"Why = a(x - h)² + k?" -- EVERYTHING ... ABOUT A QUADRATIC
Area and the Quadratic |

Intro Area is the number of square units. In arithmetic, the numbers used to compute areas are constants. In algebra, the numbers used to compute area are often variables or variable expressions, linear expressions, like one more than a number, x+1, or double an number, 2x. In algebra, (line)(line) = (quadratic) so the quadratic is important in expressing area. |

Maximize the Area in Algebra II The perimeter of a rectangle is 24 cm. Find the maximum area of the rectangle. The 6 by 6 square yields the solution, an area of 36 square units In algebra II, the y-value of the vertex of the parabola ALWAYS yields the MAXIMUM or MINIMUM value depending on the concavity of the parabola. |

The area of a rectangle is (base)(height). Use x for the base and the fact that the perimeter is 24 units. The area becomes (x)(12-x) or -x² +12x. |

If you've studied the pages on composition of functions and dilation, note that this vertex happens when the two factors are equal -- when (x) equals (12-x), the solution of x=12-x, the value 6. |

If you've studied the use of the calculator, you will appreciate that in the CALC menu, the maximize function produces the best results it can. It states the side as 5.9999994 but states the accurate area of 36 as the maximum value of the function. |

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