determine which of the following statements are true or false regarding the graph of f(x) = x^3 +bx^2 +cx +d?

Question

it tells you b, c, and d are nonzero numbers. and you have to provide a reason for each of your conclusions for the parts below.
a. it contains the origin.
b. it intersects the y-axis in one and only one point
c. it intersects the x-axis in at most three points
d. it intersects the x=axis at least once
e. for x very large, it behaves like the graph of f(x) = x^3
f. it is symmetric with respect to the origin

Answer 1

a) False. f(0) = d. Since d is not zero, then f(0) is not zero. Since the graph of a function hits only one value on a vertical line, and that value here is (0, d), then the graph of f doesn't pass through (0, 0).

b) True. This is true of any function which is defined at x = 0. In this case, as noted in part a, the point is (0, d).

c) True. Any nonzero polynomial of degree 3 has at most three x-intercepts (that is, it intersects the x axis at most 3 times).

d) True. When x is large negative, the "x^3" part of the function will be much more influential than "bx^2 + cx + d", and force the function to be negative. When x is large positive, likewise the function will be positive. Since the function changes from negative to positive, it has to change somewhere--and it crosses the x-axis there.

e) This is true, in some sense. If x is much larger than b, c, or d, then x^3 will be much larger than bx^2, cx, or d--in fact, x^3 will be the only thing that really matters much in the behavior of the function when x is very large.

f) False. Suppose f were symmetric about the origin. Then, for every value of x, we would have f(-x) = -f(x); that is, -x^3 + bx^2 - cx + d = -x^3 - bx^2 - cx - d. Adding x^3 + cx to both sides gives bx^2 + d = -bx^2 - d. Adding bx^2 + d to both sides gives 2bx^2 + 2d = 0. There is no way that this is true for all values of x if b and d are nonzero; so f cannot be symmetric about the origin.

Answer 2

a. is false because a graph only passes through the origin which if each term contains an x. So pretty much for that particular equation just take f(0) and you get back d,

f(0) = 0^3 +b(0)^2 +c(0) + d
f(0) = d

since d is non zero the graph doesnot pass through the origin.

b. true because f(x) is a function there fore it has to pass the vertical line test, the y axis is a vertical line there fore it can only pass through the once. You could also just plug in zero into the function and you just get back d, which is just one constant number which means it only hits the y axis once.

c. true because the function is a degree three equation therefore it can have at most 3 real zeros.

d. true because, that equation is continous, the end behavior for any degree three equation is, one end of the graph point up and the other end points down, therefore it much touch the x axis at least once.

e. true again because of the end behavior type

d. false, since b,c,d are constants assign them values, plug in 1, and -1, into the equation, you will get back 10 and -2. in order for a function to be symmetric with respect to the origin, plugging in any number "w", and then pluging in any number "-w", you should get back a number that is positive and then a negative.So that sounds a bit unclear so what i mean is

the function x^3 is symmetric to the origin, so if the plug in 2 i get back 8, and if the plug in -2 i get back a -8, so that function is symmetric to the origin.

hope i helped
email for any questions

Answer 3

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Answer 4

a) false as f(0) is =d non zero
b) is a 3rd degree equation so it can have 1 or more than one real roots( till 3)
c) true (sea 2)
d) true it has at least one real root as complex roots appear by pairs
e)true as lim f(x)/x^3 for x==> infinity is 1
f) false as f(-x) is not -f(x) unless b and d are 0