Analyze and graph the following function: y = x(9-x^2)^(1/2)

Question

Analysis must include domain and range, symmetry, x and y intercepts, all asymptotes, first and second derivatives, critical numbers, intervals at which the function is increasing and decreasing as well as concave up and concave down, all relative extrema, and all possible points of inflection.

Answer 1

Let the function is .

Testing for symmetry.

x - axis : Replace y with - y.

Result is not a equivalent equation, so symmetric with respect x - axis is failed.

y - axis : Replace x with - x.

Result is not a equivalent equation, so symmetric with respect y - axis is failed.

Origin : Replace y with - y and x with - x.

Result is a equivalent equation, so symmetric with respect origin.

To find the x - intercept, substitute y = 0 in the original function.

.

To find the y - intercept, substitute x = 0 in the original function.

.

The function is .

Apply first derivative with respect to x.

Apply second derivative with respect to x.

To find the critical numbers, or does not exist.

.

Relative extrema :

If then is called relative maximum of f or f has a relative maximum at .

If then is called relative minimum of f or f has a relative minimum at .

Answer 2

continued ----->

Intervals at which the function is increasing and decreasing.

interval
Test Value
Sign of
Conclusion	Decreasing	Increasing	Decreasing

To locate possible points of inflection, determine the values of x for which or does not exist.

does not exit at , so possible points of inflection at .

Intervals at which the function is concave up and concave down.

interval
Test Value
Sign of
Conclusion	Concave downward	Concave upward	Concave downward

 

Answer 3

Continued ------

Make the table, Choose different values of x and obtain random y - values.

x		(x, y)
- 4		imaginary
- 3
- 2
- 1
0		(0, 0)
1
2
3
4		imaginary

Draw a coordinate plane.

Plot the points and connected these points with a smooth curve.

Observe the graph, the domain is and range is .

Find the all asymptotes :

If f(x) approaches infinity (or negative infinity) as x approaches c from the right or left, then line x = c is a vertical asymptote of the graph of f. Observe graph, the function does not approaches to infinity, so the function f(x) has no vertical asymptote.

The line y = L is a horizontal asymptote of the graph of f if .

So, the f(x) has no horizontal asymptote.