Graph each function.

Question

ƒ (x) = 2^{(} - ∣ x ∣)

Accepted Answer

Welcome back everyone. In this problem, we're going to draw the graph of the function F of X equals four to the power of negative absolute value of X. In our answer choices, we have possible graphs for a solution that are accompanied by the values for their points. And if we're going to figure out which one of these are the right ones, then we'll need to get a list of the points for ourselves. And then we can try to plot OK. So let's head back up to our question. OK. Our problem statement and let's see if we can try to figure out what the possible values are for or what, sorry, what the possible solution is for F of X. Now, first we can start by finding the Y intercept of F of F of X. OK. And to find the Y intercept, all we need to do is to substitute the value of X 40. In other words, we are finding F of zero. So that's going to be four to the power of negative absolute value of zero, which is four to the power of negative zero, which is four to the zero. And that equals one. So that tells us 01 is the Y intercept of our graph. The horizontal Asymptote is the X axis. So in other words, whether X increases or decreases, since we're dealing with the absolute value, it's going to go as close as it ca F FX will go as close as it can to the X axis, but it will never touch the X axis. OK. So that's one thing so that we know about our graph, it's going to be asymptotic. Now, if we're going to figure out which one of these graphs is the correct one, we can find a few more ordered pairs. So let's set up a table of X and F FX values. OK? And let's do them from negative three all the way up to positive two. OK. Now that we have these table or we have this table with these values, we can go ahead and solve for our F of X values and then we can do it by the same way that we found the Y intercept. In other words, when X equals negative three, then we would find F of negative three. So that would be four to the power of negative absolute value of three for X equals negative two. We're going to find F of negative two just forward to the power of negative absolute value of negative two. I should correct this here. That should have been negative three as well. OK? And now essentially we're going to go ahead and do the same for all of our values. OK. So that is for F of negative one F of zero F of one and F of two. Now for F of negative three, the absolute value of negative three is three. So that's four to the power of negative three which equals 164. So that's our value when X is negative three, when X is negative two, the absolute value of negative two is two. So that's four to the power of negative two, which equals 1/16. Likewise, uh you, you could continue the trend here for the rest of the terms. OK. And when you do that, when X is negative one FF negative one is 1/4. When X is zero, we already know F FX is one when X is one, F FX is 1/4. OK? And when X is two, F FX is 116. So now we have a few values that we can use to help us plot. OK? And we can plot by doing a quick sketch here. At least the sketch can give us an idea of how our graph is supposed to look. So if we have our graph of X and Y values, OK, then we know that. And if we, let's say this is negative three, negative two, negative 1012, OK. Now when X is negative three, F FX is 1 64th, we could say that it is right here. When it's negative two F FX is 1/16 then it's going to, when it's negative one FX is 1/4. OK? Then when it's zero FX is one, let's say that this is one, then when it's one, it's back to 1/4 when it's two, it's don not 1/16. So if we were to go ahead and try to plot our graph, then we can say that our graph is probably gonna look something like this. OK? What? Let me just again, this is just a sketch. We're not gonna get it perfectly. OK? So I can, I can just adjust my values here. My points here because the point OK is that we know our graph is going to look something like this with the values that we've mentioned. So to figure out which one of our values is correct, we can take a look. Now when we look at a, OK, notice that when X is negative two F of X is 33 16th. So we know that this one could not be correct. Neither is the Y intercept correct. When we go to answer choice B, notice that all of the values on B fit our table that we found for a table of values. So that means B is the correct answer to make sure we can look at answer choices C and D. When we look at answer choice C, we know that when X is negative two Y is negative 3116. So that can't be correct. OK. And we also see that our graph is below the X axis. And when we look at the plot for answer choice B, it doesn't look like the sketch that we have. OK. So we know for sure that answer choice B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Graph each function. $ƒ (x) = 2^{(} - ∣ x ∣)$

Key Concepts

Exponential Functions

Absolute Value

Graphing Techniques

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