in this video we're going to focus on graph in radical equations so let's start with the basics let's start with the parent function y is equal to the sare root of x so this graph starts at the origin and it increases at a decrease in 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 zero and the highest is infinity so it's 0 to Infinity now the range you need to analyze the Y values the lowest yvalue is zero and the highest is infinity so the range is also 0 to Infinity so that's the graph of yal the < TK of X what's the general shape forga < TK 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 AIS 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 AIS now if you put let's say two negative signs one in front and on the inside of the square root uh 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 can 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 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 3 and Quadrant 4 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 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 two to theare root of x how is it going to affect the graph in this case the graph is going to start two units above the x-axis so it's shifted up two units now if we were to let's say subtract it by one it's going to shift down one unit but it's still going to go towards quadrant one 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 two units what you can do is set the inside equal to zero and solve for x in that case it's going to start two units to the right and it's going to go this way so likewise let's say if we have the square root of x + 3 the graph is going to be shifted three 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 0 0 now to find the next point it's important to understand that the square root of 1 is one so starting from the origin as you travel one to the right go up one unit now the square root of four is two so starting from the origin as you travel four to the right you need to travel up two units so therefore the next point is going to be 4 comma 2 the square root of n is three so if you want another point you can plot 9 comma 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 two in front of the square root now everything's going to be the same don't only difference is the Y values will double so the origin is going to be the same 0 0 now as we travel one to the right instead of going up one we need to double the up one part it's going to go up two as we travel four units to the right instead of going up two we're going to go up four 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 two now let's say if we have this function the square root of x -1 + 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 comma 2 now to find the next Point as we travel one to the right we need to go up one unit so that's going to take us to 2 comma 3 and if we travel four to the right it's going to go up two units so that's going to take us to five comma 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 one the highest is infinity therefore the domain is going to be 1 to 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 two and the highest goes up to infin inity so therefore the range is 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 < TK 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 four 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 2 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 four to the right and three units up so it starts at this point uh 4 comma 3 and we know that the graph is going to go towards Quadrant 4 so when we find the next Point as we travel one unit to the right instead of going up one unit we need to go down one unit due to the negative sign so that's going to take us to the point 5 comma 2 and since the square root of four is two as we travel four to the right we need to go down two so that's going to take us to the point 8 comma 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 four to Infinity since four is theow lowest x value and it includes four and as for the range the lowest actually the highest y value is three 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 three including three and so that's that's it for this problem