## Thursday, July 21, 2011

### The Problem of the Week Day 4: Week of 7/17 - 7/23

We have completed our sequences! For the easy one, we will just be using n and b, and the hard one only requires your equation. For the hard problem, the equation should be quadratic, or you made a mistake. If it is linear, go over Tuesday's work and find your mistake. Then, catch up so you can do today's work.

Easy Problem: To find the perimeter of a polygon, add up its sides. If you have a rectangle with sides 5, 5, 13, and 13, add them all up to get a perimeter of 36. On a rectangle, you can determine the perimeter with the formula P = 2b + 2h with b being the base and h being the height of the rectangle.

If you have a rectangle with n as the base and b as the height, what is the perimeter of the rectangle?

p = ___

Hard Problem: Quadratic equations have two forms they can be written in. The one we used is called "standard form," with the equation in the form ax^2 + bx + c. The other form is called "vertex form," being in the form a(x - h) + k. This is called vertex form because the vertex, or turning point, of the parabola (the graph of a quadratic equation, looking somewhat like the letter U) is (h, k).

To go from standard form to vertex form, you do something called "completing the square." Say you had the equation y = 2x^2 + 4x - 6. First, you factor your a term out of the equation to get y = 2(x^2 + 2x - 3). Then, you complete the term by dividing your x coefficient (not x^2) by two and squaring it. 2/2 = 1 which squared is 1. That means that x^2 + 2x + 1 is a perfect square trinomial. Therefore, we need to add one to the x^2 + 2x, which means we also have to subtract one to even it out. This gives us y = 2(x^2 + 2x + 1 - 1 - 3). Now, we make the x^2 + 2x + 1 a square, with the h term being the square root of the number you added and subtracted. This gives us y = 2((x + 1)^2) - 1 - 3). Then, we combine the -1 and -3 to get y = 2((x + 1)^2 - 4). By distributing the 2 over to the -4, we get our equation to y = 2(x + 1)^2 - 8.

Tip: Vertex form is y = a(x - h) + k. It is NOT x + h.

1) Put your equation from yesterday into vertex form. It will take less time to do.

2) Find the vertex of the equation.

h = ___
k = ___