Question

Construct the graph of the following curves : (i) \(y^{2}=8 x^{2}-x^{4}\) (ii) \(y^{2}=(x-1)(x+1)^{-1}\).

Short answer

Answer: (i) The domain for the curve \(y^2=8x^2-x^4\) is all real numbers (-∞, ∞), and its range is also all real numbers (-∞, ∞). The curve is symmetric about both the x-axis and y-axis. (ii) The domain for the curve \(y^2=(x-1)(x+1)^{-1}\) is all real numbers except \(x=-1\), i.e., \((-\infty, -1)\cup(-1,\infty)\). Its range is all real numbers (-∞, ∞) and it is symmetric about the x-axis.

Step-by-step solution

Determine domain and range

The domain of this curve can be found by determining the values of \(x\) for which the equation is defined. Since we can square any real number, this function is defined for all real values of \(x\). Therefore, the domain is all real numbers (-∞, ∞). To find the range, we must consider the equation and observe that when \(x\) is equal to 0, \(y\) is also equal to zero. As \(x\) becomes larger, the value of \(y^2\) becomes smaller, so the function must be symmetric about the x-axis. We can find the minima of the function by taking the partial derivative with respect to \(x\) and setting it equal to zero: \[\frac{d}{dx}\left(8x^2 - x^4\right) = 16x - 4x^3 = 0\]which implies that the function has its minima at x = 0 and x = ± √2. As x approaches ±√2, y approaches zero, so the range is also all real numbers (-∞, ∞).

Identify possible symmetry and properties

Since the equation is given in terms of \(y^2\) and all the powers of \(x\) are even, this function is symmetric about both the x-axis and y-axis, as well as being symmetric about the origin.

Sketch the graph

Now that we have identified the domain, range, and symmetry of the graph, we can sketch the graph. We know the graph will be symmetric about both axes and will have minima at x = 0 and x = ± √2 with y = 0 at those points. (ii) Given curve: \(y^2=(x-1)(x+1)^{-1}\)

Determine domain and range

The domain of this curve can be found by determining the values of \(x\) for which the equation is defined. To avoid division by zero, we must have \((x+1)\neq 0\) so \(x\neq -1\). Therefore, the domain is all real numbers except \(x=-1\), i.e., \((-\infty, -1)\cup(-1,\infty)\). For the range, since \(y^2\) is present, the function is symmetric about the x-axis. We can see that the numerator \((x-1)\) can take any real value, while the denominator \((x+1)\) ensures that the fraction never becomes too large. Thus, the range of the function is all real numbers, i.e., (-∞, ∞).

Identify possible symmetry and properties

Notice that if we replace \(y\) with \(-y\) in the equation, the equation remains the same. This means the function is symmetric about the x-axis.

Sketch the graph

Now that we have identified the domain, range, and symmetry of the graph, we can sketch the graph. The only restriction on the domain is \(x=-1\), and the symmetry of the function in the x-axis tells us that the graph will have an asymptote at \(x=-1\). Taking into account the domain, range, and symmetry, we can sketch the graph. To find more points on the graph, we can choose values of x and solve for y. For example, when x = 0, we have y^2 = -1, which means there are no real solutions for y, so the curve does not intersect the y-axis.