Math Reflection pg 39

1) The area of a rectangle can illustrate distributive property. In expanded form, the equation of a rectangle is simply the added areas of the smaller rectangles that make up the main rectangle. For example, one of the rectangles we examined had an equation A=x^2+10x+25. The same rectangle had a factored equation that looked like this: A=(x+5)(x+5). These equations show that the same rectangle had both A=(x+5)(x+5) and A=x^2+10x+25 for equations. These are equal because the equation A=(x+5)(x+5) distributed is A=x^2+10x+25. This shows what distribution is, simplifying one equation by distributing terms.

2)a.) When an equation is in factored form you can easily find out it's expanded form. You simply distribute. For example: x(x+5)=x^2+5x because by distributing x you can find x*x=x^2 and x*5=5x.

2)b.) You can find an equations factored form by looking at it's expanded form. What you have to do is find a way to split the equation into two mini-equations in parentheses. to do this you split the "X's" into two parts. When split, the X's need to add up to their original value, and multiply to the y-intercept or the number being added to the X's. here's an example:
x^2+13x+12 this equation need to be split into factored form
the 13 x needs to be split
If you split the 13x into 12x and 1x, you have a sum of 13x and when you multiply 12 and 1 you get 12
So you do x(x+1)+12(x+1) or (x+12)(x+1)
that is equal to the original equation x^2+13x+12.

3) You can recognize a quadratic function by its equation because it has a degree no larger than a two.
4) You can tell the y-intercept, xintercept, line of symmetry and minimum or maximum point by looking at a quadratic function's equations. You can find the y-intercept when the equation is in expanded form. It is the number being added or subtracted from the equation. In the equation y=x^2-25 , the -25 is the y-intercept. In the equation y+2x^2+43, 43 is the y-intercept. You can find the x-intercept when the quadratic expression is in factored form. It is the number you add to the summed parentheses to make it equal 0. Often there are two x-intercepts because the parabola of a quadratic function often intercepts the x-axis twice. Here's an examples of finding x-intercepts.
y=(x+5) (x-4) The x-ints are -5 and 4 because you add 5 to -5 to get 0 and 4 to -4 to get 0. It could also be seen as the opposite of whatever number is being added/subtracted from the x.
To find the line of symmetry you average the two x-intercepts. For example, above the x-ints are -5 and 4. -5+4= -1. -1÷2= -1/2. The line of symmetry is -1/2. Then to find the minimum point you plug -1/2 as x into the equation because the line of symmetry's equation is x=-1/2. So to find the minimum point you do y=(-1/2+5) (-1/2-4)=-20.25. -20.25 is the minimum point.

by Kate :-)