I think if you consider speed of the car as whole than the quadratic function gives a better estimate than the linear functions. But, if you consider the time interval individually then the linear function gives better estimates at specific times of the journey of the car.

Three linear functions have to be modelled in order to give a good accurate estimate of the car speed, but it is much easier to model the three linear functions because the linear models are simpler.However it doesn’t consider the fact that the car will sometime may slow down, for example towards the end of the journey or that at some point the car has to stop completely. The quadratic function takes in account that the car may slow down towards the end of its journey unlike the linear functions.

However, saying that the quadratic function is complicated and several steps of calculations are needed to work out the function and that it also doesn’t model the gear changes during the journey.So, considering both cons and pros of both linear and quadratic function, I feel that quadratic function gives us a better estimate of the car speed towards the end of the journey, in comparison to the original speed of the car. The linear functions give better and more accurate estimate of the speed through out the journey until when the car start to slow downs towards the end, which is after 12 seconds.Q10. To find an estimate speed of the car at 5 seconds it would be better to use the linear functions, because at this particular time the linear model is more accurate than the quadratic model. According to the linear model the speed of the car is 36.25mph at 5 seconds, this is after extending the 1st linear model up to 5 seconds.

 As you could see from the graph below the estimate has an error of 2.7%, compared to original speed at this particular time, which was 35.30mph. Therefore, it is valid to estimate with the first model, as it has a very small percentage error.

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 According the quadratic function the speed of the car at 5 seconds is 37.40mph, which has a percentage error of 5.9%. The percentage error is higher then the 1st linear functionsIt is better to use the quadratic function to predict the speed of the car at 18 seconds, because after 12 seconds the speed of the car decline on the original graph and towards the end of the journey the quadratic function also decline. Therefore, it is better to use the quadratic function to predict the speed of the car at 18 seconds. To predict the time at this particular time I extended the quadratic function to 19seconds.

 At 18 seconds the speed of the car is 66.78mph according to the quadratic function, which is shown below. It is not valid to use the linear model to predict the speed of the car at 18 seconds, because it will have a very high percentage error when compared to original speed.It is not possible to attain a percentage error for this particular speed for the quadratic function, as there is no speed value from the original graph to compare to.

 I can predict when the car stops, when the car journey reaches 30s, according to the extended quadratic function below. I think it will be better to use both linear functions and the quadratic function in order to estimate the speed of the car.To predict the speed you should use the appropriate function that best suits the time interval, for example towards the ends it will be appropriate to predict the speed using the quadratic function, and in the beginning it will be wiser to use the linear functions. Therefore, I find my three predictions to be as accurate as possible using both models. This is because the error is not that large in comparison to original speed of the car.ConclusionConsidering both linear and quadratic functions, I think the linear functions fits this situation better than the quadratic function. This is due to that the linear functions have lower percentage errors, and the quadratic function has higher percentage errors in comparison to the original speed of the car.

However, the quadratic function seems to better fit the speed of the car towards the end of its journey.Therefore, I think in order to estimate the speed of the car at any time interval you should consider which function fits the situation better, and then use the function that you think is better to predict the speed. I think it is better to use the linear function when predicting the speed in the beginning, for example in between the gear change that took place and use the quadratic function when predicting the speed during later stages of the car journey.