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.

$h (x) = - 4 x^{^{2}} - 4 x + 12$

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 G of X is equal to -2 multiplied by X2 + 8 X minus 6 using shifts and scalings. Awesome. So it appears for this particular problem, all we're asked to do is to create a graph for this specific function of GFX. Awesome. So with that in mind, right now, our function for GFX isn't in the proper formatting for us to effectively graph or graph. When I say effectively graph it, right now it's in its standard form. We need to format it so we could see what exactly is happening from a shift in scaling perspective. So in order to change the formatting for GFX, we need to recall and use the completing the square method. So we need to take that the function of G of X, we need to take G of X. And we need to say that GFX is equal to -2. Multiplied by X2, and then we need to add 8 X minus 6. Awesome. So what we're trying to do now is we're going to be recalling and using completing the square, and we're applying it to our function of G of X. So when we first start to apply the completing the square method, we will find that it's equal to -2 multiplied by X2 minus 4 X + 3. But in an effort to save time not going through each individually, each individual step because it could be a little bit tedious, just know we're using the completing the square method, which is like algebraic in nature. So ultimately we need to keep simplifying this. Down to its most simplified form. So this expression in its most simplified form, using the completing the square method, will say will mean that it is equal to -2 multiplied by X minus 2 to the power of 2 + 2. Fantastic. So, with that said, so let's move this over just a little bit so we have some space here, since we still need to put our graph off to the side soon. Awesome. So, With that said, our next step Once again, is we need to start identifying the original function and then applying the necessary shifts and scalings. So, as we should note and recall, in order to identify the original function, since it appears that we're dealing with a parabolic based equation, that would mean that the original function, and let's write this in red, will mean that F of X, our original function. is equal to X2, which is a parabola, which has its vertex at the origin, which is 00. So it's at its vertex is at 00, and it's opened upwards, so that means that the curve is pointing upwards. So now that we've identified our original function, we can now recall and apply our scaling methods. So we need to recall and note that the negative 2 that is in front of the X2 will indicate a vertical scaling and reflection. Specifically, the graph of F of X equals X2 will be stretched by a factor of 2 in the vertical direction and flipped over the X-axis because of this negative sign out front. So this will mean that for every value of X, the value of Y will be 2 times what it would be in the original parabola, but in the opposite direction. So it's gonna be the curve's gonna be facing downwards instead of upwards for this specific case because of this negative symbol once again. So now, Our next step is we need to recall and apply our horizontal shift. Which since 2 is subtracted from X, we need to shift the graph of -2 multiplied by X2 to the right by 2 units to obtain our graph of -2 multiplied by X minus 22. So, our next step is we need to identify and figure out what vertical shift is occurring, and we need to note that sense. We have a value of -6 and this value of -6 is being Subtracted By our problem, that means we're gonna be using it. So actually, I skipped a few steps here. So, let's rewind a little bit. So now we're focusing on our vertical shifts, so we need to focus on our negative 2 multiplied by X minus 2 to the power 2 + 2. And because we have a + 2 is added to the function, that will mean that the graph of -2 X2 will be shifted up by 2 units in order to obtain our overall graph of -2 multiplied by X minus 2 to the power 2 + 2. So now that we've established what kind of scaling and shifts are occurring, we can now officially sketch our graph. So I've gone ahead and done gone ahead and grafted our graph for us so we can have a fancy. Digital representation of our sketches. So, in red, we have our original parabola, which is F of X equals X2 0.0. So we have the origin point at 0.0. And in blue is representing our graph of G of X. So let's make a note. So I wrote in red, so red means F of X. So let's make a note of that. So F of X is our graph in red, and our graph in blue is our G of X function. So, as we should note and recall now, the required shift on the basic function to draw the graph of GFX equals -2 multiplied by X2 + 8X minus 6. So we now need to check with our graphing utility, which I went ahead and made a copy of the digital graph that I got from my graphing software. So you're more than welcome to use your graphing calculator to solve for this particular problem, or you can use whatever graphing software online that you're used to and comfortable using to make sure yours like make sure your sketch for this particular prom is legit and follows all of the Rules set to us by the function itself because we need the function to help us figure out the rules for the shifts and scalings. Awesome. So in order to confirm our sketch, we need to use our graphing calculator or software by inputting our function, which is G of X equals -2 multiplied by X2 + 8X minus 6, and we need to observe the graph, and it should match our sketch. Which is awesome. So, our blue curve once again is our modified graph. This is our final answer that we're ultimately trying to solve for is our curve in blue, and it's, it appears that our sketch matches our graphing software because our curve is upside down, which is perfect because we have a negative 2 out front, and we can see. That it's not as wide as our original function for F of X, it's narrower because of the shift that occurred. And that's it. Thank you so much for watching. Hopefully that helps, 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.

$h (x) = - 4 x^{^{2}} - 4 x + 12$

Key Concepts

Graph Transformations

Quadratic Functions

Graphing Utilities

Watch next