Karla Velazquez A01196893

*What is a QE?

An equation where the highest exponent of the variable (usually "x") is a square (2).

It is usually written ax²+bx+c = 0

Example: 2x²+5x-3 = 0

*What is the Vertex?

The vertex in a quadratic expression is the point where the function begins in other way the minimum or maximum of a quadratic function.

*What is the axis of symmetry?

The line that runs down its 'center'. This line divides the graph into two perfect halves.

*What is concavity?

-The concavity of functions may be determined using the sign of the second derivative. For a quadratic function f of the form 

(+) In a quadratic expression when the x is positive the concavity will be pointing upward

(-)In a quadratic expression when the x is negative the concavity will be pointing downward

*2 examples

*What are the roots?

Where a function equals zero.
In this example, −2 and 2 are the roots of the function x² − 4
But sometimes "root" is used as a quick way of saying "square root", for example "root 2" means √2

*How do you find the x values?

-Factorizing: found 2 numbers that multiply between themselves gives you C and if you add or subtract them they give you B. when you have this you make two parenthesis with this two numbers together with the X for example (x+5}{x-&7)=8

-quadratic formula

To use the quadratic formula first you have to have put in the formula x^2+bx+cx=d and when you have the you can use the general formula

*no se pudo dar copy/paste

-completing formula

2 peak in the curve, it will be pointing downwards or upwards depending on the

sign of the x² term.

when the concavity is upward it means that it is pointing up

when the concavity is downward it means that the concavity is pointing down.

5 A real number x will be a solution or a root if it can satisfy the quadratic equation

6 Factorizing: When you factorize, you have to put the variables in one side and numbers in the other. Quadratic formula: you are finding the x-intercepts of the graph. You have to solve for =0 and then factorize. Completing the square: First, remember that finding the x-intercepts means setting y equal to zero and solving for the x-values.

Bibliography

 Factoring. (n.d.). Retrieved March 2, 2015, from http://www.mathsisfun.com/definitions/factoring.html

Math Homework Help - Answers to Math Problems - Hotmath. (n.d.). Retrieved March 2, 2015, from http://hotmath.com/

Solutions or Roots of Quadratic Equations. (n.d.). Retrieved March 2, 2015, from http://www.sosmath.com/algebra/quadraticeq/root/root.html

Vertex - math word definition - Math Open Reference. (n.d.). Retrieved March 2, 2015, from http://www.mathopenref.com/vertex.html

Leithold, L.(2001).Ecuaciones cuadradicas en una variable. Algebra y Trigonometria.Mexico.

A funtion is a rule of correspondence that associates with each object "x" in one set, a single value F(x) from a second set.

2. A vertix is the possition in which a curve starts in a graph.

3. Is the exact middle point in a graph between the "x" or "y" axis.

4. The state of being concave or inward like the inside of a sphere.

a) when is upward (+) if a point of the tangent line of a graph is below the vicinity of the point. It looks like a right-side up curve.

b) when is downward (-) if a point of the tangent line of a graph is above the vicinity of the point. It looks like an upside-down curve. 

5. A root is any value elevated in an equation. And the "x" value is any missing value in an equation.

6. There are three way to find the "x value: by factorizing, using the quadratic formula, or completing the square.

References:

Varberg, D. and Purcell, E. J. (1992). CALCULUS With Analytic Geometry. Sixth Edition: New Jersey, US.

Desmos. Graphic Calculator. Retrieved from https://www.desmos.com/calculator

Math is Fun. (2013). Quadratic Equation. Retrieved from:

http://www.mathsisfun.com/definitions/quadratic-equation.html

Word Reference. (2015). Concavity. Word Reference. Retrieved from: http://www.wordreference.com/definition/concavity

Purplemath. (2014). Completing the Square: Finding the Vertex. Retrieved from: http://www.purplemath.com/modules/sqrvertx.htm

José Valdéz A01197032

1. its a function where the variable is squared (x^2), also called degree 2

2.a vertex is any point of intersection

3.the line that runs out the center of the parabola

4.concavity is the fact if the graph of a curve is concave or convex.

6. you can find

A quadratic function is a polynomial function in one or more variables in which the highest-degree term is of the second degree.

2- The vertex of a parabola is the place where it turns; hence, it is also called the turning point.

3- Axis of Symmetry: A line of symmetry for a graph. The two sides of a graph on either side of the axis of symmetry look like mirror images of each other.

4- Concavity is the state of being concave. Concave upward is when the slope increases, while concave downward is when the slope decreases.

• QE: is an equation of the form ax² + bx + c = 0, where a, b and c are real numbers and a ≢ 0  (494)

• FX: is a function that can be written in the form f (x) = ax² + bx + c for real numbers a, b, and c, with a ≢ 0 (531)  

• 2. What is a vertex?

• When a Parabola opens upward, the vertex is the lowest point on the curve. When the Parabola opens downward, the vertex is the highest point on the curve (531)

• 3. What is the Axis of Symmetry?

• A line of symmetry for a graph. The two sides of a graph on either side of the axis of symmetry look like mirror images of each other.

• 4.What is Concavity?

• A shape or solid which has an indentation or "cave". Formally, a geometric figure is concave if there is at least one line segment connecting interiorpoints which passes outside of the figure.

1. When is Upward (+)

• A graphor part of a graph which looks like a right-side up bowl or part of an right-side up bowl.

1. When is Downward (-)

• A graph or part of a graph which looks like an upside-down bowl or part of an upside-down bowl.

5. A solution to the quadratic equation. A quadratic equation has always two roots. Because a quadratic can always be factored as (x − a)(x − b), and a, b are the two roots.

6. To find the roots you can factor the equation.

REFERENCES

References:

Runde, A. (2011). Quadratic Functions. In Intermediate Algebra For Collage Students (8th ed., p. 703). Pearson Education.

7.

A: QE, the equation of which the roots are given by the “Quadratic formula”. Fx, is the relationship between an input and an output.

2. What is vertex?

A: Vertex typically means a corner or a point where lines meet. For example a square has four corners, each is called a vertex.

3. What is the axis of symmetry?

A: This is a line about which a shape is symmetrical. That is, the shape on one side of the line is a mirror image of the shape on the other.

4. What is concavity? - Opens upward if a>0 - or downward if a<0

5. What are the roots/“x” values? A real number “x” will be called a solution (or a root) if it satisfies the equation: ax^2+bx+c=0

6. How do you find the roots/“x” values? -By factorizing: finding what to multiply to get the quadratic -By quadratic formula: x= -b±√b^2-

4ac/2a -By completing the square: where the standard form equation is turned into a(x+d)^2+e=0

7. Examples (other posts)

8. APA References

Math Is Fun. (2014). Quadratic equations. Math Is Fun. Retrieved from:http://www.mathsisfun.com

Chudov, I. (n.d.). Lesson introduction into quadratic equations. Algebra. Retrieved from:http://www.algebra.com

Education, P. (2015). What is the vertex of a quadratic function. Virtual Nerd. Retrieved from: http://www.virtualnerd.com

Stapel, E. (2015). Graphing Quadratic Functions. Purplemath. Retrieved from:http://www.purplemath.com

Marks, E. (2004). Functions, Modeling, Change. The Family of Quadratic Formulas p. 213

A quadratic equation is any equation having the form where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0. If a = 0, then the equation is linear, nor quadratic.  An equation where the highest exponent of the variable (usually "x") is a square (2).

The minimum or maximum point of a quadratic function.

What is the axis of symmetry?

Symmetry with respect to the x-axis means that if the Cartesian plane were folded along the x-axis, the portion of the graph above the x-axis would coincide with the portion below the x-axis. Symmetry with respect to the y-axis or the origin can be described in a similar manner.

WHAT IS CONCAVITY?

·When is upward (+)

A graph is said to be concave up when the vertex is at the bottom of the function.

·When is downward (-)

A graph is to said to be concave downwhen the vertex is at the top of the function.

What are the roots/”x” values?

A real number x will be called a solution or a root if it satisfies the equation, meaning. It is easy to see that the roots are exactly the x-intercepts of the quadratic function , that is the intersection between the graph of the quadratic function with the x-axis.

How do you find the “x” values?

You can find it with the formula

2. The vertex/point of inflection/node is the point of the curve (parabola) at which the sign of the curvature changes.

3. The axis of symmetry is the axis where the portion above/beside the x/y axis coincides with the portion below/beside the x/y axis when the Cartesian plane is folded.

4. Concavity is the direction of the graph's curvature.

a. concave up: a>0

b. concave down: a<0

5. Examples:

a. (red) 2x^2-4x-2 (concave up)

b. (blue) -8x^2-2x+8 (concave down)

6. The roots/x values are the solutions (real or complex) for the equation, the points where the graph intersects the x axis.

7. In order to find the roots/x values you can use:

a. Quadratic formula

b. Factorisation

c. Square completion

References:

• P. Garrett (n.d.) Inflection points, concavity upward and downward. Math Insight. Retrieved from: http://mathinsight.org/inflection_points_concavity_upward_downward_refresher
• Larson, R. (n.d.) Algebra and Trigonometry. Symmetry p. 80

• Weisstein, E. (n.d.) Quadratic Equation. Mathworld, Wolfram. Retrieved from: http://mathworld.wolfram.com/QuadraticEquation.html

Where A, B and C are constant real numbers and A isn’t 0, it’s called polynomial equation of second degree or quadratic equation of the X variable.

2.

Is where the axis of symmetry is born / Parabolas have a highest or a lowest point, called the vertex.

3.

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

The term refers “concavity” and “inflection point” as the directionality of the curve.

-Concave up (convex):  the curve is bent upward

-Concave down: the curve is bent downward

5.

Is any real number that intercepts in the x axis with the quadratic function.

Factoring: The process of breaking any equation into separate terms

Quadratic fórmulas: (image attached )

 Completing the square : Is when we turn an equation like this: ax^2 + bx + c = 0 to this:  a(x+d)^2 + e = 0

8. References:

Wikipedia . (Modified 12 February 2015). Quadratic Equations . 2/03/15, de Wikipedia Project Sitio web: http://en.wikipedia.org/wiki/Quadratic_equationVirtual Nerd. (n.d). How do you find the axis of symmetry for a quadratic function?. 2/03/15, de Pearson Sitio web: http://www.virtualnerd.com/algebra-1/quadratic-equations-functions/graphing/graph-basics/axis-symmetry-exampl

Pierce, Rod. (17 Sep 2014). "Real World Examples of Quadratic Equations". Math Is Fun. Retrieved 2 Mar 2015 from http://www.mathsisfun.com/algebra/quadratic-equation-real-world.html

Book Reference:

Louis Leithold. (1994). 2.3 Ecuaciones Cuadraticas en un variable . En Álgebra y Trigonometría con Geometría Analítica(899). México: Oxford University Press Harla Mexico, 1994.

2. A vertex is the point where the parabola "turns around", it is the peak of the curve , which will be pointing upwards or downwards.

3. The axis of symmetry is the line that passes in the right center of the parabola that cuts it in a half. The vertex lies on the axis of symmetry.

4. Concavity is the "direction of bending" of a graph. And it can be either concave up or concave down.

-Concave up: The curve of the graph bends upwards. In the equation, the coefficient of the quadratic equations is positive. y=x^2

-Concave down: The curve of the graph bends downwards. In the equation, the "flipped" version will result in having the coefficient of the quadratic equation negative. y=-x^2

5. A root is a value for which a given function equals zero. When the function is plotted on a graph, the roots are points where the function crosses the x-axis.

6. Ways of finding the roots:

-Quadratic formula (attached at the end).

-Completing the square: make the needed operations so we can factor and square-root both sides of the equation.

-Factoring: We need to have the form ax^2+bx+c=0, then we factor the left side, set each factor equal to zero and finally we solve them.

7. Examples in other post.

8. Bibliography:

• Karush, W. (1989). Webster's New World: Dictionary of Mathematics. New York: Prentice Hall Press
• Math Insight. (n.d.). Inflection points, concavity upwards and downwards. Math Insight. Retrieved from: http://mathinsight.org
• Math Open Reference. (2009). Vertex. Math Open Reference. Retrieved from: http://www.mathopenref.com
• Free Math Help. (2013). Finding Roots. Free Math Help. Retrieved from: http://www.freemathhelp.com

2. What is the vertex? In an equation, is the minimum or maximum point of the equation.

3.What is the axis of symmetry? The axis passes through and divides the parabola in two congruent halves.

4. What is concavity? - Opens upward if a>0 - or downward if a<0

5. What are the roots/“x” values? A real number “x” will be called a solution (or a root) if it satisfies the equation: ax^2+bx+c=0

6. How do you find the roots/“x” values? -By factorizing: finding what to multiply to get the quadratic -By quadratic formula: x= -b±√b^2-4ac/2a -By completing the square: where the standard form equation is turned into a(x+d)^2+e=0

7. Examples (other posts)

8. APA References

Math Is Fun. (2014). Quadratic equations. Math Is Fun. Retrieved from: http://www.mathsisfun.com

Chudov, I. (n.d.). Lesson introduction into quadratic equations. Algebra. Retrieved from: http://www.algebra.com

Education, P. (2015). What is the vertex of a quadratic function. Virtual Nerd. Retrieved from: http://www.virtualnerd.com

Stapel, E. (2015). Graphing Quadratic Functions. Purplemath. Retrieved from: http://www.purplemath.com

Marks, E. (2004). Functions, Modeling, Change. The Family of Quadratic Formulas p. 213

-Concave Up: x^2+2x+2

-Concave Down: -x^+2x+2