Graphs and transformations (1 Viewer)

[miester]

every time i do a past hsc paper i always lose marks in the transformations section ie when you're give y=f(x) and you sketch y=f(x^2), y=[f(x)]^2 etc etc...the only ones i can get out are the absolute value ones, all the others cost me marks can anyone suggest a way to do those type of questions? it would REALLY be appreciated!

 

[D94]

So using the function squared questions as an example... you have .

Let therefore getting . Now, the curve is the given curve (usually a diagram). The u axis is just the y axis.

Draw the , and now we use this graph and the to compare critical points, eg. x-intercept/s, y-intercept/s, asymptotes.

So for example, when , , when , . Therefore we get

...when , , when , . Therefore we get

...when , , when , . Therefore we get

... and so on. Then the curve will usually take its shape.

It can work for sine, cosine, inverse, absolute values, exponentials, logs... you name it

 

[miester]

thanks, that really helped

what approach do you take when sketching 1/f(x) and f(x^2)?

 

[b3kh1t]

There is generally just a set of rules for 1/f(x). They are

f(x)=0 then (therefore when the original function crosses the x-axis it will become asymptote)

then f(x)=0 (so it will cross the x-axis however this is not certain so when it reaches the x-axis we draw an open circle)

if you draw the line y=1 then the same points at which the line intersects with the original graph, 1/f(x) will also pass through those points.

For the f(x^2) you can use the same method D94 has shown you.
So let u=x^2, now say you take the point A(2,4), x=2 therefore u=4. However the y value has not changed, the point A has just moved from (2,4) to (4,4).
Another example is point B(3,8), x=3 therefore u=9. The y value still remains unchanged, so the point B just moves from (3,8) to (9,8).
So the f(x) graph just becomes stretched when you do f(x^2).

 

[D94]

what approach do you take when sketching 1/f(x) and f(x^2)?

For I tend to do the same process, ie. , as the hyperbola can be easily drawn, and letting .

With , you can do the same process, but now, you have as your given curve (ie. your x axis is now the u axis), and let/draw , where this is just a parabola.

 

[miester]

that's great! thank you so much for your help