Biology Forums - Study Force

a) y = x(x-4)(x-1)
-> 3
-> positive
-> x = 4, 1 (zero's)
-> y = 0(0-4)(0-1) (y-intercept)
  = 0
-> 4 + 1/2 = 2.5 -> -5.63 (min)
This is what I did, in order to sketch the graph and I just wanted to know if I did my calculations right before I graph the whole thing, because my textbook doesn't show the full answer to this.

Start by solving all zeros... Notice that the function is completely factored, which makes it easy, especially for a cubic. You solve by setting y = 0

0 = x(x-4)(x-1)

0 = x

0 = x - 4
x = +4

0 = x - 1
x = +1

Therefore, the function crosses:

(0, 0), (4, 0), and (1, 0)

Next, find the y-intercept, by setting x = 0. You should get a y-intercept at (0, 0)

Now determine the end behavior of the function (video link below). You need to write it in standard form, so multiply all factors out:

y = x(x-4)(x-1)
y = (x^2 - 4x)(x - 1)
y = x^3 - x^2 -4x^2 + 4x
y = x^3 - 3x^2 + 4x

The highest degree given that it is a cubic - if you multiply it all out - is 3. 3 is odd. The leading coefficient is also positive, therefore the end behavior will move up on the right side. Since the highest power is odd, it'll go down to left.

My sketch is shown below:

(https://biology-forums.com/gallery/42/6_24_11_20_1_55_58.png) (https://biology-forums.com/index.php?action=gallery;sa=view&id=42198)

Video on end behavior:

https://www.youtube.com/watch?v=HIBMpqcbQVY

Video on solving cubics:

https://www.youtube.com/watch?v=rFmdefYF5O4

how about finding the estimated max and min values?

For that, take the average between each zero.

For example, the average between 0 and 1 is 0.5. The average between 1 and 4 is 2.5. Substitute those values into the function, and you'll get your estimated max and min values

for both max and min? because that's usually for finding only min and that's what I did above, not sure if it is right though.

Yes, it should help you find both. Sub those values in, get an output. You should get an estimate of what the local max or min is

Okay so by doing that I got a result of -5.63 for both the max and min, would you agree?

This is more in line of what I was thinking

(https://biology-forums.com/gallery/42/6_24_11_20_10_57_16.png) (https://biology-forums.com/index.php?action=gallery;sa=view&id=42199)

Sorry for the late reply, I believe I got it now, I did all of c) and just wanted to know if I did it correctly, since my textbook answers are confusing/inaccurate. 

Looking good, but did you see that the multiplicity the zero (3, 0) is 2? In case you're curious:

Yes I did notice that, thanks :)

Mark as solved?

One more thing, how about if it was a quartic function and there is a variety of zero's, how would you find the max and min once again? Like for example for c) I ended up getting 0,0, -1/2, 3, 5.

Post Merge: 3 years ago

and also when finding the max and min with the zero's does it matter what zero you're using to find it, I think that's where I'm getting confused.

Hi patrick19 (https://biology-forums.com/index.php?action=profile;u=956591)

You'd do something like this:

(https://biology-forums.com/gallery/42/6_24_11_20_8_01_32.png) (https://biology-forums.com/index.php?action=gallery;sa=view&id=42200)

https://www.youtube.com/watch?v=tQdE4L6teBA

https://www.youtube.com/watch?v=m9UhoF6JeHA

does it matter which zero your adding up with to get the max/min?

Absolutely, they need to be adjacent zeros, otherwise it makes no sense

Sorry just one more thing, do you have any videos where you have to find the equation of a graph by using the given zero's, for example one of my questions says to find the equation of the graph with zero's -3, 0, 2 or another one -1, 4 (order 2), etc.

If you're given the zeros as: -3, 0, 2

y = x(x+3)(x-2)

Where you given anything else along with the zeros. Would be best if you upload the exact question, because I don't want to be advising something if it doesn't correlate with the follow-up question.

How would you know if it was negative or positive though, when graphing it? And I will post the question, I will mark this one solved.

You need to show me the exact question