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Problem 56 Hard Difficulty

Find an expression for the function whose graph is the given curve.

Answer

$y=\frac{-3}{2} x-3, y=\frac{3}{2} x-3, y=\sqrt{4-x^{2}}$

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Video Transcript

all right. We want to write an expression that represents the graph we see here Notice that the graph has three pieces. We have a piece of a line, a piece of a circle on a piece of a line, so we're going to write a piece vice function. Okay, so let's calculate the equation of the line segment on the left. So we noticed that it has a slope of negative three halves because it goes down three and right to so we can use that as our slope. And then we can use the point negative to zero. And let's use point Slope form to find the equation of the line. So why minus zero equals negative? Three halves times X minus negative, too. So why equals negative three halves, Times X plus two. So why equals negative three halves? X minus three is the equation of that line. So that's our first piece. And the domain for that. Peace is just from X equals negative for two X equals negative, too. Okay, let's go ahead and find the equation of the other line. So it has a slope of positive three halves, and it goes through the 0.20 so we can do the same kind of thing. Why minus zero equals three halves, times X minus two and we get why equals three halves X minus three. Okay, so we can put that into our piece wise function as well. And for that piece, it's only going from X equals 22 X equals four so we can put in that domain positive two to positive for now, for the piece in the middle, it goes from X equals negative 22 X equals positive, too. And I'm just going to put less than signs not less than or equal to signs because we already used the equal to signs when we did the first piece on the third piece. Notice that the middle pieces half a circle and the equation of a circle is X minus h quantity squared. Plus why minus k quantity squared equals R squared. Where HK is the center, The center of this circle is 00 So we just have X squared plus y squared equals and the radius is two. So we have two squared for what we want to do with this is we want to solve it for why, and just take the top half of the circle. So X squared plus y squared equals four. We isolate why in that equation, subtract X squared from both sides and square root. When we square root, we have plus or minus the square root of four minus X squared. Now the plus part, that's the top half of the circle and the minus part. That's the bottom half of the circle. We want the top half of the circle, so we want the square root of four minus X squared, and there's the function that matches the graph.