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# Chapter 2:Variations

### Variations and Graphs

Question | Answer |
---|---|

What are some examples of a constant of variation? | r=5c, r=10. They are all a form of y=kx^n where k is a nonzero constant and n is a positive number. |

Ex: The weight w of an adult animal of a given species is known to vary directly with the cube of its height h. a. Write an equation relating w and h. b. Identify the dependent and independent variables. | Solution: An equation for the direct varation is w=kh^3 b. Because w is given in terms of h, The dependent variable is w and the independent variable is h. |

What is an inverse-variation function? | Its a function with a formula of the form y=k/x^n, with k not equaling 0 and n being greater than 0. |

Ex: The number n of oranges you can pack in a box is approximentely inversely proportional to the cube of the average diameter d of oranges. Write an equation to express this relation. | Solution: The cube of the diameter of d^3. So, n is = k/d^3 |

The Fundamental Theorem of Variation | a. If y varies directly as x^n( That is, y=kx^n), and x is multiplied by c, then y is multiplied by c^n. b. If y varies inversely as x^n( That is, y=k/x^n), and x is multiplied by a nonzero constant c, then y is divided by c^n. |

Formula for Slope of a Line | =changes in vertical distance/ change in horizontal distance = change in dependent variable/ change in independent variable =rise/run |

Domain and range k>0 | The domain of the function with equation y=kx^2 is the set of all real numbers. When K > 0, the range is the set of nonnegative real numbers, and the parabola opens up. |

Domain and range k<0 | The range is thet set of nonpositive real numbers and the parabola opens down. That is, the vertex of the parabola is its maximum point. |

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