In Exercises 47–52, graph functions f and g in the same rectangul...

Question

In Exercises 47–52, graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3^x and g(x) = (1/3). 3^x

Accepted Answer

Hello. Everyone in this video, we're going to be looking at this practice problem where we want to graph the functions F of X is equal to eight to the X and G of X is equal to one provided by 512 times A X. And when we graph it, we want to provide the equations for the ascent towards that we see. So in this case we're working with exponential functions and you can tell that we're working with exponential functions because the variable is in the exponent. So when you think of exponential functions, they follow the form why is equal to B to the X. And in this case exponential functions have special characteristics characteristics associated with them. So whenever B is positive, if I plug in a negative value Into Xa -2, this doesn't make why negative? It turns it into a fraction. So this would become one divided by B squared. So that means that if I just have a constant raised to X, I can never reach zero because every single time I increase the value of X, I'm simply making the fraction larger. So even if I plugged in B to the negative 1000, I would still just get one divided by B to the 1000. So if we look at our functions, you can see that F of X is eight to the X. So that means I know that all the values for F of X will be positive G of X is a little bit different because it has this fraction but the fraction is positive. So I can't assume that all the values for G. F X will also be positive also because there's nothing affecting the vertical translation of G of X. Since there's no plus or minus after the exponents. So that means I'm looking for a graph that has only positive Y values. So if we look at our answer choices, you can see that B has these negative values here. So now that B is already disqualified and C also has these negative values here. So it is already disqualified. But let's finish graphing or understanding these equations and start graphing them. So when I think of graphing them we should try to find the why intercept. So for the Y intercept that means that X is equal to zero. So if I have F. Of zero that is equal to eight to the zero power, which is equal to one. So that means for my F. Function, my x intercept is 01. I do the same for my G function G of zero is equal to one Divided by 512 times 8 to the zero. That becomes one divided by 512 times one, which is equal to one divided by 512. So the x intercept for G of zero is zero One divided by 5 12. Well this doesn't really help me much because if I were to look at that on the Y axis, say this is one And this is zero. My f function would have a point here. But my G function would have this tiny point here. And because they're exponential functions that would pass through the point and start to increase very quickly. So it could be hard to tell what I'm graphing. So just based on this preliminary graph, I am expecting my G function to be on the right of my F function. But we can take a look at this further Because if we look at five 112 this is equal to eight to the third. So I can rewrite my G function G of X is equal to one divided by 5 12 times eight X. I can rewrite this as one divided by eight to the third times eight X. And if I Right one divided by 8 to the third as eight to the negative three times eight X. You can see that I have two exponents both with a base of eight which means I can combine them To give me 8 to the power of X -3. So with this -3 I can see that there's a horizontal translation in this exponential function And because it's negative three, I know I need to shift the graph three units to the right. If it had been positive then that would be a shift to the left But I know that the -3 tells me three units to the right. So that means that if I'm looking at the Equation without the one divided by 5 12 in front. So F of X. I just need to shift it three units to the right, so 123. And that gives me a point for G of X at 31. So now I have more clearly written out my or graphed out my functions. And if we look at option choice a. You can see that we have the expected point at And if we go over three units to the right, we have the second expected point at 31. So answer choice A becomes our final answer. And you can also see that it labels out the Y ascent or the horizontal Y equals zero because both of our functions cannot equal zero. So there will be an ascent there. So A becomes our final answer. So hopefully this video helped and see you next time.

In Exercises 47–52, graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3^x and g(x) = (1/3). 3^x

Key Concepts

Exponential Functions

Asymptotes

Graphing Utilities

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