Quadratic+Functions

= << ²Quadratic Functions² >>  = Basic shape of a quadratic function is f(x) = 1/1*(x – 0)^2 + 0 or also known is the basics as f(x)=x^2 is shown below: The basic shape of the f(x)=x^2 can be describes as a parabolic figure staring in a vertex (0,0) and from the right of the vertex meaning positive side of x the function in increasing and from the left of the vertex meaning it the negative side of x the function is decreasing. Also expressed y = a(b(x – h))2 + k when a>0 the parabola open upward when a<0 parabola opens downwards. Also when | a | increases, the parabola becomes narrower; and when | a | decreases, the parabola becomes wider.
 * 1) **// BASIC SHAPE OF THE FUNCTION //**


 * // 2. BASIC EQUATION //**

The basic equation of a quadratic function is f(x)=x^2 but in a more complex way and a way to work with it and tranformation is f(x) = 1/1*(x – 0)^2 + 0 or y = a(b(x – h))2 + k.

//** 3. DOMAIN & RANGE **// The //**domain**// of the function f(x)=x^2 is -∞ < x<∞ The //**range**// of the function f(x)=x^2 is 0<y

//** 4. DESCRIBE ANY POSSIBLE APPLICATIONS OF THE FUNCTION TO REAL WORLD SITUATIONS. **// Quadratics in real life situations:

Any object that has a parabolic traveling shape uses quadratic equations in order to know many important factors of the object like altitude, distance traveled etc. Using the equation, engineers can construct better the different objects that use parabolic shapes in their traveling like

Airplanes may use some quadratics even though their traveling shape isn't fully a parabolic shape as the airplane goes up then down up down and so on.

Mortars with artillery pieces are probably the objects that most use quadratics. A mortar projectile is a perfect example of a parabola, as

Artillery pieces are also objects that are dependent in quadratics as the equations determines its altitude, its traveling distance and its maximum distances as its maximum altitude.

//** 5. HOW TRANSFORMATIONS (shifts, stretches, and reflections) AFFECT YOUR FUNCTION? **// Transformations: Function y = a(b(x – h))2 + k Changing the h in the equation y = a(b(x – h)) ^2 + k affects the function by it creates a movement along the x axis, then there are negative numbers for X the function moves to right meaning to the positive side of the x axis. This movement that I describe means that from the vertex the whole function move along the x axis and the basic shape doesn’t change.

Changing the h in the equation y = a(b(x – h)) ^2 + k changing b affects the function by it stretches the function from the vertex in the x axis direction but it keep being positive if the value of b is positive, if value of b is negative then the parabola would be decreasing to the negative value of y and once b becomes further away from 0 and is negative its again stretches in the x axis directions. So when b equal -1 it’s the reflection of original parabola along the x axis. Your an tell the difference one is a stretch other is compression.

Faltan estas dos y lo de mas de tranformations

//**6. GRAPH OF THE INVERSE FUNCTION(an equation, if possible, and including any necessary restrictions)**//

If we start with the basic quadratic functionas mentioned before; **f(x)= x ² ** then the inverse of this function is very straight forward. ** ƒ −1 (x) = √x ** and if we were to graph this inverse, the result would be the blue parabola shown in the following picture.



As we can see, if we would do the vertical line test to the inverse of the result would be that the inverse is not a function because for example, for a given value of x, there can be more than 1 value for y as shown in the graph if x=4, then y=8 and y= -8. Since the inverse of x squared is not a function, then it must have some restrictions in order to be a function, If we __**restrict**__ the domain of the function **f(x)= x ² ** to be x ³ 0, this restriction would let the inverse of the original quadratic function, to be a function, and it would look like this:

//**7. INCLUDE ANY KEY FEATURES (intercepts, asymptotes, vertices, etc.)OF THE GRAPH.**//

The inverse of a quadratic function does not have any asymptotes, but we can find the intercepts of this inverse by giving ** ƒ −1 (x) = 0, **then the x-intercept would be 0 and 0. The same for y-intercepts, if we give ** ƒ −1 (0) = √x, ** then we obtain ± **√0 ****. **


 * Th **e Vertex in the other hand, for ƒ −1 (x) = √x is the same as the one of f(x)= x ² but inverted. For example, if the vertex of the function **f(x)= 2x ² + 1 is (1,3) ( ** hypothetically), then the vertex of the inverse would be ** (3,1) **

//**<span style="font-family: arial,helvetica,sans-serif; font-size: 13px; font-weight: normal; line-height: 19px;"> LA Bibliography **// //picture shape of quadratic function// //Quadratic Function. Digital image. Wikipedia. Web. 24 Mar. 2011. [].//

//"Quadratic Functions." Oklahoma City Community College. Web. 28 Mar. 2011. [functions.htm].//

//"Function Flyer." Interactivate. Web. 27 Mar. 2011. <http://www.shodor.org/interactivate/activities/FunctionFlyer/>.//

//Picture of Inverse function by Roberto Lopez, 2011 All rights reserved.//

Back to Wiki Assignment