How do I sketch the graph of $y=x(8-2x)(22-2x)$

Question

The entire question is to sketch the graph and state an appropriate domain given that $y\, \text{cm}^3$ is the volume of a cuboid with height $x\, \text{cm}$, length $(22-2x) \, \text{cm}$ and width $(8-2x)\, \text{cm}$...I get the domain part but not sure how to graph the given question

Answer 1

I would simplify a little, taking out two factors of $2$ as a constant: $$y=4x(4-x)(11-x)$$

Now it depends how accurate you want your sketch to be. Normally you will want to identify where the graph crosses the co-ordinate axes ($y=0, x=0$) and what happens at these points eg whether the function moves from being positive to negative or negative to positive.

What happens for large values of $\pm x$ is also important.

If you are dealing with rational functions rather than polynomials you will also be needing to consider what happens when the denominators are close to zero - locating asymptotes, and the sign of the function on either side of an asymptote.

This is sufficient for a fairly accurate general sketch showing the rough shape. For more accuracy you will want to consider the derivative and establish the values of the function at local maxima and minima. A sketch would generally identify such critical points, and consider what happens at all points where the derivative is zero.

Answer 2

A few guidelines:

• Find the values of $x$ that will make $y=0$.
• Find the value of $y$ when $x=0$.
• This is a cubic polynomial, it is continuous and differentiable.
• What happens as $x$ goes to positive infinity.
• What happens as $x$ goes to negative infinity.
• You should have enough data to have a rough sketch from the information above. If necessary, you can also find the turning points.

For the quantity to makes sense, you want height, length, width to be positive.

To check your answer after you attempt it, you can type the equation into google search and you will obtain the right curve.