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Now consider a cubic function f ( x ) = a x 3 + b x 2 + c x + d . If a > 0 , then f ( x ) as x and f ( x ) −∞ as x −∞ . If a < 0 , then f ( x ) −∞ as x and f ( x ) as x −∞ . As we can see from both of these graphs, the leading term of the polynomial determines the end behavior. (See [link] (b).)

An image of two graphs. The first graph is labeled “a” and has an x axis that runs from -4 to 5 and a y axis that runs from -4 to 6. The graph contains two functions. The first function is “f(x) = -(x squared) - 4x -4”, which is a parabola. The function increasing until it hits the maximum at the point (-2, 0) and then begins decreasing. The x intercept is at (-2, 0) and the y intercept is at (0, -4). The second function is “f(x) = 2(x squared) -12x + 16”, which is a parabola. The function decreases until it hits the minimum point at (3, -2) and then begins increasing. The x intercepts are at (2, 0) and (4, 0) and the y intercept is not shown. The second graph is labeled “b” and has an x axis that runs from -4 to 3 and a y axis that runs from -4 to 6. The graph contains two functions. The first function is “f(x) = -(x cubed) - 3(x squared) + x + 3”. The graph decreases until the approximate point at (-2.2, -3.1), then increases until the approximate point at (0.2, 3.1), then begins decreasing again. The x intercepts are at (-3, 0), (-1, 0), and (1, 0). The y intercept is at (0, 3). The second function is “f(x) = (x cubed) -3(x squared) + 3x - 1”. It is a curved function that increases until the point (1, 0), where it levels out. After this point, the function begins increasing again. It has an x intercept at (1, 0) and a y intercept at (0, -1).
(a) For a quadratic function, if the leading coefficient a > 0 , the parabola opens upward. If a < 0 , the parabola opens downward. (b) For a cubic function f , if the leading coefficient a > 0 , the values f ( x ) as x and the values f ( x ) −∞ as x −∞ . If the leading coefficient a < 0 , the opposite is true.

Zeros of polynomial functions

Another characteristic of the graph of a polynomial function is where it intersects the x -axis. To determine where a function f intersects the x -axis, we need to solve the equation f ( x ) = 0 for .n the case of the linear function f ( x ) = m x + b , the x -intercept is given by solving the equation m x + b = 0 . In this case, we see that the x -intercept is given by ( b / m , 0 ) . In the case of a quadratic function, finding the x -intercept(s) requires finding the zeros of a quadratic equation: a x 2 + b x + c = 0 . In some cases, it is easy to factor the polynomial a x 2 + b x + c to find the zeros. If not, we make use of the quadratic formula.

Rule: the quadratic formula

Consider the quadratic equation

a x 2 + b x + c = 0 ,

where a 0 . The solutions of this equation are given by the quadratic formula

x = b ± b 2 4 a c 2 a .

If the discriminant b 2 4 a c > 0 , this formula tells us there are two real numbers that satisfy the quadratic equation. If b 2 4 a c = 0 , this formula tells us there is only one solution, and it is a real number. If b 2 4 a c < 0 , no real numbers satisfy the quadratic equation.

In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the x -axis. In some instances, it is possible to find the x -intercepts by factoring the polynomial to find its zeros. In other cases, it is impossible to calculate the exact values of the x -intercepts. However, as we see later in the text, in cases such as this, we can use analytical tools to approximate (to a very high degree) where the x -intercepts are located. Here we focus on the graphs of polynomials for which we can calculate their zeros explicitly.

Graphing polynomial functions

For the following functions a. and b., i. describe the behavior of f ( x ) as x ± , ii. find all zeros of f , and iii. sketch a graph of f .

  1. f ( x ) = −2 x 2 + 4 x 1
  2. f ( x ) = x 3 3 x 2 4 x
  1. The function f ( x ) = −2 x 2 + 4 x 1 is a quadratic function.
    1. Because a = −2 < 0 , as x ± , f ( x ) −∞.
    2. To find the zeros of f , use the quadratic formula. The zeros are
      x = −4 ± 4 2 4 ( −2 ) ( −1 ) 2 ( −2 ) = −4 ± 8 −4 = −4 ± 2 2 −4 = 2 ± 2 2 .
    3. To sketch the graph of f , use the information from your previous answers and combine it with the fact that the graph is a parabola opening downward.
      An image of a graph. The x axis runs from -2 to 5 and the y axis runs from -8 to 2. The graph is of the function “f(x) = -2(x squared) + 4x - 1”, which is a parabola. The function increases until the maximum point at (1, 1) and then decreases. Both x intercept points are plotted on the function, at approximately (0.2929, 0) and (1.7071, 0). The y intercept is at the point (0, -1).
  2. The function f ( x ) = x 3 3 x 2 4 x is a cubic function.
    1. Because a = 1 > 0 , as x , f ( x ) . As x −∞ , f ( x ) −∞ .
    2. To find the zeros of f , we need to factor the polynomial. First, when we factor x out of all the terms, we find
      f ( x ) = x ( x 2 3 x 4 ) .

      Then, when we factor the quadratic function x 2 3 x 4 , we find
      f ( x ) = x ( x 4 ) ( x + 1 ) .

      Therefore, the zeros of f are x = 0 , 4 , −1 .
    3. Combining the results from parts i. and ii., draw a rough sketch of f .
      An image of a graph. The x axis runs from -2 to 5 and the y axis runs from -14 to 7. The graph is of the curved function “f(x) = (x cubed) - 3(x squared) - 4x”. The function increases until the approximate point at (-0.5, 1.1), then decreases until the approximate point (2.5, -13.1), then begins increasing again. The x intercept points are plotted on the function, at (-1, 0), (0, 0), and (4, 0). The y intercept is at the origin.
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Questions & Answers

questions solve y=sin x
Obi Reply
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
Ryan Reply
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
NIKI Reply
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
follow algebraic method. look under factoring numerator from Khan academy
moe
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
The

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