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Quadratic Functions/Camila de la Vega, Marcela Treviño, Eugenia Sanchez. 

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Quadratic Functions/Camila de la Vega, Marcela Treviño, Eugenia Sanchez. 

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1. Define a Quadratic Function: is a polynomial function in one or more variables in which the highest-degree term is of the second degree.

2.  What is the difference between a linear and a quadratic function: A linear function is represented by the equation y=mx+b and it is simply a straight line. You can use linear functions for finding things like distance, slope, midpoint, etc. Quadratic functions are represented by the equation ax^2+bx+c=0 and when graphed are called parabolas.

3.  x-intercepts: You can use factoring, or completing the square , or the quadratic formula

4.  Vertex: The Vertex of a parabola is the point at the bottom of the "U" shape (or the top, if the parabola opens downward), y=a(x–h)²+k,

5.  Axis of symmetry: The axis of symmetry of a parabola is the vertical line through the vertex. For a parabola in standard form,y=ax²+ bx + c,

6.  Concavity: ifa>0, thenfis concave upward everywhere, ifa<0, thenfis concave downward everywhere. , isconcaveif for any two points on this surface A(x1, y1) and B(x2, y2), the line segment AB is completely located beneath this surface or on this surface.

7.  Domain and Range: thedomainof a quadratic functionf(x) is the set ofx-values for which the function is defined, and therangeis the set of all the output values (values off).

8.  Examples:x intercept- x2+2x+1, vertex- 3x2 + x – 2, axis of sym- y = x2 – 4x + 2, domain- x2 – x – 2 = 0

9.  Examples:x intercept: set y=0 0 = x2 − 4 x2 = 4 x = 2 or −2 The points are (2,0) and (−2,0) y intercept: set x=0 y = 02 − 4 y = −4 The point is (0,-4)

(n.d.). Retrieved June 22, 2015, from http://www.rsu.edu/faculty/vkyrylov/JavaApplets/OperationsResearch/QuadraticFunction/ConvexVsConcave.htm

(n.d.). Retrieved June 22, 2015, from http://www.rsu.edu/faculty/vkyrylov/JavaApplets/OperationsResearch/QuadraticFunction/ConvexVsConcave.htm

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Quadratic Function:

A quadratic equation is any second-degree polynomial equation — that’s when the highest power of x, or whatever other variable is used, is 2. The solution or solutions of a quadratic equation.

What is the difference between a linear and a quadratic function?

A linear function is represented by the equation y=mx+b and it is simply a straight line. Quadratic functions are represented by the equation ax^2+bx+c=0 and when graphed are called parabolas.

Elements of a Quadratic Function:

X-intercepts: The X intercepts are the point that the quadratic equation goes by the X axis.

y = 3(0)2 + (0) – 2

= 0 + 0 – 2 = –2

0 = 3x2 + x – 2

0 = (3x – 2)(x + 1)

3x – 2 = 0  or  x + 1 = 0

x =  2/3  or  x = – 1

Vertex:  The vertex of in a quadratic equation it’s the minimum or maximum point of the equation.

Axis of symmetry: The axis of symmetry is the vertical line that goes through the vertex of a quadratic equation.

Ex → y = ax2 + bx + c, where a ≠ 0. Find the axis of symmetry for y= x2 + 6x + 5.

1. Label a and b.

a= 1, because 1 is the coefficient of x2

b= 6, because 6 is the coefficient of x

2. Plug the numbers into the axis of symmetry formula.

x= -(6)

2(1)

3. Simplify.

2x= -6

x= -3

Concavity: A curve inwards.

-If the coefficient of a is positive, then the graph opens upward.

-If the coefficient of a is negative, then the graph opens downward.

Example 1: Determine the concavity of f(x) = x3 − 6 x2 −12 x + 2 and identify any points of inflection of f(x).

Because f(x) is a polynomial function, its domain is all real numbers.

Testing the intervals to the left and right of x = 2 for f″(x) = 6 x −12, you find that

hence, f is concave downward on (−∞,2) and concave upward on (2,+ ∞), and function has a point of inflection at (2,−38)

Domain and Range: Domain: The domain of a function is the set of all possible input values

Range: The range is the set of all possible output values

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Quadratic functions. Federico de Zamacona, Carlos Jasso, Luis Valencia

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Quadratic functions. Federico de Zamacona, Carlos Jasso, Luis Valencia

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A Quadratic Function is a polynomial function which can be described as f(x)= ax^2 + bx + c and a is not equal to zero.

Ex.1

-10^2-6x+6=0

Difference between Linear Equations and Quadratic Functions:

Linear equations are graphed in a lines and quadratic functions are graphed as parabolas.

The x-intercept is where the x-axis crosses.

Synonyms:

zero, solutions.

y = x^2-3x-10=0

(x+2)(x-5)

x=-2 or x=5

Vertex:in a parabola is the point at the top or bottom of it

y= a(x-h)^2+k

Axis of symmetry: the symmetry axis is a vertical line that crosses through the vertex line of a parabolla

ax^2+bx+c

Concavity: if a>0, thenf is concave upward everywhere, if a<0, thenfis concave downward everywhere. , isconcaveif for any two points on this surface A(x1, y1) and B(x2, y2), the line segment AB is completely located beneath this surface or on this surface.

Domain & Range: in a function the domain is all the posible input values. And the range all the output values.

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Define Quadratic Function: is a polynomial function in one or more variables in which the highest-degree term is of the second degree.

What is the difference between a linear and a quadratic function?

A linear function is represented by the equation y=mx+b and it is simply a straight line. Quadratic functions are represented by the equation ax^2+bx+c=0 and when graphed are called parabolas.

x-intercepts: You can use factoring, or completing the square , or the quadratic formula.

Vertex:  The vertex of in a quadratic equation it’s the minimum or maximum point of the equation.

Axis of symmetry: The axis of symmetry of a parabola is the vertical line through the vertex. For a parabola in standard form,y=ax2+ bx + c.

Concavity

Concavity: if a>0, thenf is concave upward everywhere, if a<0, thenfis concave downward everywhere. , isconcaveif for any two points on this surface A(x1, y1) and B(x2, y2), the line segment AB is completely located beneath this surface or on this surface.

Domain and Range:

the domain of a quadratic functionf(x) is the set ofx-values for which the function is defined, and therangeis the set of all the output values (values off).

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Quadratic Equations: are parabola, and the degree or the highest exponent in the equation is 2.

An example of a quadratic equations:

- 5x^2-3x+3=0

The linear equations is graph as a line and the quadratic function is square and it is graph as a parabola.

x-intercept

- Definition: are where the graph cross the x-axis.

- Synonyms- roots, solutions, zeroes.

-y= 0 for the x-intercepts so:

25x^2+4y^2=9

25x^2+4(0)^2=9

25x^2+0=9

x^2=9/25

x=(+/-)3/5

Vertex

-Definition: it's the minimum or maximum point of the function.

-Formula: -b/2a

5x^2-3x+3=0

3/2(5)=Vertex

Axis of symmetry

- Definition: is the vertical line that divides the parabola into 2 congruent halves.

- Formula: -b/2a

- How to find it: same as vertex but you draw a straight line in the vertex.

Concavity

- Definition: when the parabola is upward or downward

- How to distinguish: is upward when the leading coefficient is positive and it is downward when leading coefficient is negative.

Domain and Range

- Definition: the domain are all the "x" values and range are all the "y" values.

MathisFun. (2012). Quadratic Equations. 22-june-2015, Sitio web: https://www.mathsisfun.com/algebra/quadratic-equation.html

Stapel, Elizabeth. (2010). x- and y-Intercept. 22-June-2015, Sitio web: http://www.purplemath.com/modules/intrcept.htm

http://hotmath.com/hotmath_help/topics/axis-of-symmetry-of-a-parabola.html

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Quadratic Equations: are parabola, and the degree or the highest exponent in the equation is 2.

To solve a quadratic inequality, follow these steps:

Solve the inequality as though it were an equation.

The real solutions to the equation become boundary points for the solution to the inequality.

Make the boundary points solid circles if the original inequality includes equality; otherwise, make the boundary points open circles.

Select points from each of the regions created by the boundary points. Replace these “test points” in the original inequality.

If a test point satiA quadratic function is a function of the form $f(x) = ax^2 + bx + c$ . The graph of a quadratic function is a parabola.sfies the original inequality, then the region that contains that test point is part of the solution.

Represent the solution in graphic form and in solution set form.

The graph of $f(x) = x^2 - 4x + 5$ opens upward, because the coefficient of $x^2$ is $+1$ .

An inequality in which one side is a quadratic polynomial and the other side is zero.

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Quadratic Functions. Paola de la Garza/Mariana Juárez/Santiago Caballero

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Quadratic Functions. Paola de la Garza/Mariana Juárez/Santiago Caballero

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Quadratic Functions- is a polynomial function in one or more variables, in which the highest-degree term is of the second degree, a, b, and c are numbers and a is not equal to zero.

y=ax²+bx+c

2. The vertex is the lowest or highest point of a parabola.

3. The axis of symmetry is the vertical line that divides the parabola into two identical halves. It passes through the vertex.

4. Concavity can be determined when working with the form f(x)=ax²+bx+c .If a is positive, then f is positive and the graph of f is concave up. If a is negative, then f is negative and the graph of f is concave down.

5. Roots are the x-intercepts or zeros. In a function there can be one, two, or zero roots.

6. To find the roots you only have to set a value for x and then solve the equation: ax^2+bx+c=0

Example:

0 = 3x ² + x – 2
0 = (3x – 2)(x + 1)
3x – 2 = 0  or  x + 1 = 0
x = ²/₃  or x = – 1

Bibliography:

http://www.purplemath.com/modules/grphquad3.htm

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Quadratic Funcion Mario Villanueva, Zofia Garza, Mauricio Leal

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Quadratic Funcion Mario Villanueva, Zofia Garza, Mauricio Leal

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https://docs.google.com/presentation/d/1qdaJ_31pQa42bJCcaGMOJrtBx5VrR_QkjShA-VF4SkU/edit?usp=sharing