# Evaluating and Graphing Polynomials

## Evaluating Polynomial Functions: A Comprehensive Guide

Polynomial functions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving equations and recognizing different types of graphs. In this article, we will delve into the method of solving polynomials by graphing and identifying the various types of graphs based on their degree.

### What Are Polynomials?

Before diving into evaluation techniques, let's define what a polynomial is. A polynomial is a mathematical expression containing a variable raised to positive whole-number exponents, with each term being multiplied by a coefficient.

### The Standard Form of Polynomial Functions

Polynomial functions follow a standard form, as shown below:

**P(x) = ax ^{n} + bx^{n-1} + ... + cx + d**

It's important to note that polynomial functions are typically written in descending order, with the highest exponent first. Additionally, the exponents in polynomial functions must be positive whole numbers. Negative exponents, such as **x ^{-2}**, do not fit the criteria for polynomial functions.

### Evaluating Polynomial Functions: The Two Methods

When evaluating polynomial functions, there are two primary approaches: direct substitution and synthetic substitution.

#### Direct Substitution

Direct substitution involves substituting a given value for the variable and solving for the result. Here's an example:

- Evaluate P(x) when x = 2:
**P(2) = 3(2)**^{3}+ 14(2)^{2}+ 5(2) + 1**P(2) = 24 + 56 + 10 + 1 = 91**- Thus, P(x) = 91 when x = 2.

#### Synthetic Substitution

Synthetic substitution is a more efficient method for evaluating large polynomial functions. Let's use the same example from above to illustrate this technique:

- Step 1. Write down the coefficients of each term, with the given value of x to the left of the leading coefficient:

**| 2 | 3 | 14 | 5 | 1**

- Remember to add any missing terms with a coefficient of zero (0).
- Step 2. Multiply the leading coefficient by the given value of x and bring it down:

**| 2 | 3 | 14 | 5 | 1****| | 6 | | |****| 2 | 9 | 14 | 5 | 1**

- Add the results and write the sum below the horizontal line:

**| 2 | 9 | 14 | 5 | 1****| | 6 | | |****| 2 | 9 | 20 | 5 | 1**

- Continue this process until you reach the final value below the line, which is the solution for P(x) when x = 2:

**| 2 | 9 | 20 | 5 | 1****| | 6 | | |****| 2 | 9 | 62 | 5 | 1****| | 12 | | |****| 2 | 9 | 74 | 5 | 1**

- Therefore, P(x) = 74 when x = 2.

### Graphing Polynomial Functions

Graphs are visual representations of polynomial functions, and there are various types based on the degree of the function, which is the highest exponent. Here are some examples:

- Linear graphs have a degree of 1 and appear as straight lines.
- Quadratic graphs have a degree of 2 and form parabolas.
- Cubic graphs have a degree of 3 and are more curved than quadratic graphs.
- Quartic graphs have a degree of 4 and exhibit even more curvature than cubic graphs.
- And so on, with the degree increasing as the graph becomes more curved and changes direction.

### Evaluating Polynomial Functions by Graphing

As the degree of a polynomial function increases, its graph becomes more complex. It's crucial to use key features to accurately represent these functions. These key features include finding the x-intercepts (or zeros) of the function and determining the number of direction changes in the graph.

## Understanding Zeros, Y-Intercepts, and End Behavior of Polynomial Functions

Polynomial functions are expressions that consist of multiple terms with a variable raised to positive whole-number exponents and each term may have a coefficient. These functions can be evaluated, graphed, and analyzed to determine important characteristics such as zeros, y-intercepts, and end behavior.

### Finding the Zeros of a Polynomial Function

The zeros of a function are the values of x that make the function equal to zero. In order to find the zeros, the function is set equal to zero and various methods like factoring, division of polynomials, completing the square, or using the quadratic formula can be used.

For example, let's consider the polynomial function **P(x) = 2x ^{3} - 11x^{2} + 20x - 12**. Using the factoring method, we can rewrite this as

**P(x) = (x-2)(2x-3)(x+2)**. Thus, the zeros or x-intercepts of this function are 2, 3/2, and -2.

If a zero appears multiple times, the graph will touch the x-axis at that value and then change direction.

### Finding the Y-Intercept of a Polynomial Function

The y-intercept of a polynomial function can be found by substituting x = 0 into the original function. This will give us the y-coordinate where the curve crosses the y-axis.

### Determining the End Behavior of a Polynomial Function

The end behavior of a polynomial function refers to the behavior of the graph on either end. This can be determined using the leading coefficient test.

**Leading Coefficient Test**

The leading coefficient of a polynomial is the term with the highest exponent. To determine the end behavior, we need to look at the exponent of this term and the sign of its coefficient.

**Odd function (i.e. )**

**Positive leading coefficient:**The function will point downwards on the left and upwards on the right end of the curve.**Negative leading coefficient:**The function will point upwards on the left and downwards on the right end of the curve.

**Even function (i.e. )**

**Positive leading coefficient:**The function will point upwards on both ends of the curve.**Negative leading coefficient:**The function will point downwards on both ends of the curve.

The end behavior of a polynomial graph can be summarized using the table below:

Leading CoefficientOdd/Even FunctionLeft EndRight EndPositiveOddDownwardsUpwardsNegativeOddUpwardsDownwardsPositiveEvenUpwardsUpwardsNegativeEvenDownwardsDownwards

## Sketching a Polynomial Graph

To graph a polynomial, we can either use the leading coefficient test to determine the end behavior and draw a continuous and smooth curve, or we can make a table of values using direct substitution, plot the points on a coordinate plane, and connect them with a curve.

## Evaluating Polynomials

There are two methods for evaluating polynomials: direct substitution and synthetic substitution. The degree of a polynomial corresponds to the number of direction changes in its graph and the number of x-intercepts. The leading term can also help us determine the end behavior of the curve.

## What Are Polynomials in Math?

Polynomials are expressions with multiple terms containing a variable raised to positive whole-number exponents, and each term may have a coefficient. They can be evaluated, graphed, and analyzed to determine their characteristics.

## How to Evaluate a Polynomial?

Evaluating a polynomial means finding its solution for a given value of x. This can be done using direct or synthetic substitution.

## Examples of Evaluating Polynomials

**Problem:** Evaluate f(x) = 3x^2 + 4x - 5 at x = 2

**Solution:** f(x) = 3(2)^2 + 4(2) - 5 = 12 + 8 - 5 = 15

## What Does a Polynomial Graph Look Like?

The degree of a polynomial determines the shape of its graph. A polynomial with a degree of 2 or more will have a continuous and smooth line with possible maximum or minimum points in the middle and approaching positive or negative infinity on either end.

## Mastering Polynomial Functions: A Step-by-Step Guide to Graphing

Graphing polynomial functions may seem intimidating, but with these straightforward steps, you can confidently graph any polynomial function.

- Step 1: Find the Roots

The first step in graphing a polynomial function is to find its roots. This can be done through various methods such as factoring, using the quadratic formula, or completing the square.

- Step 2: Determine the y-Intercept

After finding the roots, the next step is to determine the y-intercept. This is where the graph intersects with the y-axis.

- Step 3: Conduct the Leading Coefficient Test

To determine the end behavior of the polynomial function, conduct the leading coefficient test. This involves considering the degree and leading coefficient of the function.

- Step 4: Sketch the Function

Using the information gathered in the previous steps, sketch the function on a graph. Begin by plotting the roots and y-intercept, then use the leading coefficient test to determine the end behavior of the graph.

Alternatively, you can also create a table of values by substituting various values of x and plotting the corresponding points on the graph. This method helps identify the middle section of the graph, and the curve can then be drawn smoothly, using the leading coefficient test to determine the end behavior.

It is essential to note that the polynomial function does not need to be factored for this approach to work.

Now equipped with these steps, you can confidently conquer polynomial functions. Happy graphing!