In Exercises 39–50, graph the given functions, f and g, in the sa...

Question

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x, g(x) = x + 3

Accepted Answer

Welcome back. I am so glad you're here. We are given two functions. The first is F. Of X. Which is equal to the square root of three times X. And that X is not inside the radical with the three. The second function we're given is G of X. And that is very similarly the squared of three times X. But then plus five. We are asked to graph both of these functions. And then describe the relation between the two functions on the graph. I'm going to draw a quick coordinate plane here. It's going to have a horizontal X. Axis of vertical Y axis and they come together at the origin in the middle. Now for graphing that F of X, I'm going to create an Xy table, plug in some values for X. Put those into the function. Figure out the corresponding Y values and then plot those points. I am going to choose negative 10 and positive one for the X values. Now plugging in negative one. So we have the square root of three times not X. But now negative one And -1 times the square root of three is the negative squared of three. Now plugging in zero. The squirt of three times not X but zero. Anything times zero is zero. So we've got a point at 00. Now plugging in one. The squared of three times one. Well that is just going to be the squared of three. Now we can graph, we can plot these points and keep in mind that for the squared of three, that's going to be somewhere in between one and two because we know that the squared of one is one and the square root of four is to the squared of three will be somewhere in between those. So now the first point negative one negative squared of three from the origin at 00 in the middle. We're going to go to the left one unit for our X value of negative one. And then we're going to go down negative squared of three values for our Y value. That's approximately here for the next 0.0, that is right at the origin. I will put that there and then label it 00 because that is both our X and Y intercepts. Now for the last 0.1 squared of three from the origin. We're going to the right one unit for X value of one. And then we'll go up squared of three units for our Y value of squared of three. That's approximately here. We can connect those points and there is our F of X. Great. One of them is graft. Now let's work on graphing our G of X. Again, we can create an xy table and we can plot the negative 10. And one point Again we could also figure out what our X intercept is going to be by putting a zero in for Y. So let's start off with negative one for our X value. That's going to be the squirt of three times not X. But negative one. And then plus five. Well again the squared of three times negative one is negative squared of three. But plus five. I'm just going to leave it like that actually negative squared of three plus five. So that's something between negative one and negative two plus five. Now plugging a zero in. So we are going to have the squirt of three times not X but zero plus five while the squirt of three times zero is zero. So this is zero plus five which will give us five. Now plugging one in the squared of three times one plus five, that will equal the squirt of three plus five. And I'm going to leave it like that. The squared of three plus five. Now plugging in zero for Y. That means R. G of x will be zero. So we have zero equals the square root of three times X plus five. Now solving for X, we need to subtract five from both sides. So on the left will have negative five equals and on the right we have the squared of three times X. To get X by itself. We're going to divide both sides by the squared of three. And I know this is kind of a messy solution having the squared of three in the denominator but it will work for what we're doing here just to find it on the graph. So we have an X value of negative five divided by the square of three. And now to plot these points for, I want to start actually with the 0.5. So from the origin we are going to go neither left nor right for our X value of zero. And we're going to go up five units for our Y value of five. So that will be approximately here. We'll label this 05 because that is our why intercept now note for these other two for the negative one X. Value and the positive one X. Value. Those are very similar to our affects Y values where X was negative one and positive one. The only difference is that they're adding five. So we can go from those other two points that we plotted and just move up five units from them. And that will be our negative one comma negative squirt of three plus five and our positive one squared of three plus five values. And now plotting finally our X intercept here from the origin, we're moving to the left negative five divided by squared of three units. And that's approximately here and then we're going neither up nor down for that Y value of zero. So this should be a parallel line. I didn't draw it perfectly but it would be a parallel line to our affects here is our G of X. The only difference here and this is how we could describe the relation between them is that the G of X shifted five units up from the F of X. So that's the key difference here. Now in trying to find the appropriate answer choice, we want to look for an F of X with an X and Y intercept at 00. And we want to find a G of X with a Y intercept at 05. And then the description is that G. Of X shifted five units upward from F of X. Alright, looking at answer choice, A F of X has an X and Y intercepted 00. Yay G of X has a Y intercept at 05. And it even has which I didn't label on this graph but it has the X intercept at negative five divided by the square root of three comma zero. And that's what they've got on this graph. The description says the graph of G. Is the graph of F shifted five units upward. Well, that's fantastic. Answer choice A is looking pretty good. Let's just make sure we haven't missed anything and rule out B. C. And D. So answer choice B says the graph of G. S. The graph of F shifted five units downward, nope, sorry, we didn't shift down, we shifted up answer choice C. Says the graph of G. S. The graph of F shifted square root of three units upward. No, we didn't shift squared of three units. We shifted five units upward. And D says the graph of G. S the graph of F shifted squared of three units downward, nope was neither the square root of three units, nor was it downwards. So answer choice A checks out and that is the correct answer choice. Well done, we'll catch you on the next one.

In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x, g(x) = x + 3

Key Concepts

Graphing Functions

Transformation of Functions

Coordinate System

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