Finding Formulas for Functions

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Question

Finding Formulas for Functions

Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.

Accepted Answer

Welcome back, everyone. A. X. Y lies on the curve Y equals square root of X + 6. Let L represent the distance between X, Y and the fixed. 3.0. Express L as a function of Y. So for this problem we're given two points. Let's label the first one as X1. Y1. We know that this point is simply X, Y, right? This is the general expression of the point, and our second point is X2. Y2. It is exact. It is 3.0. We're looking for the distance between these two points, so we have to recall the distance form. Danced is equal to square root of x2 minus x 12 plus Y2 minus Y12. So what we're going to do is simply use that distance formula and we're going to label it as L, right, because this is going to be our distance as well. We're just going to express it in terms of Y later on. But for now it is sufficient to substitute the coordinates. So x2 is 3. We're simply taking 3 minus X 1 is X, so we're taking 3 minus X2 plus. Now Y2 is 0, and we are subtracting Y1, which is simply Y. Now let's simplify. We're going to get 3 minus X, all of that squared. So applying the formula we first we'll square 3, which gives us 9. We subtract 2 multiplied by 3 multiplied by X, which is 6 X, and then we add the square of X. Now, we're adding 0 minus Y2, which is simply negative Y2, and that's Y2. So now from here, it is really important to understand that we're trying to express L. As a function of y and we know that y. is equal to square root of x + 6. So what we can do is simply square both sides of this equation, and we're going to see that Y2 equals x + 6, and from here we're going to solve for x. We can show that x is equal to Y2 minus 6, right? And also we can show that x2, notice that we also have X2, is equal to y2 minus 62. So let's substitute all of those in. We get L of Y equals square root of 9 minus 6 X, which is 6 incy2 minus 6. Plus x 2 which is y2 minus 62 + y2. Well done, so we already have our expression. All that we have to do is simplify. So let's take our square root. We get 9 minus 6 Y2 + 36. So here we are simply distributing and we're going to apply the formula for the square of a difference which gives us the square of the first term. That's why it's the 4th power. We are then subtracting 2 multiplied by Y2 multiplied by 6, so that's 12 Y squad, and then we're adding the square of 6, which is 36. And finally, we need to add Y2. Now, let's combine the like terms. First of all, we have to notice that we have our radicals, so that's we're root. We're going to arrange our terms, starting with Y to the power of 4. Now, considering Y2d, we have -6 minus 12, that's -18, 18 + 1 gives us -17, so we end up with -17 Y2. And now considering the three constants, we have 9 + 36, that's 45, and 45 + 36 gives us 81. So we're simply adding 81. And now we have the complete expression for L as a function of a Y. Thank you for watching.

Finding Formulas for Functions

Consider the point (x,y) lying on the graph of y = √(x − 3). Let L be the distance between the points (x,y) and (4,0). Write L as a function of y.

Key Concepts

Distance Formula

Function Representation

Substitution in Functions

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