Need help with sketching square root of cubic functions

[vicrost]

Anybody who has studied maths with me will know that my least favourite topics are those that deal with sketching graphs. I absolutely loathe sketching questions and always found it very confusing and hard to remember how to sketch the various types of functions that could be asked. However,since they are quite important in the HSC, I've decided to step it up and try my best to master the art of sketching all the types of graphs that could be asked.

So I was practicing how to sketch complex square root relations and came across this question  (question 2g) ) :



My attempt to sketch the graph, note this is just rough sketching as I was just practicing my ability to get the key points and shape:



As you can see, I got the starting point , -3 , and the idea that the other half of the graph is a reflection across the x-axis, correct. But then this is the answer shown in the text book:



My question is how should I figure out that the graph will behave the way it did for the small values to the right of -3. The other part is when it curves after crossing the y-axis. What technique am I missing? Or are we just expected to test points and get the gist of the shape of the graph through that?

Thank You

[fun_jirachi]

Hello

You shouldn't have to test points outside of getting the feel of what the graph should look like or testing some points afterwards to make sure your graph is drawn correctly.

For \(y^2 = x^3 + 27\), you can just think about it as \(y=\sqrt{x^3+27}\) then reflect it in the x-axis as you have already noted. Then, you can apply the normal principles of the graph of \(y=\sqrt{f(x)}\):
- Recall that \(\frac{d(\sqrt{f(x)})}{dx} = \frac{f'(x)}{2\sqrt{f(x)}}\). Clearly in this case, we have a stationary point at \(x=0\).
- If \(f(x)>1\), then \(\sqrt{f(x)} < f(x)\) and vice versa. They will intersect at \(f(x)=1\).
- \(\sqrt{f(x)}\) will have an endpoint on the x-axis wherever \(f(x)\) crosses the x-axis.
- etc.

As for the behaviour of the graph that you've pointed out:
- analyse the derivative a little bit. It should be evident that for larger positive values of x \(\sqrt{f(x)}\) will grow faster.
- The behaviour close to the stationary point is mostly done to exaggerate my previous point 'If \(f(x)>1\), then \(\sqrt{f(x)} < f(x)\) and vice versa. They will intersect at \(f(x)=1\).' You ideally want to signpost certain features of the transformation (though probably not to that extent -- I think this is overkill and will probably lose you marks if you draw something like this) to get awarded marks. Also, realise that in that particular solution, the scale of the y-axis isn't actually the same as the scale of the x-axis.

Hope this helps

[vicrost]

Thnx, I understand the reasoning for the behavior of the graph now. I forgot the benefit of using the derivatives to find certain features (stationary points, etc) of a graph.

I just want to ask, if you don't mind, that do you recommend finding the derivative for all square root relation questions or just cubic ones so that I don't miss key points like the stationary point?

Also, can you suggest some tips you found very useful for sketching graphs and especially complex graphs. Like during the exam, do you think sketching questions should be given a decent chunk of time or is it better to prioritize other questions first. What I do normally now is I would skip all the sketching questions, finish the the paper then come back to them so that I can draw them calmly, knowing that I've at least solved other questions first. Thank you

[fun_jirachi]

I recommend at least having a look at the derivative regardless of what the transformation or function is. It points to a lot of features that you should be signposting on your graphs, as markers will be looking for them (and they will often be included in the question).

There's not a lot of tips I can give since personally it's mostly based on intuition. But at least try to:
- Know key features of common transformations and how to identify them. I'd include exponentiation, reciprocation, simple translations, dilation, logarithms, powers.
- Practice
- If in doubt, as a last resort, plot some points

It doesn't matter what I say or think about time allocation for particular questions because different students excel and struggle with different topics. In general, you should be allocating more time for objectively harder questions, then subjectively harder questions, then whatever else is left. If it works for you go ahead, but you should try to experiment and see what exam strategy lets you maximise your marks; if it so happens to be leaving sketching questions last use it, otherwise try something else. This goes for every other subject too. As an example, I didn't spend a lot of on sketching because I left my time for the questions towards the back end of the paper. Use your reading time well too; you should be identifying what questions you want to spend more time on (sketching, if need be in your case) and planning accordingly.