Beginner
what are the steps to graph a polynomial function?
Step1: Determine the end behavior of the graph
Step2: Find the x-intercept(s) or the zero(s) of the function.
Step3: Find the y-intercept of the function.
Step4: Graph
What is the use of polynomial functions in real life?
Polynomial can use in modeling, industries, construction, physics, etc. it can also use in expressing energy, inertia , and voltage.
Which of the following is a Polynomial Function?
a) P(x)= 6x-4-4x2+3
b) P(x)= 5x3-3x8-5
c) P(x)= 6x5-7x3+4
The correct answer is C.
Given f(x)=-3x3+5x2+x-5, What is the behavior of the function?
The behavior of the graph increases on the left and decreases on the right.
which best describes the graph of the function f(x)= x3+3x2-4x-12?
It goes down to the far left and up to the far right.
How can you identify if it is even function or odd function?
By the degree of the polynolmail
How can you identify the end behavior of the graph?
The end behavior of the graph of a polynomial function can also be determined through the use of a leading term test. The leading term text indicates the eventual rise or fall of the function as x increases or decreases without bound
Find all the zeros of f(x)= -x4+4x3-4x2.
the zeros of f(x)= -x4+4x3-4x2 are 0 and 2.
What is the behavior of the following graph: f(x)=x3-2x2-3x.
the behavior of the graph increases on the left and decreases on the right.
Which best describes the graph of the function f(x)= -2x3+6x-4?
It goes up to the far left and down to the far right.
how can you tell if a graph is a polynomial function?
Use the leading coefficient test to find the end behavior of the graph of a given polynomial function. Find the zero of a polynomial function. Find the multiplicity of a zero and know if the graph crosses the x-axis at the zero or touches the x-axis and turns around at the zero. If there is no variable in the denominator of the function and no variable under a rational.
Is the function a polynomial?
h(x)=7x^3+2x^2+1/x
No, because there is a variable in the denominator.
how many zeros does the polynomial f(x)=7x4+6x3-5x+7 have?
4
Given f(x)=(x-4)(x+1)(x+3) Find all the zeros.
x=4 , x=-1 , x=-3.
Find all of the zeros, the multiplicity of each zero, and whether or not the graph crosses the x-axis or turns around at each zero of the function:
f(x) = x3 - 12x2 + 45x - 54
The correct answer is: One zero at x = 3, with a multiplicity of two, and turns around at the x-axis, and another zero at x = 6, with a multiplicity of one, and crosses the x-axis.
To find all of the zeros, the multiplicity of each zero, and whether or not the graph crosses the x-axis or turns around at each zero of the function:
f(x) = x3 - 12x2 + 45x - 54
factor the equation:
f(x) = x3 + 45x - 12x2 - 54
f(x) = (x2 - 6x + 9) (x - 6)
f(x) = (x - 3)2 (x - 6)
0 = (x - 3)2 (x - 6)
(x - 3)2 (x - 6) = 0
(x - 3)2 = 0 or (x - 6) = 0
x - 3 = 0 or x - 6 = 0
x = 3 or x = 6
What is the degree of the polynomial f(x)=-2x^3(x-1)^2(x+5)
degree of 6
Use the leading coefficient to determine the end behavior of the polynomial function.
f(x)=5x^3+7x^2-x+9
falls left and rises right
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers
f(x)=x^3-x-1: between 1 and 2
f(1) = –1 f(2) = 5 The sign change shows there is a zero between the given values.
Find all of the zeros of the polynomial:
f(x) = x^3 - 7x^2 - 4x + 28
You found all of the zeros of the polynomial:
f(x) = x3 - 7x2 - 4x + 28
by factoring:
f(x) = x2(x - 7) - 4(x - 7)
f(x) = (x - 7)(x2 - 4)
f(x) = (x - 7)(x + 2)(x - 2)
Setting f(x) = 0, to find the zeros, we have:
(x - 7) = 0 or (x + 2) = 0 or (x - 2) = 0
x = 7 x = -2 x = 2
The correct answer is: x = -2, 2, 7
Give the turning points of the following polynomial function:
f(x) = x^4 - 8x^2 + 16
The correct answer is: (-2,0), (0, 16), (2,0)
We can get a clue about the turning points of the polynomial function:
f(x) = x4 - 8x2 + 16
by letting u = x2, then
= u2 - 8u + 16
= (u - 4) (u - 4)
(u - 4) (u - 4) = 0
u - 4 = 0 or u - 4 = 0
u = 4
x2 = 4
x = +/- 2
Now we have a clue where the graph approaches or crosses the x-axis.
Next, use your calculator to look for maxima and minima, giving the three turning points (n - 1) of (-2, 0), (16, 0), and (2, 0)
Find the zero for the polynomial and give the multiplicity for each zero
f(x)=2(x-5)(x+4)^2
f(x)=2(x-5)(x+4)^2
x = 5 has multiplicity 1; The graph crosses the x-axis. x = –4 has multiplicity 2; The graph touches the x-axis and turns around.
Find the zero for the polynomial and give the multiplicity for each zero
f(x)=4(x-3)(x+6)^3
f(x)= 4(x-3)(x+6)^3
x = 3 has multiplicity 1; The graph crosses the x-axis. x = –6 has multiplicity 3; The graph crosses the x-axis.
Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers
f(x)=2x^4-4x^2+1
f(–1) = –1 f(0) = 1 The sign change shows there is a zero between the given values.
Find all of the zeros of the function:
f(x) = 9x3 - 24x2 + 16x
To find all of the zeros of the function:
f(x) = 9x3 - 24x2 + 16x
factor:
f(x) = x(9x2 - 24x + 16)
f(x) = x(3x - 4)(3x - 4)
To find the zeros, where the graph crosses the x-axis, set y = f(x) to zero, and solve for "x":
0 = x(3x - 4)(3x - 4)
Using the zero product property:
x = 0 or x = 4/3
The correct answer is: x = 0, 4/3
f(x)=3x^2-x^3
(a)Give the graphs end behavior
(b) Find x-intercepts and state if the graph crosses or touches the axis
(c) Find the y-intercept
a)rises to the left and falls to the right
b) -x^3+3x^2=0
-x^2(x-3)=0
x = 0, x = 3 The zero at 3 has odd multiplicity so f(x) crosses the x-axis at that point. The root at 0 has even multiplicity so f(x) touches the axis at (0, 0).
c) f(0)=-(0)^3+3(0)^2=0
The y-intercept is 0.