Question

Sketch the graph of the function. Label the intercepts, relative extrema, points of inflection, and asymptotes. Then state the domain of the function. \(y=x \sqrt{4-x^{2}}\)

Short answer

The x-intercepts are -2, 0, and 2; the y-intercept is at 0. The single relative maximum is at \(x=0\). There are no points of inflection and asymptotes. The domain is \(-2 ≤ x ≤ 2\).

Step-by-step solution

Find the x and y intercepts

To find the x-intercepts, set \(y=0\) and solve for \(x\). This gives \(x = 0\) and \(x = ±2\). To find the y-intercept, we can substitute \(x=0\) into the equation to get \(y=(0) \sqrt{4-0^{2}} =0 \). So the x-intercepts are -2, 0, and 2; the y-intercept is at 0.

Determine the relative extrema

To find the relative extrema, we will need to take the derivative of the function, set it equal to zero, and then solve for \(x\). The derivative of \(y\), \(y' = \sqrt{4-x^2} - \frac{x^2}{\sqrt{4-x^2}}\), and by equating to 0 we get \(x = 0\) as a relative extreme.

Find the points of inflection and asymptotes

To find inflection points, we take the second derivative of \(y\) and set it to zero. But for this function, the second derivative is very complex and does not have real roots, so there are no inflection points. As the function is polynomial-like, there are no asymptotes.

State the domain of the function

The function is defined for values of \(x\) such that \(-2 ≤ x ≤ 2\). That is the domain of the function.

Sketch the graph

Now with all features determined, sketch the graph of the function. With \(y=0\) at \(x=-2, 0, 2\), a peak at \(x=0\), no asymptotes, and no inflection points, the resulting graph resembles an approximated bell shape.