Investigation: Graphs of Exponential Functions
Graphing 'Simple' Exponential Functions
Move the slider below to change the base of the exponential function .
After change the value of , you can compare you graph with by 'ticking' the 'Show ' option.
Question 1.
a. What does the dashed line represent, y=0, represent?
b. Compare the graphs andand describe in your own words how transforms as increases.
c. Compare the graphs andand describe in your own words how transforms as decreases.
d. Try to plot , what happens and why?
e. Why is it important to plot the coordinates of a second point (e.g when )?
(Hint: Imaging comparing two graphs using only the y-int)
Transforming Exponential Graphs
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!
That is , let's investigate the graph of where and can only be or and .
Question 2
a. Play around with the three transformations. Which is your favorite?
b. How does the equation of the asymptote change as the graph is vertically translated?
c. How does the equation of the asymptote change as the graph is horizontally translated?
d. Typically when performing transformations on graphs the order of transformations is significant.
Is the order of transformations significant is the graph above? If so what is the order of transformations?
e. Describe the difference between and .
f. Compare the 'basic' graph and the graph of when reflected in the axis. What do you notice?
Try explaining why using index laws?
g. What conditions are required for the exponential graph to intersect the axis? (Consider more than one transformation).
Extension (Optional):
Question 3
Using a CAS calculator or a Geogebra Calculator (https://www.geogebra.org/calculator):
a. Explore graphs of where and are not or .
b. Explore exponential graphs transformed by horizontal transformations. i.e. .
c. Bring it all together and explore graphs of the form .
(Bonus: Why is there now a where used to be?)