What is Quadratic function: Definition and 46 Discussions
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.
For example, a univariate (single-variable) quadratic function has the form
f
(
x
)
=
a
x
2
+
b
x
+
c
,
a
≠
0
{\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0}
in the single variable x. The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the y-axis, as shown at right.
If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.
The bivariate case in terms of variables x and y has the form
f
(
x
,
y
)
=
a
x
2
+
b
y
2
+
c
x
y
+
d
x
+
e
y
+
f
{\displaystyle f(x,y)=ax^{2}+by^{2}+cxy+dx+ey+f\,\!}
with at least one of a, b, c not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a hyperbola).
A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant:
f
(
x
,
y
,
z
)
=
a
x
2
+
b
y
2
+
c
z
2
+
d
x
y
+
e
x
z
+
f
y
z
+
g
x
+
h
y
+
i
z
+
j
,
{\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,}
with at least one of the coefficients a, b, c, d, e, or f of the second-degree terms being non-zero.
In general there can be an arbitrarily large number of variables, in which case the resulting surface of setting a quadratic function to zero is called a quadric, but the highest degree term must be of degree 2, such as x2, xy, yz, etc.
Can someone please tell me where I am wrong. I am learning how to write ##a^{2} + bx + c## in this form ##f(x)= a(X-X_0)^{2} +Y_0##.
The method used in my textbook is a reduction to the perfect square. And it goes like this:
##f(x)=ax^2+bx+c##
##=a[x^2+\frac{b}{a}x]+c##
##=a\left [...
The graph's turning point of a quadratic function f(x)=ax^2+bx+c is over the X-axis. If the coordinate of the turning point is (p, q) and a > 0, the correct statement is ...
A. c is less than zero
B. c is more than zero
C. q is less than zero
D. q equals zero
Since the point (p, q) is over the...
Homework Statement
##x,y,z \in \mathbb{R}##, find the minimal value for $$x^2+2y^2+z^2-6x+4y-10z+17$$
Homework Equations
None
The Attempt at a Solution
First I try to use”complete the square” method to make the polynomial something like:
$$(x-3)^2+(\sqrt2y+\sqrt2)^2+(z-5)^2-19$$
Then I am...
Homework Statement
For which values of b will the quadratic function ƒ(x) = x2-2bx+7 have a minimum value of 6?
Homework Equations
y = ax2+bx+c
y = a(x-h)2+k
b(first one) = -2ah(second one)
c(first one) = ah2+k(second one)
The Attempt at a Solution...
I can't come up with this function for hours:
I just started to learn quadratic functions so there must be an obvious solution for this which I clearly missed...
The problem is:
f(x)=-2x^2-(1/2)
Determine if this statement is true of false:
The axis of symmetry is x=-2.
What is the axis of symmetry? How can you figure out the axis of symmetry without a b value, since the formula for it is x=-b/2a
Homework Statement
Show that \Gamma_T is maximum at E_a = \frac{N_aE}{N_a+N_b}
Homework Equations
The expression for \Gamma(E) when N\gg 1
\Gamma_T = C_aC_b exp \left( -\frac{E_a^2}{2N_a\mu_B^2h^2} \right) exp \left[ - \frac{(E-E_a)^2}{2N_b\mu_B^2H^2} \right]
where C_a and C_b are...
Hey, everyone! I'm helping a friend through his calculus course and we've come across something that has stumped me (see: the title). When I learned calculus, our treatment of the epsilon-delta definition of the limit was, at best, brief. Anyway, here is the problem:
Given ##\lim_{x \rightarrow...
HI! I'm not sure if this can go in precalculus or not because I'm from Australia, and our Maths subjects don't get that specific until university level.
1. Homework Statement
For my assignment on quadratic functions, I have to find the equation (the the form of ax^2+bx+c) for a table of...
Hi!
I am trying to build a better understanding of quadratic functions. So I had this idea of trying to fit a quadratic function to two arbitrary points, but I am not sure how this really works.
Given the points (-4,1) and (12,1) on a Cartesian plane, how can a quadratic function be defined...
Fundamental theorem of calculus: \int_a^b \frac{df}{dx} dx = f(b) - f(a)
All right? Everybody knows...
BUT, BUT, exist some analog of the FTC for this case:
\int_a^b \left( \frac{df}{dx} \right)^2 dx = ?
Hum?
Hello, I'm having some trouble on this question, and I'd imagine it's just because I'm looking at it incorrectly.
The problem statement is:
The quadratic function f(x) = p + qx - x2 has a maximum value of 5 when x = 3 (i.e. vertex at (3, 5), right?)
Find the value of p and the value of q...
Homework Statement
A player hits a volleyball when it is 4 ft above the ground with an initial vertical velocity of 20 ft/s (equation would be h = -16t2 + 20t + 4). What is the maximum height of the ball?
Homework Equations
quadratic formula
The Attempt at a Solution
t = -20 ±√202 - 4(-16)(4)...
Find the difference quotient f(x+h)-f(x)/h
Where h\ne 0, for the function below
f(x)=5x^2+4
Simplify your answer as much as possible.
How do I do this?
Homework Statement
We want to determine the coefficients of a polynomial of the form:
##p(x)=c_{1}x^2 +c_{2}x+c_{3}##The polynomial ##p(x)## must satisfy the constraint ##p(1)=1##.
We would also like ##p(x)## to satisfy the following 4 constraints:
##p(−1)=5##
##p(0)=−1##
##p(2)=6##...
Hmm I'm having issues with an optional problem for some review for a quiz later tonight.
Determine the Quadratic Function for the Graph
the points labeled are X intercepts =(-3,0) (5,0); Y-Intercept = (0,-30), Local Min/Vertex = (1,-32)
It's the parabola looking problem at the bottom right...
These 3 equations all describe the same quadratic function. What are the coordinates of the following points on the graph of the function? From which equation is each point most easily determined?
y = (x - 5) (x + 1)
y = x ^ 2 - 4x - 5
y = (x - 2) ^ 2 - 9
X-intercept, what are the points...
Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct :)
Part B
1) A golfer hits a nice iron shot and the ball's height above the ground is given by h(t) = -16t^2 + 80t, where t is the time in seconds since the ball was...
Homework Statement
The part I'm having problems with is where the last two expressions in 4.13 are equated. Why is xtKx*equal to x*^tKx*?
The Attempt at a Solution
xtKx* is an inner product and due to symmetry is equal to x*^tKx, but wouldn't equating x to x* mean every <x,y> = <x,x> =...
Here is the question:
Here is a link to the question:
Problem Solving - Math? - Yahoo! Answers
I have posted a link there to this topic so the OP can find my response.
Consider the map f : \mathbb{R}^2 \rightarrow \mathbb{R}^2
defined by
(x,y) \mapsto (xy-x^2, xy-y^2)
I'm interested in figuring out the range of this function, but I keep thinking myself in circles. What would be a systematic method for approaching something like this?
Homework Statement
The profit, P, of a video company, in thousands of dollars is given by P=-5x^2+550x-5000, where x is the amount spent on advertising in thousands of dollars.
a) Determine the amount spent on advertising that will result in a profit of $0.
Homework Equations...
Homework Statement
Given f(x) = x^2 Sketch the graph of each of the following, state the domain and range.
- 1/2f(2x+6) -2
Homework Equations
y=x^2
The Attempt at a Solution
I Simply have a question. The question is, that this 2x can be factored out from the brackets...
Hi all,
Hope some here can help me with this math problem.
Given,
y1 = ax^2 + b.
y2 = cos (y1).
where a and b are constants. Is y2 periodic with respect to x.? Visually using example grpah, seems to be periodic. How do u find the exact period of such a function?
Thanks in advance...
Hey So i need to write down the first order necessary condition (F.O.C). and i need to find out where does a stationary point exist
I know how to solve for the F.O.C when i am given an equation in the standard quadratic polynomial form but here the equation is
f(x) = 1/2 xT Qx - cTx
where T...
I am trying to find the domain of this function f(x)=sqrt(x^2 - 2x +5). i am supposing that y=x^2 - 2x +5 and y must be greater of equal to 0. in otherwords where the graph is touching or above x axis. to find where the graph is touching x-axis i am trying to find the x intercept and i get...
Homework Statement
......49ac-12b2
(4α-3β)(3α-4β) =------------------------
.......a2
Deduce that, If 12b2< 49ac< 49b2/2
then β lies between 3α/4 and 4α/3
Homework Equations
α+β = -b/a
αβ = c/a
The Attempt at a Solution
Homework Statement
What is the condition, for a quadratic function of the form
ax2 + bx + c = y
to be a perfect square? (x, y are real here)
There's a question of this type in a book I'm working with, and I'd just like to have some general conditions for any quadratic...
The Attempt...
Hello, I am sorry for removing the template but it is not an actual problem that I need help with. Well it is, but it should be pretty quick and straight forward.
The problem is located here: http://www.cdli.ca/courses/math3103/unit05_org01_ilo04/images/3-less9.gif
(It's a bit difficult...
Homework Statement
For the function y = 1 / ((x-1)^2)
what is:
a) the sign of slope
b) change in slope (increasing or decreasing)
for the interval x > 1
Here is the graph:
http://img7.imageshack.us/img7/3204/12491454.jpg
Homework Equations
None
The Attempt...
Hi guys what's up. I just have a small question i wish to ask. If you are given vertex(-2,3) and it passes through point(2,5) how do you determine its equation? Its been around 6 months since i last did some math so i am having a hard time remembering. I am not asking you guys to solve it but...
Hi everybody,
I'm trying to minimize a function with multiple variables. My goal is to approximate on the L2 norm a matrix by the outer product of 2 vectors (or is it called tensor product ?).
So I have to determine a vector y = (y1,...,yn) and a vector x = (x1,...,xm) such that their...
Hello PF,
Ok so here is my problem. I have an array of data points which are the chi^2 values for a fit I have performed. I wish to calculate the 1-sigma error for this fit, so I figure I could find the lowest chi^2 value, and the two points either side that value, and fit a quadratic...
Homework Statement
It Costs a bus company $225 to run a minibus on a ski trip, plus $30 per passenger. the bus has seating for 22 passengers, and the company charges $60 per fare if the bus is full. For each empty seat, the company has to increase ticket price by $5. How many empty seats...
Homework Statement
A company sells running shoes to dealers at a rate f $40 per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum...
Quadratic function...help!
Hi
I have an exam (next week) on quadratic functions! (10th grade) Any online websites where I can practice?I need problematic situations(there's a problem - you analyze - calculate stuff,etc)... By the way, I'm studying in a FRENCH HIGH SCHOOL, therefore the...
P = 87x - 700 -x^2
1. Draw a graph of P against x and estimate the maximum value of P and also calculate the coordinates of the stationary point and verify if it is maximum.
I am not very good at graph drawing or stationary points... any help people?
SOLVED
Homework Statement
While going through a question, I came across a function f(x)=5^{x}+5^{-x}
I saw its graph through a software. It was a parabola with minimum value 2.
Now a question arises in my mind.
Every function of the type g(x)=ax^{2}+bx+c is a parabola.
Can I assume...
If one plots the graph of
y = x2-10 x+24
which is a parabola, one observes that it is above the x axis, i.e. y > 0, for two intervals of the real line. (The union of these intervals constitute the set of solutions of the quadratic inequality at the beginning of this question.)
Find...
[SOLVED] Quadratic Function
Homework Statement solve (f(x)=-3 x^2 + 9 x + 1/4 )
fx=(-3)(x^2-3 x)+1/4
fx=(-3)(x^2-3 x+9/4)+1/4+27/4
fx=(-3)(x^2-3 x+9/4)+28/4
fx=(-3) (x-3/2)^2+7
Homework Equations
The Attempt at a Solutionthe solution is already given in the book, but I don't understand why...
Homework Statement
Find a quadratic function f(x), that satisfies the given conditions: f(1) = 2,
f(-1) = 4, f(3) = 8.
Thanks
Homework Equations
f(x) = k(x-s)(x-t)
The Attempt at a Solution
I tried entering the points as the the x and f(x) while estimating as what the...
Let f(x) be a quadratic function such that f(0) = -4 and
\frac{f(x)}{x^2(x-5)^8}dx
is a ration function.
Determine the value of f'(0).
f'(0)=______
i don't really have a clue on how to do this. I can only think of integrating the function then find the derivative of it and plug in 0...
f(x) = -2x^2 + 12x + 12
I was able to express it in standard form and find the maximum value, but I can't seem to get the x-intercepts
I got -2x^2 + 12x + 12 = 0
-2 (x^2 - 6x + 6) = 0
does that mean I need to use the quadratic formula or did I do something wrong?