# 10.6 – Graphing Polynomials

## Key Terms

• Extreme Value – A value of the function y = f(x) that is the greatest or least value of y.
• The greatest value is called the maximum, and the least value is called the minimum.
• Graph of the Polynomial – The graph of an equation in which the right side is a polynomial.
• The graph is a curve with no breaks or sharp corners.
• Maximum – The highest point on the graph of a function, or the greatest value in a data set.
• Minimum – The lowest point on the graph of a function, or the smallest value in a data set.
• Roots – Values for which a function equals zero.
• The roots are also called zeros of the function.
• Any x-value at which the graph of a function crosses the x-axis is a root of the function.
• A root is shown on a graph when the y-value equals zero (y = 0)
• Ex. (4, 0), (-3.2, 0), (100, 0), etc.

## Review

Remember…
• The degree of a polynomial is its greatest exponent.

## Notes

Graphs of Polynomials
• The graph of any polynomial is continuous and smooth.
• It doesn’t have a break in it, and it doesn’t have any sharp corners.
• The graph of the equation shown is made up of all the points that satisfy it.
• The graph is a picture of the solution set.

• Polynomial graphs increase or decrease from left to right, and they will change direction if they have a degree of 2 or higher.

How to Graph a Polynomial
• Step 1: Make a table of x- and y-values for the polynomial.
• Substitute the given x-value into the equation.
• Solve the equation for y.
• Step 2: Plot those points in the xy-plane.
• Step 3: Connect the points with a smooth curve.

• Example

Higher Degree Polynomials
• As the degree of a polynomial increases, its graph gets more complicated and it can change direction more often.
• Tips for Graphing Higher-Degree Polynomials
• Plot more points before you connect them.
• Choose some bigger x-values.
• Polynomials with odd number degrees (3 or larger) have s-shape graphs.
• They usually have one hill and one valley.
• The ends start and end in opposite directions.
• Positive coefficients: starts down and ends up
• Negative coefficients: starts up and ends down
• Polynomials with even number degrees have u-shape graphs.
• Sometimes they look like w or m-shaped graphs.
• The ends start and end in the same direction.
• Positive coefficients: start and end up
• Negative coefficients: start and end down

• Examples

Extreme Values
• A extreme value of a polynomial is a value for which the polynomial is bigger or smaller than any other nearby values.
• A point on the graph of the polynomial that has a greater or smaller y-value than any nearby points on the graph
• Polynomials of degrees greater than 2 can have more than one maximum or minimum value.
• The largest possible number of minimum or maximum points is one less than the degree of the polynomial
• Maximum – A polynomial’s greatest / biggest value
• The highest point on a polynomial’s graph
• Looks like the top of a hill
• Minimum – A polynomial’s least value
• The lowest point on a polynomial’s graph
• Looks like the bottom of a valley
• Minimum Value of a Polynomial
• A point on the graph of the polynomial that has a smaller y-value than any nearby points on the graph

• Maximum Value of a Polynomial
• A point on the graph of the polynomial that has a greater y-value than any nearby points on the graph

Roots of a Polynomial
• The points where the graph crosses the x-axis
• The x-values are the roots when the y-values equal zero
• Roots are also called zeros of the function

• Follow these steps to use a graph to estimate its roots:
• Step 1: Use the scale and graph to identify a range for each root.
• Step 2: Try different x-values within that range.
• Step 3: When you find the x-value that gives y = 0, you have found the root.
• See examples at the bottom of this lesson to practice estimating the range of the roots of polynomial graphs.

Solutions of Graphs
• The point(s) of intersection – where the two graphs cross.
• If they never cross, there are zero solutions.
• If they cross once, they have one solution.
• If they cross twice, they have two solutions.