Simple Graphs question (1 Viewer)

[kurt.physics]

I just have some questions on the theory in the fitzpatrick textbook.

- i) (graphs of cubic function) two distinct turning points if 4b^2 - 12ac > 0

i dont know how they got that. To me, it looks like the discriminant of a quadratic, but its for cubic.

- (Graphs of rectangular hyperbola)

whats the difference between a rectangular hyperbola and a normal hyperbola (or are they the same thing)

why do we want to ensure xy is positive (by putting it equal to c^2)


And with graphing with "addition of ordinates",

what do yous do when you graph them, do you measure the two and add or is there some special trick.

Thanks,
-Kurt

 

[lolokay]

1. yes, it is the discriminant of a quadratic
do you know calculus yet?

2. a rectangular hyperbola is one where the asymptotes are perpendicular (it's a type of hyperbola)

the only reason to have xy positive is so it's always in the same quadrants (I would think)


3. when adding the ordinates, i think you would just look at the general trend of what's going on

 

[Trebla]

Consider y = ax³+bx²+cx+d
dy/dx = 3ax² + 2bx + c
For distinct turning points there must be two solutions to this quadratic so Δ>0
i.e. 4b² - 4(3a)(c) > 0 => 4b² - 12ac > 0

A rectangular hyperbola is a special case of the normal hyperbola. The normal hyperbola takes the form x²/a² - y²/b² = 1. A rectangular hyperbola is the special case when a = b. It is also the same as xy = c² because the asymptotes are perpendicular to each other in both cases.
The form xy = c² is used because in this course we are only concerned with a hyperbola with branches in the first and third quadrant.

Addition of ordinates is just adding y-values of two graphs. There is no "special trick", you just have to know how to add ordinates over whole continuous functions. e.g. y = x + sin x can be interpreted as the small values of x shift the ordinates of y = sin x up a little bit and as x gets larger, the shift upwards of y = sin x increases...so you end up with a slanted version of y = sin x.

 

[kurt.physics]

Thanks lolokay and trebla!!!

just another question. When i see logarithmic and exponential functions i dont really feel comfortable working with them, see i dont know much about them and their graphs. What can i do? How do i learn the specifics of their graphs.

 

[Js^-1]

Thanks lolokay and trebla!!!

just another question. When i see logarithmic and exponential functions i dont really feel comfortable working with them, see i dont know much about them and their graphs. What can i do? How do i learn the specifics of their graphs.

Just use them a lot. Do a lot of questions that involve the log and exponential graphs. Even simple 2 and 3 unit graphs will help. Graph y = e^x and then graph y = e^-x just so you get used to them. Try subbing in points to understand why it is that value at the certain place. Just experiment with different equations and see what happens when you modify it, say, by multiplying the whole thing by 2 or whatever.

 

[kurt.physics]

Cheers.