[b][size=150]Graphing 'Simple' Exponential Functions[/size][/b][br][br]Move the slider below to change the [i]base [/i][math](a)[/math]of the exponential function [math]y=a^x[/math].[br][br]After change the value of [math]a[/math], you can compare you graph with [math]y=2^x[/math] by 'ticking' the 'Show [math]y=2^x[/math]' option.

[b]Question 1.[br][/b][br][b]a.[/b] What does the dashed line represent, y=0, represent?[br][br][b]b. [/b]Compare the graphs [math]y=2^x[/math] and[math]y=5^x[/math]and describe in your own words how [math]y=a^x[/math] transforms as [math]a[/math] increases.[br][br][b]c. [/b]Compare the graphs [math]y=2^x[/math] and[math]y=\left(\frac{1}{2}\right)^x[/math]and describe in your own words how [math]y=a^x[/math] transforms as [math]a[/math] decreases.[br][b][br]d.[/b] Try to plot [math]y=(-2)^x[/math], what happens and why?[br][br][b]e.[/b] Why is it important to plot the coordinates of a second point (e.g when [math]x=1[/math])? [br] ([i]Hint: Imaging comparing two graphs using only the y-int[/i])[br]

[b][size=150]Transforming Exponential Graphs[/size][/b][br][size=150][br][size=100]During our study of parabolas in semester 1 we learned of three different types of transformations: dilation, reflection, and translations. Let's try applying the reflection and translation transformations to our 'simple' exponential functions! [/size][br][br][size=100]That is , let's investigate the graph of [math]y=b\times a^{n\times x}+k[/math] where [math]b[/math] and [math]n[/math] can only be [math]+1[/math] or [math]-1[/math] and [math]k\in\mathbb{R}[/math].[/size][/size]

[b]Question 2[br][/b][br][b]a.[/b] Play around with the three transformations. Which is your favorite?[br][br][b]b. [/b]How does the equation of the asymptote change as the graph is vertically translated?[br][br][b]c. [/b]How does the equation of the asymptote change as the graph is horizontally translated?[br][br][b]d. [/b]Typically when performing transformations on graphs the order of transformations is significant. [br] Is the order of transformations significant is the graph above? If so what is the order of transformations?[br][br][b]e. [/b]Describe the difference between [math]y=-2^x[/math] and [math]y=\left(-2\right)^x[/math]. [br][br][b]f.[/b] Compare the 'basic' graph [math]y=2^x[/math] and the graph of [math]y=\left(\frac{1}{2}\right)^x[/math] when reflected in the [math]x-[/math]axis. What do you notice?[br] Try explaining why using index laws?[br][br][b]g.[/b] What conditions are required for the exponential graph to intersect the [math]x-[/math]axis? (Consider more than one transformation).[br][br][br]

[b][size=150]Extension (Optional):[br][br][/size]Question 3[br][/b][br]Using a CAS calculator or a Geogebra Calculator (https://www.geogebra.org/calculator):[br][br][b]a.[/b] Explore graphs of [math]y=b\times a^{n\times x}+k[/math] where [math]b[/math] and [math]n[/math] are not [math]+1[/math] or [math]-1[/math].[br][br][b]b. [/b]Explore exponential graphs transformed by horizontal transformations. i.e. [math]y=a^{x-h}[/math].[br][br][b]c. [/b]Bring it all together and explore graphs of the form [math]y=b\times a^{\frac{1}{n}\left(x-h\right)}+k[/math].[br] [i](Bonus: Why is there now a [/i][math]\frac{1}{n}[/math][i] where [/i][math]n[/math][i] used to be?)[/i]