: (3,−4)

These points represent where the roller coaster track changes direction, either peaking at a maximum or dipping to a minimum.

Roots of Polynomial Functions

Roots are the values of x that make a polynomial function equal to zero. They're also known as zeros or x-intercepts. The places where the graph of the function crosses the x-axis.

Understanding Roots

To find the roots of a polynomial, you need to solve the equation where the polynomial is set equal to zero. For example, let's look at a simple quadratic polynomial:

f(x) = x² - 5x + 6

To find the roots of this polynomial, we need to solve for x when:

x² - 5x + 6 = 0

Factoring the Polynomial

One way to solve this equation is by factoring the polynomial. We look for two numbers that multiply to give the constant term (6) and add up to the middle coefficient (-5).

We know that -2 and -3 work because:

• (-2) × (-3) = 6
• (-2) + (-3) = -5

So, we can rewrite the quadratic as:

(x - 2)(x - 3) = 0

Now, to find the roots, we set each factor equal to zero:

• x - 2 = 0 → x = 2
• x - 3 = 0 → x = 3

Therefore, the roots of the polynomial f(x) = x² - 5x + 6 are x = 2 and x = 3.

Solved Examples of Polynomial Function

Example 1:

Evaluate the polynomial function f(x) = 2x² + 3x + 5 for x = 2.

Step 1: Identify the polynomial function.
We are given f(x) = 2x² + 3x + 5.

Step 2: Substitute x = 2 into the polynomial.
f(2) = 2(2)² + 3(2) + 5.

Step 3: Simplify the expression.
f(2) = 2(4) + 3(2) + 5
f(2) = 8 + 6 + 5.

Step 4: Add the values.
So, f(2) = 19.

Example 2:
Find the value of the polynomial f (x) = x³ - 4x² + 6x for x = -1.

Step 1: Write down the polynomial.
f(x) = x³ - 4x² + 6x.

Step 2: Substitute x = -1 into the polynomial.
f(-1) = (-1)³ - 4(-1)² + 6(-1).

Step 3: Simplify the expression.
f(-1) = -1 - 4(1) + (-6)
f(-1) = -1 - 4 - 6.

Step 4: Add the values. 

So, f(-1) = -11.

Example 3:
Evaluate the polynomial f(x) = 3x⁴ - 2x³ + x² - 5x + 7 for x = 1.

Step 1: Write down the polynomial.
f(x) = 3x⁴ - 2x³ + x² - 5x + 7.

Step 2: Substitute x = 1 into the polynomial.
f(1) = 3(1)⁴ - 2(1)³ + (1)² - 5(1) + 7.

Step 3: Simplify the expression.
f(1) = 3(1) - 2(1) + 1 - 5 + 7
f(1) = 3 - 2 + 1 - 5 + 7.

Step 4: Add the values.
So, f(1) = 4.

Example 4:
Find the value of the polynomial  f(x) = 4x³ - 3x² + 2x - 6 for x = -2.

Step 1: Write down the polynomial.
f(x) = 4x³ - 3x² + 2x - 6.

Step 2: Substitute x = -2 into the polynomial.
f(-2) = 4(-2)³ - 3(-2)² + 2(-2) - 6.

Step 3: Simplify the expression.
f(-2) = 4(-8) - 3(4) + 2(-2) - 6
f(-2) = -32 - 12 - 4 - 6.

Step 4: Add the values.
So, f(-2) = -54.

Polynomial Function Assessment

1. Identify the degree and type of the polynomial:

• f(x) = 5x⁴ - 3x² + 2
• What is the degree of this polynomial?
• Is this a quartic, cubic, or quadratic function?

2. Graph the polynomial function:

• f(x) = x³ - 2x² + x - 3

3. Find the value of the polynomial function for a given x:

• f(x) = 2x³ - 3x² + x - 5
• What is f(2)?