The following equations implicitly define one or more function...

Question

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

y² = x²(4 − x) / 4 + x (right strophoid)

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says two graph the functions Y is equal to F1 of X and Y is equal to F2 of X that are implicitly defined by the equation Y2 is equal to 3x2 multiplied by the quantity of 5 minus X in quantity, divided by the quantity of 6 + 2 X in quantity. Use a graphene utility. Now, the first thing we need to do is solve our equation that we were given in the problem for Y. And so to do this, we just need to take the square root of both sides. In doing so, we will get Y is equal to plus or minus the square root. Of the quantity of 3 X squared multiplied by the quantity of 5 minus X in quantity, divided by the quantity of 6 + 2 X in quantity. And we can actually simplify this expression by factoring out a square root of X2 in the numerator, which will give us the absolute value of X. That means that this quantity is going to be equal to, plus or minus, the absolute value of X. Multiplied by the square root of the quantity of 3, multiplied by the quantity of 5 minus X in quantity, divided by the quantity of 6 + 2 X in quantity. So here we can actually break this up into two functions. We can have Y is equal to F1 of X, and here we will choose the positive root. So this is going to be equal to the absolute value of X, multiplied by the square root of the quantity of 3, multiplied by the quantity of 5 minus X in quantity, divided by the quantity of 6 + 2 X in quantity. And we will also have Y is equal to F2 of X, and here we will choose the negative root. So this is going to be equal to minus the absolute value of X. Multiplied by the square root of the quantity of 3, multiplied by the quantity of 5 minus X in quantity, divided by the quantity of 6 + 2 X in quantity. So now what we need to do is graph these two functions using a graphing utility. And when we do so, we will get the following graph. Where the curve that is above the x axis is due to our F1 of X function, and the curve that is below the x-axis is due to our F2 of X. So this graph here is the answer to our problem. So I hope that this explanation has been useful, and I'll see you in the next video.

The following equations implicitly define one or more functions.
c. Use the functions found in part (b) to graph the given equation.
y² = x²(4 − x) / 4 + x (right strophoid)

Key Concepts

Implicit Functions

Graphing Techniques

Strophoid Curves

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