Exponential and Logarithmic Functions
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Today we are looking at more examples of exponential functions and specifically graphing them. Common problem is to ask you to graph a transformed exponential function. That’s what we are going to do today. But here I have a problem that says comparing graph 2 exponential function y equals ¼ times 2 to the x, and y equals 2 to the x minus 2.
Just remember. don’t forget your properties of exponents. You should always keep those in mind because it may turn out that using the properties of exponents. you can simplify a function, make it easier to graph. I’m going to try to use some properties right now on this guy.
Y equals ¼ of 2 to the x. Now ¼ is 2 to the -2. And by the product of power’s rule, this is 2 to the negative 2 plus x. Which of course is the same as 2 to the x minus 2. And what I’ve discovered is that, this function, y equals ¼ times 2 to the x, is exactly the same as this function. So I only need to graph one function.
Let me write that down y equals ¼ times 2 to the x, is the same as y equals 2 to the x minus 2. So I’m just going to graph the 1 function. Before I graph, I should make a table. And again you don’t need to plot too many points when you are graphing an exponential function.
Let me graph. let’s see, let’s start with 0. I’ll go 1, 2, 3 let’s see. Now I’m choosing these points based on what’s going to be easy to evaluate. I’m thinking about if I plug 0 in here I’m going to get 2 to the -2 that’s ¼. I don’t want to put negative numbers in for of x because I’m going to get a negative exponent that’s too small. I’m going to get a really tiny number that’s hard to plot so I’m sticking with numbers that are easy to work with.
When x equals 1, I get 2 to the 1 minus 2, that’s 2 to then -1, ½. When x equals 2 I get 2 to the 0 that’s 1. When x equals 3, I get 2 to the 3 minus 2, 2 to the 1 which is 2. So I’ve got 4 good points. I’m going to plot these and that will give me my graph.
So let’s start with (0, ¼). (1, ½) (2,1) and then (3, 2). And just connect these with a nice smooth curve and you’ve got your graph. This is y equals 2 to the x minus 2. So again, make sure you remember your properties of exponents. If you do, you might notice that one function is more easily graphed in the form of another.
Also, remember the transformations of exponential functions. Sometimes one transformation is the same as another. In this case, y equals ¼ times 2 to the x, a vertical compression, is exactly the same as, y equals 2 to the x minus 2, a horizontal shift.