In Exercises 55–58, graph each function, not by plotting point...

Question

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
y = - √(1 + x/2)

y = - √(1 + x/2)

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Graph the function Y equals negative square root of x divided by 2 + 4 by starting with the graph of Y equals the square root of X and applying approximate transformations. Awesome. So it appears for this particular problem we're ultimately trying to graph this function of Y, which the specific function of Y is equal to the square root of X divided by 2 + 4, and we're asked to graph this specific function of Y by starting out with our graph of Y equals the square root of X, and we're also asked to accomplish creating this graph of. Why? By applying approximate transformations. Awesome. So now that we know what we're ultimately trying to solve for, what we need to do first off is to start off with our parent function, which we're going to focus our attention on Y equals the square root of X. So Y equals the square root of X. So let us note that the domain of Y equals the square root of X will be X is greater than or equal to 0. So now we need to evaluate the parent function at arbitrary values of X within our domain. So when we have it at X equals 0, that will mean that Y is equal to the square root of 0, which is going to be equal to 0. And in an effort to save time moving forward is that saying square root of any value of X over and over and over again, I'm just going to give you what the value of Y is equal to. So when X is equal to 1, And when X is equal to 2, when X is equal to 3, when X is equal to 4, And when X is equal to 9, fantastic. So, when X is equal to 1, that will mean that Y is equal to 1, and when X is equal to 2, that will mean that Y is approximately equal to 1.41, and when X is equal to 3, that means that Y will be approximately equal to 1.73. And when X is equal to 4, that will mean that Y is equal to 2. And finally, when X is equal to 9, that will mean that Y is equal to 3. Fantastic. So moving right along here. So we now can plot these points and connect them through a smooth curve. But before we do that, let's take a moment here. Before we plot that curve, let's try to consider our new graph that we're trying to plot. So we are told that Y equals the square root of X is our parent function. So let us also consider our graph for Y that we're ultimately trying to solve for. So Y is equal to negative square root of X divided by 2 + 4. So now, with that in that case, Let us replace the X and Y equals the square root of X by saying Y is equal to the square root of x divided by 2. So this will help us sketch the graph horizontally, or rather, we need to note that this, as I was saying sketch, so. Looking at our original parent function, which is Y equals the square root of X, when we say that Y is equal to the square root of X divided by 2, this will mean that it will take our original curve for this parent function, and it will stretch it horizontally by a factor of 2. So that will mean that the X coordinates obtained previously must be multiplied by 2. So for 0.0, that will mean that it's going to be stretched 22 multiplied by 0.0, which will give us, when we simplify, that will mean that our new coordinate point will be 0.0. And once again, we're multiplying, since we're stretching our graph horizontally by a factor 2, so we're multiplying our values of X by 2. So that would mean that 1.1 will become 2.1. That will mean that 2.1.41. Will become 4 com. 1.41, fantastic, moving right along here. That will mean that 3.1.73 will become 6.1.73 and and don't forget that all of our corner points are inside parentheses, and 4.2 will become. 8.2. Fantastic. So now, We have enough information to plot our second curve, but we're not quite done yet, so we're breaking up the elements of our function of Y that we're trying to graph ultimately into the different parts to see how the graph transforms. Awesome. So with that in mind, our next step now is we need to replace the X by X + 8 in our Y equals the square root of X divided by 2. And when we do that, that will mean that our new equation will become Y equals the square root of X + 8 divided by 2, which will be equal to the square root of X divided by 2 + 4. Fantastic. So, we need to note that since we replaced X by X + 8 or X minus multiplied. So essentially, let's make a note of this off to the side. So since we replaced. X by X + 8 or X minus minus and then parentheses 8, because a minus minus becomes a plus. This means that it will translate our graph of Y equals the square root of X divided by 2 to the left by 8 units. So that will mean That We need to note that our original graph will just be moving 8 units to the left. So let's make a note of that so. 8 to the left, so 8 units to the left, and I did a capital L for left. So lastly, the point we need to consider for our graph is we need to consider Y is equal to this negative symbol, so Y is equal to negative, square root of X divided by 2 + 4, and because we have that negative symbol outside of our square root, that will mean that this will reflect our graph of Y equals, so note once again we're trying to graph Y equals the square root of X divided by 2 + 4. So it's gonna take this graph. Or rather the curve for the square root of X divided by 2 + 4, and because we have this negative sign means that's ultimately what we're trying to solve for. So the negative sign is outside of the square root. We'll take our graph of Y equals the square root of X divided by 2 + 4, and it will flip it, rather reflect it upside down across the X axis. So with that said, we now have enough information to create our final graph. So let's plot our points that we found earlier onto our graph. And when we do that, we should gather the following information. So, let us note that our curve for Y equals the square root of X is represented in green, and when we transform it to be Y equals the square root of X divided by 2, which is going to be represented in red, we will see that it slightly shifts down just a little bit. And when we move our second curve, which is the Y equals the square root of X divided by 28 units to left, you'll see that it goes into the negative second quadrant. And then we'll take our curve, which I now turned it into a dotted line. To represent our curve that's been translated 8 units to the, or rather, has been shifted 8 units to the left. And what we're going to do is we're going to reflect it over the X axis because it has that negative sign outside of our square root. So we flip it over upside down, and when we do that, we will find our final answer. So, our final graph will have to be the following. Which all it is is our graph for Y equals the square root of X divided by 2 + 4, but we flipped it over the, I'm sorry, we flipped it over the X axis for our for our new graph of Y. So that will mean that our final answer starts at So we have -5, -6, 78, so it's at. 0.8, and then it starts to branch outward and curve and stretches into the into the negative 4th quadrant. Awesome is it goes 123, and 4 when you look at your different squares on your coordinate plane. Awesome and that's it. We've officially solved for this problem. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
y = - √(1 + x/2)

Key Concepts

Standard Functions

Function Transformations

Square Root Function

Watch next

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.y = - √(1 + x/2)

Key Concepts

Standard Functions

Function Transformations

Square Root Function

Watch next

In Exercises 55–58, graph each function, not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.15–1.17, and applying an appropriate transformation.
y = - √(1 + x/2)