Use shifts and scalings to graph the given functions. Then che...

Question

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.

$g (x) = - 3 x^{2}$

Accepted Answer

Hello there. Today we're gonna 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 GFX equals -5 multiplied by X2d using shifts and scaling. Awesome. So it appears for this particular problem we're going to be trying to graph this function of G of X using the shifts and scaling methods. Awesome. So right off the bat, so first off, we need to recall and use our shifts in scaling methods. So we need to get that in our minds. So in order to do that, in order to graph this function of GFX, we need to start by identifying the original functions, and then we need to apply the necessary shifts and scalings to the graph. So we're going to start out with our basic. graph, which is the original function which let's pretend that the negative 5 doesn't exist and we just focus on X2. So as we should recall, a function that is X2 all by itself, that is a parabolic equation. So we're dealing with a parabola. So that is our first step, so we're identifying the original function, which once again, the original function is this X square. And let's write that down, shall we? And I'll write it in blue. So our original function, let's call it F of X, and F of X is equal to X2, and this is a parabola that opens upwards since it's positive. If it was a negative X2, then it would be inverted. It would be flipped upside down. But since it's, since we're just focusing on X2, it's positive, that means it's upright and it's opening upwards. Awesome, and it's Vertex, as we should recall, is going to be at 00. So, So that means that it starts at 00 and then it has two edges of the graph where the curve points to the left and right and it meets at the vertex point. OK. So now looking at our original function, we can now apply scaling. So we need to note and recall that we have this coefficient of -5, and this is in front of the X2. Which will mean that this indicates a vertical scaling and reflection, so specifically, The graph of F of X equals X2 is stretched, so that I was saying, so I should backtrack here. So it's saying And let's write our new function. This is our function that we're trying to graph, which is G of X is equal to -5 x2. So specifically this graph where we have a negative 5 multiplied by X2, this will be stretched by a factor of 5 in the vertical direction, and it will be flipped upside down will be flipped over the x axis because of the negative sign as we were discussing earlier. So this means that for every value of X, the value of Y will be 5 times what it would be in the original parabola, but in the opposite direction because it has this negative symbol. So it's gonna be facing downwards, so it's gonna be flipped over, and it's going instead of it being wide, so here, let's quickly sketch this to get an idea and let's sketch it in green. Or better yet, We'll keep using. So we're going to sketch our original function in blue, and then we're going to scrap do a rough, so we're going to be sketching our original graph in blue and then we're going to be sketching the graph that we're ultimately trying to figure out in red and this is a very rough sketch. We'll have a, I'll show a more proper computer generalized graph in just a moment, but this is just to get a visualization of what's going on here. So for our original function. For F of X, we just have our standard parabola, which, as you can see, will be wide and it has its vertex at 00, which is at the center of our coordinate plane. And like I said, the curve points to the left and right, going up towards infinity, going up and up and up. Whereas, as we were discussing for our graph that we're trying to graph for G of X is gonna be narrower and it's gonna be flipped upside down, and it's gonna be skinnier. So it's gonna be narrow. Instead of it being wide, like our original function, it's going to be skinnier because instead of it going out. 1 up, 1 out 1, it's gonna be going down 5 out 1, so it's gonna be skinnier. Soda So that's just a rough sketch, but let's quickly check to make sure there's any horizontal shifts or vertical shifts. So, In order to do that, we need to recall that there will be no horizontal shifts in this particular case because there is no term added or subtracted from the X term in our given function for GFX. So this means as a result that there will be no, so no horizontal shift occurring for this particular function for GFX. And the vertex will remain for this particular parabola for G of X, it will remain at the origin point of 00. So let's make a note of that. So our vertex will also be at 00 for our GFX plot or our graph. So our next step is we need to recall a note for this particular function for GFX. There will be no vertical shift. So similarly, there is no constant term that is added or subtracted from the entire function at large, which means that this will indicate that there will be no vertical shift that will take place. So as a result, the vertex will remain at 00. So with that said, we can refer back to our sketch. So looking at our sketch, we could see that it starts at the vertex 00. And since the parabola is vertically stretched by a factor of 5, it will point, so the points that are usually 1 unit away from the vertex, like we were saying, it's gonna be going down 5 at 1. So this is just to reiterate what we were saying. And it's in the negative direction. Once again, this is very important because there's this negative symbol out in front of the X squared. And the 5 that's being multiplied by X2 is the stretching factor, and this negative sign indicates that it's gonna be flipped upside down, so it's gonna be inverted. So, with that in mind, So, like we said, it's gonna be 5 units away in the negative direction because of this reflection, and we have to draw, once again draw the parabola opening downwards because of this negative symbol outside or that's being multiplied to the X2d factor, awesome, or rather the term I should say. So with that said, that means for this particular problem once again we have to really note that we're dealing with a vertical stretch and a reflection for our function of GFX. So we can check our sketch, and once again this is a very rough sketch, we can check our sketch using a graphing utility which I've gone ahead and plugged in our GFX function into some graphing software and have produced the following graph. And as you can see, we have our original. Function for F of X drawn in blue, and then we have our function for G of X drawn in black, and as you can see, our GFX function is much narrower, so it's much skinnier compared to our original function which is in blue, which is wider, and this is because we have this vertical stretch by a factor of 5, and once again it's flipped upside down because of our negative sign. Awesome. So that's it. That is how we solve for this problem, that's how we graph our function of GFX. Hooray, we did it. So once again, as a quick recap, we need to sketch our graph vertically by a factor of 5 and reflect the graph about the X axis. So that means we're just taking our original function and we're flipping it upside down, and then we're applying a shift in scaling, which we're not shifting it any, so we can forget about shifts, but we have to scale it to note that it has a vertical stretch of 5, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Use shifts and scalings to graph the given functions. Then check your work with a graphing utility. Be sure to identify an original function on which the shifts and scalings are performed.
$g (x) = - 3 x^{2}$

Key Concepts

Function Transformations

Quadratic Functions

Graphing Utilities

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