# 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

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.

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1. ### Formula for Xo and Yo for graph of quadratic function

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 [...
2. ### MHB Determining c in Quadratic Function Turning Point

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...
3. ### Quadratic function minimal value

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4. ### Quadratic function solution method optimizing

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5. ### B Really basic quadratic function problem

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...
6. ### MHB What is the axis of symmetry of quadratic function f(x)=-2x^2-(1/2)

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7. ### Find the maximum of E_a when the entropy is at its maximum

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...
8. ### Finding Delta Given Epsilon with a Quadratic Function

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...
9. ### How to find a quadratic function from a table of values?

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10. ### MHB How to write the quadratic function.

Write a quadratic function y whose maximum value is 4 and the axis of symmetry of the graph is x=-2. Suggestions?
11. ### MHB How to fit a quadratic function to two points?

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...
12. ### Is There a Fundamental Theorem of Calculus for Squared Derivatives?

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?
13. ### MHB Finding a Quadratic Function from a Vertex

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...
14. ### Application of quadratic functions to volleyball

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)...
15. ### MHB Difference quotient for quadratic function

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?
16. ### Linear regression with least squares for quadratic function

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##...
17. ### MHB Determining Quadratic Function for a Graph

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...
18. ### How to find quadratic function

I'm looking for the function y=ax^2+c with the given points (-3,89) (2,39). Thanks

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...
20. ### MHB Application of quadratic function in kinematics

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...
21. ### Linear Algebra, unique minimizer of a quadratic function

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> =...
22. ### How To Find This Quadratic Function?

How to find the below quadratic function? y = 0, x = 0 y = 2, x = 1 y = 2, x = -1 Thank you
23. ### MHB Swag's question at Yahoo Answers regarding maximizing a quadratic function

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.
24. ### What is the range of the quadratic function f(x,y) = (xy-x^2, xy-y^2)?

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... 26. ### Graphing Quadratic Functions: Domain, Range, and Factoring Help 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... 27. ### Find the period of Cosine of Quadratic function 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... 28. ### First-order necessary conditions for a quadratic function 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... 29. ### Domain of a quadratic function under square root having no x intercept 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... 30. ### Quadratic Function Bounds for β: Solving for β in Terms of α, a, and b 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 31. ### Perfect Square (Quadratic function) 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... 32. ### Finding the inverse of a quadratic function? 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... 33. ### Reciprocal of a quadratic function - math help 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... 34. ### How to determin the equation of quadratic function? 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... 35. ### How to minimize a simple quadratic function of multiple variables ? 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... 36. ### Problem with fitting simple quadratic function to 3 data points 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... 37. ### Quadratic function profits 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... 38. ### Quadratic Function Word Problem 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...
40. ### Draw graph of quadratic function 87x - 700 - x^2

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?
41. ### Does there exists a quadratic function for every parabola?

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...
42. ### Discover the Intervals of a Quadratic Function | Find Solutions and Graph

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...
43. ### Why 9/4 in Quadratic Function SOLVED

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44. ### Finding a Quadratic Function to Satisfy Conditions

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45. ### Find the Derivative of a Quadratic Function with f(0)=-4 and Evaluate f'(0)

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...
46. ### Quadratic function - x-intercepts?

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?