In this video, we're going to focus on graphing radical equations. So let's start with the basics. Let's start with the parent function, y is equal to the square root of x. So this graph starts at the origin, and it increases at a decreasing rate.
And so it looks like that. The domain for this function, if you analyze it from left to right, the lowest x value is 0 and the highest is infinity. So it's 0 to infinity.
Now the range, you need to analyze the y values. the lowest y-value is 0 and the highest is infinity. So the range is also 0 to infinity. So that's the graph of y equals the square root of x.
What's the general shape for negative square root x. What's going to happen if we put a negative sign in front of the radical? If you put a negative sign in front of it, it's going to reflect over the x-axis.
So, it's going to look like that. And if you put a negative sign inside the radical, in front of x, then it's going to reflect over the y-axis. Now, if you put, let's say, two negative signs, one in front and on the inside of the square root function, this is going to reflect over the origin.
And so, it's going to look like this. Now, if you have a difficult time remembering all these rules, here's something that can help you. The sign in front of x is positive, and the sign that's in front of the radical, which we'll say is associated with y, is also positive.
x is positive in quadrants 1 and 4. y is positive in quadrants 1 and 2. x and y are both positive in quadrant 1. So notice that the graph points towards quadrant one So just so you know this is quadrant one Here's number two. This is quadrant three and quadrant four So looking at the second one x is positive, y is negative. x is positive towards the right, y is negative below the x-axis. So it points towards quadrant 4. Now looking at the third example.
x is negative, y is positive. x is negative towards the left, y is positive above the x-axis, so it's going to go towards quadrant 2. And for the last example, x is negative towards the left, y is negative below the x-axis. below and so it goes towards quadrant 3. That can help you to determine what direction the graph is going to go. Now what about this example?
What's going to happen if we add 2 to the square root of x? How is it going to affect the graph? In this case, the graph is going to start 2 units above the x-axis. So it's shifted up 2 units.
Now, if we were to, let's say, subtract it by 1, it's going to shift down 1 unit, but it's still going to go towards quadrant 1, so it looks like that. Now, what about this graph? Let's say if we have the square root of x minus 2. In this case, it's going to shift to the right 2 units.
What you can do is set the inside equal to 0 and solve for x. In that case, it's going to start 2 units to the right, and it's going to go this way. Likewise, let's say if we have the square root of x plus 3, the graph is going to be shifted 3 units to the left, and it's going to open towards the right.
Now let's draw a more accurate sketch using points. Let's graph these two functions. So for the first one, the first point...
starts at the origin. It's. Now to find the next point, it's important to understand that the square root of 1 is 1. So starting from the origin, as you travel 1 to the right, go up 1 unit.
Now the square root of 4 is 2. So starting from the origin, as you travel 4 to the right, you need to travel up 2 units. So therefore, the next point is going to be 4, 2. The square root of 9 is 3. So if you want another point, you can plot 9, 3. But I think this is good enough. And that's how you can get a more accurate sketch.
So let's see what happens if we put a 2 in front of the square root. Now everything's going to be the same. The only difference is the y values will double. So the origin's going to be the same, 0, 0. Now...
As we travel 1 to the right, instead of going up 1, we need to double the up 1 part. It's going to go up 2. As we travel 4 units to the right, instead of going up 2, we're going to go up 4 relative to the origin. So this graph is going to look like this.
As you can see, it's been stretched vertically by a factor of 2. Now, let's say if we have this function, the square root of x minus 1 plus 2. So you can plot points if you want to, but what I want you to do is graph it and also write the domain and range of the function. So first, let's find the new origin. The graph has been shifted one unit to the right, and it's been shifted up two units.
So therefore, we're going to start at 1, 2. Now to find the next point, as we travel 1 to the right, we need to go up 1 unit. So that's going to take us to 2, 3. And if we travel 4 to the right, it's going to go up 2 units, so that's going to take us to 5, 4. And so those are the points that we have. And the graph is going to go like that. So now, to analyze the domain of the function, we need to look at the x values. The lowest x value is 1. The highest is infinity.
infinity and it includes one so we need to use a bracket with one for infinity always use parentheses now let's analyze the range the lowest y value is 2 and the highest goes up to infinity so therefore the range it's going to be 2 to infinity So now let's work on another example. Try this one. Let's say that y is equal to 3 minus the square root of x minus 4. Go ahead and graph it, and then write the domain and range of the function using interval notation. So we can see that the graph has shifted 4 units to the right. and it's been shifted up three.
If you want to, you can rewrite it like this, if that makes it easier. Now there is a negative sign in front of the radical, so it's not going to be going towards quadrant one, rather it's going towards quadrant two. X is positive, Since there's a positive sign in front of x, so it's going to go towards the right. But since there's a negative on the outside, y is negative, so it's going to go towards quadrant 4. But now let's find a new origin. So the graph has been shifted 4 to the right and 3 units up.
So it starts at this point, 4,3. And we know that the graph is going to go towards quadrant 4. So, when we find the next point, as we travel 1 unit to the right, instead of it going up 1 unit, we need to go down 1 unit due to the negative sign. So, that's going to take us to the point 5, 2. And since the square root of 4 is 2, as we travel 4 to the right, we need to go down 2. So that's going to take us to the point 8, 1, which is about here. And so this is going to be, let's do that again.
Here we go. So that's going to be the shape of the graph. So now, the domain is going to be 4 to infinity, since 4 is the lowest x value, and it includes 4. And as for the range, the lowest, actually the highest y value is 3. The lowest is negative infinity. As this travels towards the right, it's slowly decreasing, but it continues to decrease forever.
So therefore, the lowest y value is negative infinity. So the range is going to be from negative infinity to 3, including 3. And so that's it for this problem.