**Major Project- Finding Math in Badminton**

**The Picture...**

This is a Badminton racket and birdie.

When looking at this photo you can see that the shape of the racket is an oval, which is also called an Ellipse. There are wires going across the racket and on those wires you can see shapes. The sides of the racket are also symmetrical.

Symmetry

**Graphing**

Parabola

What Does The Formula Mean?

For a better understanding, in the formula, "h" and "k" will be the coordinates for the highest or lowest point on the parabola, the vertex. "y" and "x" will represent the coordinates of a single point on the parabola. "a" represents several things: if it's positive the parabola will open upwards, if it's negative the parabola will open downwards, if it's greater than 1 the parabola will narrow and if it's less than 1 it will widen.

Symmetry is when one half of an object is the mirror image of the other half. In relation to this, the X and Y axis placed on the picture shows that the four quadrants are symmetrical according to the racket head.

Graphing is used to show where on a coordinate plane objects are placed. In the picture the Triangles placement can be shown through graphing.

A

B

C

M

N

O

On triangle ABC the coordinate points are

A (3.1 units, 5.5 units)

B (0 units, 3.5 units)

C (3.1 units, 1.9 units)

On triangle MNO the coordinate points are:

M (3.1 units, 1 units)

N ( 0 units, -1 units)

0 (3.1 units, -2.5 units)

In between each coordinate there

is line segment .

What is Parabola?

A Parabola is a graph. The graph is of a quadratic function. A quadratic function contains a squared variable. In this example the function used to determine the equation of the parabola is the vertex form. The vertex form is

If we use the racket to show a parabola the Vertex will be (0,-9), and the other point will be (5,-8).

To determine the equation of the Parabola

First we must put the coordinates of the vertex into the formula.

Then you put the other point's coordinates into the formula to solve for "a"

Next you simplify

and isolate "a"

We plug this information into the original formula:

Finding the Slope and equation of a Tangent Line

What is a tangent line?

A tangent line is a line that touches a curve at a specific point but does not cross over the curve. In this example, it is shown as the green line. Using the same data from the equation of the parabola, we can calculate the slope and equation of the tangent.

The first step is to understand the point slope formula, which is y-y1=m(x-x1). Since the tangent is at point (5,-8) we can say that these numbers are actually the variables "x1" and "y1." In the formula "m" is representing the slope of the tangent. We will be calculating this by taking the derivative of the parabola's equation. Now that we know about point slope formula we can plug in the point (5,-8).

What is a Derivative?

A derivative is a way to represent a rate of change, for example a slope. (it's the rise/run, It's changing!) In this case, we're taking the derivative of the equation of the parabola

Another way to write a function is to say f(x)= instead of y=

To say we're taking the derivative we write it as,

The first step to taking a derivative is to know that the derivative of any constant number is just zero

The next step is to look at the variable, all you do is bring down the exponent to the front of the variable and then subtract 1 from the exponent up top

When this is made to look simpler it looks like this

Because we have an x coordinate (5,-8) we can plug this into the derivative to get the slope of the tangent line

Now we can go back to our point slope formula and plug in this slope to determine the equation of the tangent line

Which is the equation for this parabola.

When you look at the value of "a" you can see that is it positive and less than 1. This matches the graph as the parabola opens up and widens.

Not dividing by 9 here,

you're adding 9 to each

side.