↑ Return to 10 – Polynomials

Print this Page

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.

Alg1B 10.6 Polynomial Graphs


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

Alg1B 10.6 Parts of a Graph

 

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

Alg1B 10.6 Plot Graph

Alg1B 10.6 Plot Graph Trace

 

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

Alg1B 10.6 Degree 3

Alg1B 10.6 Degree 4

 

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

Alg1B 10.6 Max

  • 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

Alg1B 10.6 Min

 

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

Alg1B 10.6 Roots

  • 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.

Alg1B 10.6 Solutions

Examples

  • Ex 1. The graph of a polynomial is shown below. Which of the following identifies the ordered-pair coordinates of the extreme value?

Alg1B 10.6 Q1-01

Answer: (2, 1)

  • Ex 2. The graph of a polynomial is shown below. Between which two x-values does this polynomial have an extreme value?

Alg1B 10.6 Q1-02

Answer: -3 to -2

  • Ex 3. The graph of a polynomial is shown below. At which value of x does this polynomial have an extreme?

Alg1B 10.6 Q1-03

Answer: x = 1

  • Ex 4. The equation and graph of a polynomial are shown below. The graph reaches its maximum when the value of x is 3. What is the y-value of this maximum?

Alg1B 10.6 Q1-04

Answer: 3

  • Ex 5. Below is the graph of a polynomial. Which statement about this graph is true?

Alg1B 10.6 Q1-05

The point (-2, -4) is a minimum.

  • Ex 6. Below are several points on the graph of a polynomial. Which of the following statements are true about the graph of this polynomial?

Alg1B 10.6 Q1-06

Answers:

There is a minimum between x = -1 and x = 0

The graph has a maximum value between y = 1 and y = 3.

  • Ex 7. The equation and graph of a polynomial are shown below. The graph reaches its maximum when the value of x is -2. What is the y-value of this maximum?

Alg1B 10.6 Q1-07

Answer: 3

  • Ex 8. The table shows the coordinates of several points on the graph of a polynomial.
    • x, y
    • -4, -4
    • -3, 2
    • -2, 1
    • -1, 5
    • 0, 2
    • 1, -2
    • 2, -5
    • 3, -1
    • 4, 3
    • According to the table (above), which point is a minimum of this polynomial?
      • Answer: (2, -5)
  • Ex 9. The points plotted below are on the graph of a polynomial. In what range of x-values must the polynomial have a root?

Alg1B 10.6 Q2-01

Answers:
0 to 1
4 to 5

  • Ex 10. The points plotted below are on the graph of a polynomial. In what range of x-values must the polynomial have a root?

Alg1B 10.6 Q2-02

Answers:
0.5 to 1.5
4.5 to 5.5

  • Ex 11. The points plotted below are on the graph of a polynomial. Which of the following x-values best approximate roots of the polynomial?

Alg1B 10.6 Q2-06

Answers:
x = -3.5
x = -2.3
x = -0.21
x = 1

  • Ex 12. The points plotted below are on the graph of a polynomial. How many roots of the polynomial lie between x = -4 and x = 3?

Alg1B 10.6 Q2-07

Answer: 4

  • Ex 13. How many solutions does the following set of polynomials have?

Alg1B 10.6 Q3-01

Answer: 2

  • Ex 14. How many solutions does the following set of polynomials have?

Alg1B 10.6 Q3-02

Answer: 1

  • Ex 15. How many solutions does the following set of polynomials have?

Alg1B 10.6 Q3-03Answer: 0

  • Ex 16. Which ordered pair is a solution to the system of polynomials graphed below?

Alg1B 10.6 Q3-04Answer: (1,-1)

  • Ex 17. Which ordered pair is a solution to the system of polynomials graphed below?

Alg1B 10.6 Q3-05

Answer: (1, 44)


Permanent link to this article: http://newvillagegirlsacademy.org/math/?page_id=4982