Years after 1990 |
Solar energy consumption |

0 | 60 |

1 | 63 |

2 | 64 |

3 | 66 |

4 | 69 |

5 | 70 |

6 | 71 |

7 | 70 |

8 | 70 |

9 | 69 |

10 | 66 |

11 | 65 |

12 | 64 |

Years after 1990 |
Wind energy consumption |

8 | 31 |

9 | 46 |

10 | 57 |

11 | 68 |

12 | 106 |

**Inspecting the data**

What is clear from the above data sets is the definite increase of the years from 1 to 12 and from 8 to 12. This one quality makes the data appear linear. The wind and solar energy figures are varying on the constant increase over the years. In a linear equation, “Y” would be the number of years and “X” would be the consumption on both data sets.

**Making the decision**

The scatter plots have persuaded me that the data is more of a quadratic expression than a linear one. The graph shows data a trend line that is curved, a phenomenon that is only achievable in quadratic equations when they are put into a graphical expression.

**Scatter plots**

I expected the models to have a have a positive leading. This is because I expected that energy will increase as the population increases. However, this is not the case, especially in solar energy consumption. The consumption has dropped. Wind consumption has, however, increased as expected. It may have been the reason the drop occurred in solar energy as people moved from solar energy to wind energy.

**Fitting a model to data**

Y=a(x-h)^{2} + k

The graph open downwards,a situation that implies that a<0. From the graph “h” and “k” are the minimum and maximum values, which are 60 and 70 respectively.

Y-intercept is 60 from the graph (x, y = 0, 60)

Therefore, 60= a(0-60)^{2} + 70

= a(-60)^{2} + 70

a(-60)^{2} = 60 + 70

a(-60)^{2 }= 130

-360a = 130

a = – 0.361

Standard form of equation = Y= ax^{2} –ah^{2} + k

**Regression**

See the attached excel file.

**Regression on the solar consumption model **

**Cubic model for wind consumption graph **

**Finding the vertex algebraically**

** **

Y=a(x-h)^{2} + k

Therefore, 0 = a(0-h)^{2} + k

= 0(-h)^{2} + k

0= ah^{2} + k

ah^{2 }= k

a = – 0.361

– 0.361* h^{2 = }k

K = – 0.361h^{2}

Yes. The model does agree with the vertex used in question 3.

**Comparing actual data models**

From the data presented below, the two graphs can be compared based on the regression information they display. The solar energy graph shows that R^{2 } is 0.9509 while that of the wind energy consumption graph is 0.9677. This shows that wind consumption model fits the data more than the solar energy consumption model.

**Making a decision**

From the various models used in this paper, the best fit for the data I had initially is the one that was used for the wind energy consumption in the United States. Notably, this is because the R-Square from the regression representation of both graphs in higher in the wind energy graph than it is in the solar energy consumption. The equation used is as shown in the wind energy consumption above.