[list][size=150][*]In Pattern A, the length and width of the rectangle grow by one small square from each step to the next.[/*][*]In Pattern B, the number of small squares doubles from each step to the next.[br][/*][*]In each pattern, the number of small squares is a function of the step number, [math]n[/math].[/*][/size][/list][br][table][tr][td][size=150]Pattern A[/size][/td][td][size=150]Pattern B[/size][/td][/tr][tr][td][img]data:image/png;base64,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[/img][/td][td][img]data:image/png;base64,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[/img][/td][/tr][/table][br][br][size=150]Write an equation to represent the number of small squares at Step [math]n[/math] in Pattern A.[/size][br]
Is the function linear, quadratic, or exponential?[br]
[size=150]Write an equation to represent the number of small squares at Step [math]n[/math] in Pattern B.[br][/size]
Is the function linear, quadratic, or exponential?[br]
How would the two patterns compare if they continue to grow? Make 1–2 observations.
[math]p\left(x\right)=6x^2[/math] and [math]q\left(x\right)=3^x[/math].[br][br]Investigate the output of [math]p[/math] and [math]q[/math] for different values of [math]x[/math]. For large enough values of [math]x[/math], one function will have a greater value than the other. Which function will have a greater value as [math]x[/math] increases?
Jada says that some exponential functions grow more slowly than the quadratic function as [math]x[/math] increases. Do you agree with Jada? Explain your reasoning.[br]
Could you have an exponential function [math]g\left(x\right)=b^x[/math] so that [math]g(x)[/math]<[math]f(x)[/math] for all values of [math]x[/math].