In Exercises 67-80, begin by graphing the square root function, f...

Question

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x)=√(-x+1)

Accepted Answer

Hello, today we're gonna be graphene H of X as a transformation of F of X. So what we are given is F of X is equal to the square root of three X. And we are told that H of X is equal to the square root of the quantity negative three X plus nine. So before we graph H of X, we need to get an idea of what the shape of F of X is going to look like. So for F of X, we currently have a product inside of the square root. So we can separate three X as a product of two square roots. And that's going to give us the square root of three times the square root of X. Now, here F of X is using the square root of X. So the traditional graph of the square root of X is going to look like this. We're going to have this axis and the starting point of this graph is going to be at the origin zero, comma zero. And here we're going to have a curve that stretches off to the right. So this is going to be the traditional graph for F of X, I mean for the square root of X. However, in order to go from the traditional graph of the square root of X to F of X, we need one transformation and that's going to be involving the square root of three. See here, the square root of three acts as a coefficient to the square root of X. And the square root of three is going to produce a value that is going to be greater than one. Since the coefficient has a value that is greater than one, this is going to result in a vertical stretch of the graph. So we're going to be stretching the traditional graph of the square root of X vertically along the Y axis. So now if we want to go ahead and graph F of X, so again, F of X is equal to the square root of three times the square root of X, we're still going to have our starting point at zero comma zero. But now the graph of the curve is going to be much taller at the beginning. So this is going to be the shape of F of X. Now let's go ahead and identify what transformations we need in order to go from F X two H of X. So we can go ahead and simplify H of X a little bit and we can do that by first factoring out of three from the quantity negative three X plus nine. This is going to give us the square root of three times the quantity negative X plus three. Now here, if we factor we can separate actually three from the quantity negative X plus three as a product of two square roots. And this is going to give us the square root of three times the square root of the quantity negative X plus three. And what we can go ahead and do is factor out a negative sign from this quantity. And this is going to give us the square root of three times the square root of negative one times the quantity X -3. Now, here we're going to have to transformations between F of X and H of X C. We're going to keep the same vertical stretch going from F X two H of X since they both have the coefficients of the square root of three. But the first transformation to go from F of X two H of X is going to involve the quantity X minus three C X minus three is of the form X minus H. And whenever you have an X minus H quantity, this is going to indicate a horizontal shift of the graph. Here X minus three is in the standard form of X minus H and three takes up the spot of H. So H is going to equal to three, which means that we're going to have a horizontal shift of three units. To the right, that's going to be our first transformation. Now, our second transformation is going to involve the negative sign inside of negative X minus three. And let me explain why, see the negative sign is going to cause this graph to be flipped with respect to the Y axis. And the reason for this has to be for the domain of H of X C, the domain of any square root function must be greater than or equal to zero. So if we set negative X minus three, greater than or equal to zero, we first need to divide both sides by negative one, which is going to give us X minus three is less than or equal to zero. And if we add three to both sides of this inequality X must be less than or equal to three. So that's going to result in the graph of H of X to flip along the Y axis. So that's going to be our second transformation. So let's go ahead and graph the transformations on F of X. So the first transformation is going to be a horizontal shift three units to the right. So now the graph is shifting three units to the right. The new starting point is going to be here at three comma zero. And we're still going to keep the same shape as original F of X because this is not altering the shape. So we're still going to have this stretched right. Curve. So that's gonna be our first transformation. Now, the second transformation is going to flip the graph with respect to the Y axis. So this transformation is not going to change the starting point at 3:00. But again, because the domain says that all values of X must be less than or equal to three. We're flipping the scrap with respect to the Y axis which is going to result in a shape like this. And this is going to be the graph of H F X. And the answer that most accurately represents this is going to be be. And just to double check, we have the graph of F of X who's starting point is at the origin and whose graph is shifted up vertically and H of X is going to have a horizontal shift three units to the right where the new starting point is at three comma zero and it's going to be flipped along the Y axis or with respect to the Y axis. So it's going to be heading in the opposite direction of F of X. So since be accurately represents all of this, that means B is going to be the answer to this problem. So I hope this video helps you in understanding how to graph using transformations. And I'll go ahead and see you all in the next video.

In Exercises 67-80, begin by graphing the square root function, f(x) = √x. Then use transformations of this graph to graph the given function. h(x)=√(-x+1)

Key Concepts

Square Root Function

Graph Transformations

Function Composition

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