Maintaining Mathematical Proficiency

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Exercises 11 We are given an empty table of values corresponding to the function y=ax2.Since the value of a is not specified, we can just evaluate the expression in terms of a for the different values of x given. Let's try it for the first x-value, x=1 ax2x=1a(12)Calculate powera(1)Multiplya We can do the same for the rest of the x-values and complete the table.Now we can proceed to find the differences between consecutive y-values and see if we can identify a pattern.Can you identify the pattern? Notice that each difference equals the sum of the x-values involved, times a. y(2)−y(1)=3a=(2+1)ay(3)−y(2)=5a=(3+2)ay(4)−y(3)=7a=(4+3)a⋮​ We can use this pattern to find an expression for the function's value at x=6. For this, we can star from the difference of the function's values at x=5 and x=6, which we know will be equal to (6+5)a. Then, we can isolate y(6). y(6)−y(5)=(6+5)a⇕y(6) = (6+5)a +y(5)​​ Finally, we can find y(6) since we know y(5)=25a. y(6)=(6+5)a+y(5)y(5)=25ay(6)=(6+5)a+25a Simplify Add termsy(6)−25a=(11)a+25aMultiplyy(6)−25a=11a+25aAdd terms y(6)=36a Therefore, the value for the function when x=6 is 36a.