The graph of ƒ is shown in the figure. Graph the following fun...

Question

The graph of ƒ is shown in the figure. Graph the following functions.

$f (2 (x - 1))$

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. Given the graph of the function H of X sketch the graph of the function H of three multiplied by X minus two. Awesome. So it appears for this particular problem we're taking our original function which is H of X which is plotted on our graph that's provided to us by the prom itself. And we're gonna be applying transformations to our original graph to obtain H of three multiplied by X minus two. So now that we know that we're applying transformations to our new function, let us write down the function that we're focusing on which is H of three and we're gonna be taking H of three multiplied by X minus two. So this is our new new function that we're trying to graph. So with that in mind, let us quickly focus on our different components here of our graph or rather our function that we're trying to graph. So looking at the different components of our function. So in order to graph these, as we should recall, we're gonna have to compress our graph horizontally. So we're gonna do a horizontal compression, which I'm denoting as HC by a factor of 13 because we are multiplying three by out we're multiplying three to our function. So three to the X. So we're multiplying three multiplied by X minus two. So since we're multiplying three to X, that means we have to perform a horizontal compression by a factor of one third. And then we also need to note that we have X minus two or we're subtracting two from X. And because we're subtracting two from X, that means we're gonna have to shift our graph which we have to take our graph and move it two units to the right. And we know that we're moving it two units to the right because we're subtracting two from X. Now, if it was adding two to X or is it X plus two, then that means we would be moving or shifting our graph two units to the left. But since it's X minus two, we're moving our graph two units to the right. Awesome. So with that in mind, when we plug in our new function into our graphing software, which my graphing software produced the following graph which I screenshotted and shared with you. So essentially it's the same graph as our original graph or to our original function. However, it has been shifted two units to the right. So it's located in a new location and it's been compressed, meaning it's been squished, it's been narrowed ever so slightly. So it's way more skinny compared to our original function of H of X, which is a little bit wider. So essentially, it's the same exact graph as H of X our original graph, but it's been compressed horizontally by a factor of one third and has been moved two units to the left to the right. So as you can see, since if you look at our two points, which I'll highlight in red on our original graph, which you could see it's about a little bit over one. So it's like one comma negative 30 for one of our peaks. And as you could see our same peak, if we look on it on our new graph for our new function, you can see now that it's like about two comma negative 30. As you can see since it's been shifted two units to the right and that's it we've officially solved for this problem. Hooray. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

The graph of ƒ is shown in the figure. Graph the following functions. <IMAGE>

$f (2 (x - 1))$

Key Concepts

Function Transformation

Horizontal Stretch and Compression

Graphing Composite Functions

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