Use shifts and scalings to transform the graph of

Question

ƒ (x) = \sqrt{x}

into the graph of g. Use a graphing utility to check your work.

$g (x) = 3 \sqrt{x - 1} - 5$

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. Apply the transformations on the graph P FX equals the square root of X into the graph of H of X equals four, multiplied by the square root of X minus three minus six. Check your work with the help of a graphing calculator. Awesome. So it appears for this particular problem we're using our original graph which our original graph is P FX equals the square root of X. And we're applying transformations to the specific graph using H of X equals four multiplied by the square root of X minus three minus six. And we're also told that we can check our work with the help of a graphing calculator. Awesome. So now that we know that we're applying transformations to our original graph to see what exactly is happening or how our original graph will change. Let us read off our multiple choice answers to see what our final answer might be. A is vertical stretch by a factor of four right shift by three units and shift down by six units. B is horizontal compress by a factor of one third left shift by three units and shift down by six units. C is horizontal compression by a factor of three right shift by three units and shift down by six units. And finally, D is horizontal compress by a factor of one third right shift by three units and shift up by six units. OK. So first off, we need to note that sense and this is important. So I'll highlight it in red, that sense. The number four is multiplied to X. That will mean that this will stretch the graph of. So we need to take our original graph which is P FX is equal to square root of X. And because we're multiplying by four, so we'll do a star. So we're multiplying by four, that means that a, so it's gonna be stretched vertically. So I'm gonna say, vs vertical stretch, that's what VS stands for. So because we're taking our original graph and we're multiplying it by four, we're MF multiplying four to X, that means a vertical stretch is going to occur by a factor of four. So, so factor so fo four, so factor of four. Awesome. So our next step is is like since we're stretching it vertically by a factor of four, that means our new graph which I'll write this in blue, our new graph, which we call Q of XQ of X will be equal to four multiplied by the square root of X. Awesome. Our next step is we need to focus on the X minus three. So since three is being subtracted from X, as we should recall, this will shift our graph three units to the right, since it's negative three, now, if it was positive three, then that means I'd be moving three units to the left. But since it's negative three, we're moving three units to the right. So that will mean that we'll get a new graph of, and we'll write our new graph as R of X and R of X will be equal to four multiplied by the square root of X minus three. Awesome. And lastly, we need to focus on our minus six factor and since six is subtracted from the function itself. That will mean that it will shift our graph of R of X equals four multiplied by the square root of X minus three down by six units to obtain our final graph, which we're trying to figure out how to graph it, which is H of X is equal to four multiplied by the square root of X minus three minus six. Awesome. So taking everything that we just said out loud together once we applied all of our transformations, when you go ahead and basically plug in your H of X function into your graphing software of choice or your graphic calculator, you should obtain the following graphs. So in blue, we have our original graph for the function of P FX. And in red, we have our H of X function which we were trying to figure out how our original graph will change. And as you can see our blue graph, it is going, it starts from zero and then it works its way increasing, going upwards. So it's a curve whereas our H of X it starts at a different point. So it's, it doesn't start at 00. It doesn't come start at the origin point. It starts not exactly at one. So it's like at 0.5 or 0.6 something and it almost goes straight up, but then it starts to curve to the right. And as you could see, that's because once again to quick, quickly recap, we stretched our graph vertically by a factor of four because it's four multiplied by the square root of X. And then since three is subtracted from X, that means we shifted our graph three units to the right. So that means we took our origin point which was at zero and we shifted it three to the right. And then we need to subtract since it was six is since six is subtracted from the function itself, then that will mean that the entire graph is shifted down by six units. Awesome. So essentially our original function which is P of X will become H of X. So H of X is the exact same function as P FX, but it's been transformed. That means it's changed location. It's been shifted around a bit. So it's the same graph but it's in a different location. Awesome. And that's it. We've officially solved for this problem. Hooray. So that will mean that our final answer without a doubt has to be multiple choice answer letter A which states that a vertical stretch by a factor of four right shift by three units and shift down by six units. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Hi.

Use shifts and scalings to transform the graph of $ƒ (x) = \sqrt{x}$ into the graph of g. Use a graphing utility to check your work.
$g (x) = 3 \sqrt{x - 1} - 5$

Key Concepts

Transformations of Functions

Vertical and Horizontal Shifts

Scaling and Stretching

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