# What is Quadratic functions: Definition and 39 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. ### MHB Derivatives with Quadratic Functions.

Slightly confused at what it wants me to do here?

So, I found these statements and I need your assistance to prove them since my body condition is not fit enough to think that much. 1. The quadratic equation whose roots are k less than the roots of ax^2+bx+c=0 is a(x+k)^2+b(x+k)+c=0. 2. The quadratic equation whose roots are k more than the...
3. ### MHB Quadratic functions: Which of these quadratic functions has exactly one x -intercept

5. Which of these quadratic functions has exactly one x -intercept? o A. y=x 2 −9 o B. y=x 2 −6x+9 o C. y=x 2 −5x+6 o D. y=x 2 +x−6 A 2. What are the x-intercepts of y=(x−2)(x+5) ? o A. (0, 2) and (0, -5) o B. (0, -2) and (0, 5) o C. (-2, 0) and (5, 0) o D. (2, 0) and (-5, 0) D 5. Which of...
4. ### MHB Intersection points of two quadratic functions

Justify the following by using table, graph and equation. use words to explain each representation f(X) = 2 x2 - 8x and g(x) = x2-3x+ 6 the points (-1,10) and (6,24)
5. ### Quadratic function solution method optimizing

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...
6. ### 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...

1. Determine the equation that represents the relationship between the power and the current when the electric potential difference is 24v and the resistance is 1.5 Ω. 2. Draw a graph of the parabola that corresponds to the equation found in (a). 3. Determine the current needed in order for...
8. ### Application of Quadratic Functions that involve finding equation

Homework Statement Homework Equations none The Attempt at a Solution Is this correct? Thanks.
9. ### MHB Can Absolute Values of Quadratic Functions Determine Their Discriminants?

Let f(x) and g(x) be quadratic functions such as the inequality \left| f(x) \right| \ge \left| g(x) \right| is hold for all real x . Prove that \left| \Delta_f \right| \ge \left| \Delta_g \right|. For quadratic function f(x)=ax^2+bx+c , then \Delta=b^2-4ac. I have no idea how I could...
10. ### 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)...
11. ### Application of quadratic functions to volleyball

hi! i don't quite know how to start solving for this. i understand the problem and what it's asking for but i have no idea how to start solving for it. In a volleyball game, a player from one team spikes the ball over the net when the ball is 10 feet above the court. The spike drives the ball...
12. ### Area problem, involving quadratic functions

Hi, I don't understand what this question is asking and I have idea how to do it.. any help is very much appreciated! I understand how to complete the square, parabolas and such and the concept of maximum and minimum, I just don't understand this question. A Cattle farmer wants to build a...
13. ### Extrema of Quadratic functions

Homework Statement Does every quadratic function have a relative extrema? Homework Equations Quadratic function: ax^2 + bx + c. Aka a polynomial. Polynomials are continuous through all real numbers. The Attempt at a Solution It seems as if all quadratic functions would have...
14. ### MHB Supeerstar's questions at Yahoo Answers regarding optimizing quadratic functions

Here are the questions: I have posted a link there to this topic so the OP can see my work.
15. ### MHB Difference Quotient for Linear and Quadratic Functions

Here are the questions: I have posted a link there to this topic so the OP may see my work.
16. ### A Problem on quadratic functions

Homework Statement a) Let f(x) = ax^2+ bx + c; ∀x ∈ ℝ; where a(≠ 0), b and c ∈ ℝ are constants, (i) Prove that f(k) =(64+(b^2−4ac)^2)/(64a) ; where k = −(b/2a)+(1/a)+(b^2−4ac)/(8a). (ii) Hence, without using graphs and without using your knowledge on quadratic functions and equations, prove...
17. ### Quadratic Functions Word Problem

Homework Statement Your budget for constructing a rectangular enclosure which consists of a high surrounding fence and a lower inside fence that divides the enclosure in half is 2400 dollars. The high fence costs 8 dollars per foot and the low fence costs 4 dollars per foot. Find the...
18. ### Quadratic functions: [Diagram Included] Football game scenario

Homework Statement You are the quarterback for the Quinte Saints Football team. You are in the middle of the COSSA gold medal game and you see your receiver is wide open down the field beside the sideline. If he catches the ball, you win the game. However, the biggest guy Joey from the...
19. ### Quadratic functions: [Diagram Included] Football game scenario

Homework Statement You are the quarterback for the Quinte Saints Football team. You are in the middle of the COSSA gold medal game and you see your receiver is wide open down the field beside the sideline. If he catches the ball, you win the game. However, the biggest guy Joey from the...
20. ### Non-linear function - Cubic and Quadratic functions

1. i am stuck 2. The profit function is given by P=-2x^2 + 60x - 250, where x units is the quantity and P the total profit. Find the maximum profit and the number of units where maximum profit occurs. Sketch the curve of the profit function. 3. i can't do anything, my brain just jammed...
21. ### Ranges of simple quadratic functions

Taking a general function with two quadratic eqs, Y = (ax^2+bx+c / px^2+qx+r) ax+bx+c = pyx^2+qxy+ry x^2(a-py) + x(b-qy)+(c-ry)=0 The discriminant turns out to be a+ve or a 0. So, (b-qy)^2 - 4(a-py)(c-ry) >or= 0 Now how does this equation yield a range for y?or can it?what are the next steps?
22. ### MHB Arent quadratic functions supposed to be symmetrical?

As we were talking about earlier. f(x) = x^2 + 3x + 6 f ' (x) = 2x + 3 P1 = (0, f(0)) f ' (0) = 3. This is not symetrical... I hope you guys enjoy my question bombarding. As there are a lot of maths concepts that make me want to cry. It's funny, half the people in my degree fail their...
23. ### Problem solving with quadratic functions.

Homework Statement okay I know understand that I should get a reasonable understanding about the questions I post before I post them sorry about before. An Illinois farmer will plant from 800 to 2000 acres of soybeans. The number of acres q that he will plant depends on the selling price...
24. ### Problem solving with quadratic functions.

Homework Statement Eastern Ceramics can sell up to 200 of its flower pots per day in accordance with the demand function. p=13 -.04q write revenue as a function of the quantity sold q. find the output level q that maximizes R and the selling price at this output level.Homework Equations The...
25. ### Quadratic functions skipped in pre-calculus?

Hello, I have a quick question regarding Calculus and what should be covered in pre-cal. I'm currently in a pre-cal course and according to the syllabus it seems the teacher is skipping over quadratic functions. I only noticed because for the most part her course just follows along with the...

Homework Statement Sketch the graph of y=x^2 and graph y=1/2(x+4)^2 -5 Homework Equations The Attempt at a Solution So, This is the method I use for vertical compression/expansion Step pattern of a parabola at y=x^2 is 1,3,5 Start at vertex, go out by one, go up by 1...
27. ### Calculators TI-89 Titanium Help (graphing quadratic functions)

Is there a way to input values within a certain range into my calculator? For example, if I come across something like this: "Draw a curve of y=2x between the values x=+3 and x=-3. From the curve write down the following values: etc, etc, etc" How can I make it graphic w/in a certain range?
28. ### Inflection points theorem for quadratic functions.

Hey all, I was bored as heck with my internet down and started scribbling some random math stuff. I'm sure what I've come up with is old news to most but I found some interesting properties in the following proof and I'm curious what my fellow physicsforums users can come up with to see what...
29. ### Quadratic Functions Word Problem, It's Tricky

Homework Statement Ok, the problem is you have a 30cm wire, you cut it in two and you make one into a square. The second part is a rectangle with a 2:1 side ratio. I need to find the perimeter of each of the two shapes if the outcome sum of areas were to be a minimum. The Attempt at a...
30. ### I'v quistion on Quadratic Functions

Homework Statement Let f(x)= 125x-6x2 find the maximum value of f to four decimal places graphically Homework Equations (x)= 125x-6x2 The Attempt at a Solution Homework Statement attempt solution: x=-b+\sqrt{}b2-4ac/2a x=-125+\sqrt{}1252-4(-6)(0)/2a x= 0/-12=0...
31. ### Word problems : quadratic functions

[b]1. A farmer grows orange trees. He has 20 trees/hectare with 300 oranges/tree. When adding another tree to a hectare, the amount of oranges decreases by ten. What is the max yield of trees. (They want to know how many trees they should have in a hectare to maximize their orange production.)...
32. ### Help Identifying x-intercepts of quadratic functions

Homework Statement Identify the vertex and x-intercept(s) of the quadratic function algebraically. Homework Equations f(x) = 1/2x^2 - 4 1/2 Is the coefficent if its not clear. The Attempt at a Solution Ok I do not need help finding the vertex I have already got that, I...
33. ### Quadratic Functions, and Completing the Square

I had a hard time completing the square of some problems. This is what I did with them, I would appreciate if someone could show me the mistake that I am doing and the right process in order to solve these problems. Also, here there is an article about completing the square for anyone that...
34. ### Help on tis quadratic functions question

[SOLVED] Help on tis quadratic functions question Homework Statement The speed v m/s of a particle traveling from A to B at time t s after leaving A, is given by v =10t-t². the particle starts from rest at A and comes to rest at B. Show that the particle has a speed of 5m/s or greater for...
35. ### When will the stone reach its highest elevation and hit the ground?

Homework Statement Suppose that an object is thrown into the air with an initial upward velocity of Vo meters per second from a height of ho meters above the ground. Homework Equations Then, t seconds later, its height (h(t) meters above the ground is modeled by the function h(t) = -4.9t^2...
36. ### What is the formula for the family of quadratic functions passing through (3,0)?

A family of quadratic functions passes through the points (3,0). Find the family of quadratic functions err i have no idea hwo to do it except substituting those values in ... 0=9a+3b+c what does it meant the family of quadratic functions?
37. ### Solve Quadratic Function: Find x-Intercepts & Just Touches x-Axis

1. A quadratic fraction is of the form f(x)= 2x^2+bx+5 Find the values for b where the graph of f(x): 1.just touches the x-axis 2. has two x-intercepts 3.does not cut or touch the x-axis I've tried using the quadratic formula but i have no idea how to do it when there are two unknowns...
38. ### Family of quadratic functions?

This question has been killing me for days, : Give a family of quadratic functions of the form y=ax^2+bx+c, that passes through the following points: (1,1) and (2,0) I see how we can find the family, but how do we find the specific functions that pass through both those points? I tried making...
39. ### Quadratic Function Q&A: Working w/ Numbers Below 10

2 questions 1.can a quadratic function be just set to 1 number e.g. 5 or can it be a bigger number. at the moment I am working on numbers below 10 to get used to the formulas. 2. is this correct how i am doing it? formula N --> N2--> 3N2 + 4 e.g. N =5 5 x 5 = 25 = N2...