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=Parabolas created using Geometric Sequences =

Samuel has been experimenting with quadratic equations of the form: //y //=//a x// ² +2 //b x// + //c // Samuel chose values of //a//, //b// and //c// by taking three consecutive terms from the sequence: 1,2,4,8,16,32...   Example quadratic equations might be:

//y //= //x// ²+//4// //x// + //4 // or //y //=2 //x// ²+<span style="font-family: MathJax_Main; font-size: 20px;">//8// //x// <span style="font-family: MathJax_Main; font-size: 20px;">+8

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Try plotting some graphs based on Samuel's quadratic equations, for different sets of consecutive terms from his sequence.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Do you notice anything interesting?

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Can you make any generalisations? Can you prove them?

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Samuel's sequence is an example of a geometrical sequence, created by taking a number and then repeatedly multiplying by a common ratio. Samuel's sequence starts a <span style="font-family: 'Comic Sans MS',cursive; font-size: 20px;">t <span style="font-family: 'Comic Sans MS',cursive; font-size: 160%;">1 <span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">and has common ratio <span style="font-family: 'Comic Sans MS',cursive; font-size: 160%;">2 <span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">(each number in the sequence is <span style="font-family: 'Comic Sans MS',cursive; font-size: 160%;">2 <span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;"> times the previous number).

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Create some more geometrical sequences and substitute consecutive terms into Samuel's quadratic equation.

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Here are some questions you might like to explore:

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Can you make any predictions about the graph from the geometric sequence you use to generate the equation?

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">What if the common ratio is a fraction, or a negative number?

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">What if the starting number for your geometric sequence is a fraction, or a negative number?

<span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">Can you make any generalisations? Can you prove them? <span style="font-family: 'Comic Sans MS',cursive; font-size: 140%;">