Question

Use what you know about end behavior and zeros to graph the following function.

Answer 1

\[f(x) = x(x - 5)(x - 3)(x + 2)(x + 4)\]

x-intercepts: -4, -2, 0, 3, 5

I would give it an end behavior of:
\[x \rightarrow -\infty, y \rightarrow +\infty\]
\[x \rightarrow +\infty, y \rightarrow +\infty\]

Answer 2

Also, a y-intercept of 0.

Answer 3

How do I draw this xD

Answer 4

check the left hand limit again

if x becomes a very large negative number then your function is essentially

(a large negative number) ^ 5 which ends up going to - infinity

Answer 5

since the highest power is 5 the graph has 5 - 1 = 4 "turns" or places where the function changes from increasing to decreasing or vice versa "bumps" in the function

Answer 6

Oh, the coefficient is 1 and the degree is positive.

It should be:


\[x \rightarrow -\infty, y \rightarrow -\infty\]
\[x \rightarrow +\infty, y \rightarrow +\infty\]

Answer 7

yeah

Answer 8

that's the initial setup, then I draw a graph that goes through all the intercepts and makes 4 "turns"

Answer 9

if you want to get picky about the shape you could substitute values into the equation to get more precise y-values

Answer 10

No, my teacher does not need for it to be precise.

This works. https://www.wolframalpha.com/input/?i=f(x)+%3D+x(x+-+5)(x+-+3)(x+%2B+2)(x+%2B+4)

I just did not know what to do with the x-intercepts, but I get it now. Start with the end behavior, then just do the turns.

Answer 11

And wolfram links still do not work ._.

Answer 12

yeah some of the special characters like parentheses mess it up :S

Answer 13

one more important thing, if you have a repeated root like f(x) = x(x-2)(x-2) the graph doesn't cross through x = 2 to the other side, it touches the zero and then "bounces" back in the original direction

Answer 14

Like a parabola?

Answer 15

yeah sort of

Answer 16

the graph above crosses through x = 0 but when it gets to x = 2 it "touches" x = 2 and then goes back up w/o crossing the y-axis

Answer 17

another method here (might be beyond the scope of your course) is to evaluate the critical numbers of the first derivative to model increasing and decreasing behaviour and evaluating the second derivative hypercritical points to determine concavity behaviour.

Answer 18

@shadow is this calc or pre-calc?

Answer 19

Precal, so that concept is beyond my course. Would be willing to learn it through.

Answer 20

derivatives make this process a lot easier, I think you'd enjoy it ^^ I don't think I have the the capacity to teach the fundamentals well though

Answer 21

Well, I have to go atm but we could explore/discuss this later. Should be free in roughly 20-30min.

Answer 22

Just some quick clarifications as I just came across an example of this:

"one more important thing, if you have a repeated root like f(x) = x(x-2)(x-2) the graph doesn't cross through x = 2 to the other side, it touches the zero and then "bounces" back in the original direction"

This is basically multiplicity 2? and what do you mean by "touches the zero."

@Vocaloid

Answer 23

Yeah that's multiplicity

By touches I mean the graph reaches 0 without crossing through the axis

Answer 24

So it reaches y = 0 but does not touch the x axis?

Answer 25

It touches the x axis but doesn't cross through

Answer 26

oh

Answer 27

I got it

Answer 28

Thank you