Horizontal and Vertical Shifts of the Square Root Function
WELCOME TO A LESSON ON GRAPHING THE SQUARE ROOT FUNCTION AND VERTICAL AND HORIZONTAL SHIFTS OF THE SQUARE ROOT FUNCTION OFTEN CALLED TRANSLATIONS. SO LET'S START BY CONSIDERING THE BASIC SQUARE ROOT FUNCTION F OF X = THE SQUARED OF X. WE WANT TO FIRST FIND THE DOMAIN AND THEN GRAPH THE FUNCTION. THE DOMAIN IS A SET OF ALL POSSIBLE INPUTS
OR X VALUES FOR THIS FUNCTION. AND SINCE WE'RE TAKING THE SQUARE ROOT OF X, X HAS TO BE gt; OR = TO 0 BECAUSE X CANNOT BE NEGATIVE. Â AND WE'LL GRAPH THIS FUNCTION BY COMPLETING A TABLE OF VALUES. AND AGAIN, SINCE WE'LL BE TAKING THE SQUARE ROOT OF X WE'RE GOING TO SELECT X VALUES THAT ARE PERFECT SQUARES
LIKE 0 SQUARED IS 0, 1 SQUARED IS 1, 2 SQUARED IS 4 AND 3 SQUARED IS 9. NOW, TO DETERMINE THE CORRESPONDING Y VALUES WE'LL PERFORM SUBSTITUTION INTO THE FUNCTION. SO NOTICE WHEN X = 0 Y OR F OF 0 WOULD BE THE SQUARED OF 0 OR 0, WHEN X = 1 Y WOULD BE = TO F OF 1 WHICH IS = TO THE SQUARED OF 1 WHICH IS 1.
WHEN X = 4, Y IS = TO F OF 4 WHICH IS = TO THE SQUARED OF 4 OR 2. AND WHEN X IS = TO 9, Y IS = TO F OF 9, WHICH IS = TO THE SQUARED OF 9 OR 3. NOW WE'LL GO AHEAD AND PLOT THESE FOUR POINTS TO SKETCH OUR FUNCTION, SO WE HAVE THE POINT (0, 0) (1, 1) (4, 2) AND (9, 3). SO THIS WOULD BE THE GRAPH
OF OUR BASIC SQUARE ROOT FUNCTION. NOW WE'LL TAKE A LOOK AT VARIOUS TRANSLATIONS OF THIS PARENT FUNCTION. NOW LET'S CONSIDER THE FUNCTION G OF X = THE SQUARED OF X +2. WE WANT TO GIVE THE DOMAIN AND THEN GRAPH THE FUNCTION. NOTICE THAT WE'RE STILL TAKING THE SQUARED OF X HERE ON THE FUNCTION
SO THE DOMAIN OF THIS FUNCTION IS GOING TO BE THE SAME AS THE DOMAIN OF THE BASIC SQUARE ROOT FUNCTION. WE'LL HAVE X gt; OR = TO 0, SO BECAUSE OF THIS WE USE THE SAME X VALUES IN THIS TABLE AS WE DID FOR THE PREVIOUS TABLE. SO WE'LL HAVE 0, 1, 4 AND 9. NOTICE OUR COORDINATE PLAN ALREADY CONTAINS A GRAPH OF THE BASIC SQUARE ROOT FUNCTION
Hi everyone. Today we're going to talk abouthow to graph transformations of a function. To complete this problem, we will first graphthe original function and then separately consider each of the transformations. Let'stake a look. In this particular problem, we've been askedto graph various transformations of the function y equals the square root of x. We're goingto be looking here for basic transformations but remember that there are many types oftransformations. The purpose of graphing these four in particularis to start getting comfortable with transformations but remember that if you ever run into transformationson a test, you can always just plug in points
for x and then start plotting your resultson your axes. That's your failsafe when it comes to transformations. What we're going to talk about today shouldreally just familiarize you with the basics so that you can get these done as quicklyas possible. I've graphed our original function y equals the square root of x on this firstset of axes. Over here on the second set of axes, we're going to be graphing the transformationy equals the square root of x minus 2, this function right here. What you want to realize is that you can completelyseparate the constant negative 2 from the
square root of x. That's in contrast withfor example this function down here where you have 2 times the square root of x. Youcan't completely separate them because they're multiplied together but when you have twoterms like this, the square root of x and a constant negative 2 that are added or subtractedfrom one another, you can think about being able to separate them. And when the only thingthat's transforming your function is a constant like this, that means you're dealing witha vertical transformation and your function is just going to be moving up or down withinthe same set of x values. So what we mean by that is that we have thefunction y equals square root of x minus 2.
Every point that would be on our originalgraph, y equals the square root of x, is just going to get moved down to the units. So inother words, we're going to take every point and shift it down by negative 2. Let's sayit's about here. We will take every point and shift it down by negative 2 so that ourtransformation actually looks like that. In this third example here, we have y equalsnegative square root of x. We have noticed a negative 1. Multiply it by our square rootof x term. Whenever you have a constant multiplied by your original function, that means you'reeither going to be stretching or shrinking your graph andor flipping the graph acrossan axis.
In this case, every point you plug in forx is going to return a certain y value but this time now that we've multiplied by thisnegative out in front, we're going to get the same y value but multiplied by negative1. So this is just a transformation flipped across the xaxis and the graph looks likethis. So just remember that if you've got youroriginal function, square root of x, with a negative out in front, that means you'regoing to be flipping it over the xaxis. Similarly here with our next function, y equals twotimes the square root of x, we've got a constant coefficient multiplied by our originalfunction. Same with the last one. We had negative
1 multiplied by our original function. Now we have 2 multiplied by our original functionand what that means is that we're going to be stretching the graph by a factor of2. So when we draw the graph, instead of every ycoordinate lying along the original function,the new ycoordinate we get will be doubled what the old one was. So that will look somethingroughly like this and the way you can think about it is we would have had our originalycoordinate here. Let's say that that's at 1. Well our new ycoordinate will be doublethat, a factor of 2. So the new ycoordinate will be here at 2.