Here’s a graph and chart I made to help you understand how polynomials and their turning points work. I think polynomials should always be taught this way: graphing the individual terms, and showing that they are literally the combination of all of the y values of all the terms. And all junior high and up students should be taught how to graph such terms and polynomials in the ubiquitous Microsoft Excel, not just graphic calculators–Excel should be a part of any modern junior high/high school math class as a stepping stone to lots of work they’ll be doing in the future in Excel and other more math-dedicated programs and languages.

The first column in the table below is the x value, the second column is the value of the whole polynomial 4(x-1)^3 – 12(x-1)^2 +15, the third column is the value of the first term 4(x-1)^3, the fourth column is the value of the second term -12(x-1)^2, and the fifth column is the value of the third term, the constant 15.

You can see per x value how the positiveness or negativeness of the first and second terms, plus the constant 15, how the size and sign of the first two terms combined with the constant 15 add up to make the net of the whole polynomial positive or negative (or zero) at different points on the x axis. I put in x axis grid lines so you could count off grid marks too see how for the same x value, sometimes the first term has a higher positive than the second has a negative, and vice versa (of course offset to a degree by the constant 15).

Chapter 8 of Algebra II for Dummies has some great info on polynomials and their turning points also. The chapter also has great information on finding roots of polynomials, patterns to look for in factoring, and the Rational Root Theorem, which basically helps you find rational roots (rational numbers have terminating decimals or decimals that repeat in a pattern). You may need a graphing calculator/computer to estimate irrational roots, as they don’t factor cleanly/evenly).

x | 4(x-1)^3 – 12(x-1)^2 +15 | 4(x-1)^3 | -12(x-1)^2 | +15 |

-8 | -3873 | -2916 | -972 | 15 |

-7.75 | -3583.4375 | -2679.69 | -918.75 | 15 |

-7.5 | -3308.5 | -2456.5 | -867 | 15 |

-7.25 | -3047.8125 | -2246.06 | -816.75 | 15 |

-7 | -2801 | -2048 | -768 | 15 |

-6.75 | -2567.6875 | -1861.94 | -720.75 | 15 |

-6.5 | -2347.5 | -1687.5 | -675 | 15 |

-6.25 | -2140.0625 | -1524.31 | -630.75 | 15 |

-6 | -1945 | -1372 | -588 | 15 |

-5.75 | -1761.9375 | -1230.19 | -546.75 | 15 |

-5.5 | -1590.5 | -1098.5 | -507 | 15 |

-5.25 | -1430.3125 | -976.563 | -468.75 | 15 |

-5 | -1281 | -864 | -432 | 15 |

-4.75 | -1142.1875 | -760.438 | -396.75 | 15 |

-4.5 | -1013.5 | -665.5 | -363 | 15 |

-4.25 | -894.5625 | -578.813 | -330.75 | 15 |

-4 | -785 | -500 | -300 | 15 |

-3.75 | -684.4375 | -428.688 | -270.75 | 15 |

-3.5 | -592.5 | -364.5 | -243 | 15 |

-3.25 | -508.8125 | -307.063 | -216.75 | 15 |

-3 | -433 | -256 | -192 | 15 |

-2.75 | -364.6875 | -210.938 | -168.75 | 15 |

-2.5 | -303.5 | -171.5 | -147 | 15 |

-2.25 | -249.0625 | -137.313 | -126.75 | 15 |

-2 | -201 | -108 | -108 | 15 |

-1.75 | -158.9375 | -83.1875 | -90.75 | 15 |

-1.5 | -122.5 | -62.5 | -75 | 15 |

-1.25 | -91.3125 | -45.5625 | -60.75 | 15 |

-1 | -65 | -32 | -48 | 15 |

-0.75 | -43.1875 | -21.4375 | -36.75 | 15 |

-0.5 | -25.5 | -13.5 | -27 | 15 |

-0.25 | -11.5625 | -7.8125 | -18.75 | 15 |

0 | -1 | -4 | -12 | 15 |

0.25 | 6.5625 | -1.6875 | -6.75 | 15 |

0.5 | 11.5 | -0.5 | -3 | 15 |

0.75 | 14.1875 | -0.0625 | -0.75 | 15 |

1 | 15 | 0 | 0 | 15 |

1.25 | 14.3125 | 0.0625 | -0.75 | 15 |

1.5 | 12.5 | 0.5 | -3 | 15 |

1.75 | 9.9375 | 1.6875 | -6.75 | 15 |

2 | 7 | 4 | -12 | 15 |

2.25 | 4.0625 | 7.8125 | -18.75 | 15 |

2.5 | 1.5 | 13.5 | -27 | 15 |

2.75 | -0.3125 | 21.4375 | -36.75 | 15 |

3 | -1 | 32 | -48 | 15 |

3.25 | -0.1875 | 45.5625 | -60.75 | 15 |

3.5 | 2.5 | 62.5 | -75 | 15 |

3.75 | 7.4375 | 83.1875 | -90.75 | 15 |

4 | 15 | 108 | -108 | 15 |

4.25 | 25.5625 | 137.3125 | -126.75 | 15 |

4.5 | 39.5 | 171.5 | -147 | 15 |

4.75 | 57.1875 | 210.9375 | -168.75 | 15 |

5 | 79 | 256 | -192 | 15 |

5.25 | 105.3125 | 307.0625 | -216.75 | 15 |

5.5 | 136.5 | 364.5 | -243 | 15 |

5.75 | 172.9375 | 428.6875 | -270.75 | 15 |

6 | 215 | 500 | -300 | 15 |

6.25 | 263.0625 | 578.8125 | -330.75 | 15 |

6.5 | 317.5 | 665.5 | -363 | 15 |

6.75 | 378.6875 | 760.4375 | -396.75 | 15 |

7 | 447 | 864 | -432 | 15 |

7.25 | 522.8125 | 976.5625 | -468.75 | 15 |

7.5 | 606.5 | 1098.5 | -507 | 15 |

7.75 | 698.4375 | 1230.188 | -546.75 | 15 |

8 | 799 | 1372 | -588 | 15 |

Internal tag: math

Filed under: math, Uncategorized | Tagged: algebra, algebra II, calculus, education reform, Excel, graphs, math, math reform, turning points |

me, on February 13, 2010 at 10:08 am said:Fuck of you math loving homo!

trickledown, on February 13, 2010 at 5:58 pm said:lol!

M rickshaw, on July 3, 2011 at 12:26 am said:There is a ten percent chance that you are a homo. How’s that for math?

dana, on November 16, 2011 at 5:56 pm said:Loved this. I’m a teacher and I printed this off to show my students. Ignore the guy Feb 13th..I’m sure he does a great job washing cars, but obviously anything intellectual is WAAAAY out of his league.

Yo, on January 14, 2014 at 8:49 pm said:Noobs