Mathematical Reflection 2

1. The area of a rectangle can illustrate the Distributive Property. For instance, if the lengths of the rectangle were (x+2) by (x+3), the area of the rectangle would become (x+2)(x+3). By drawing it out as a rectangle, we can break up each term separately and distribute, or multiply, them to the others.
2. a) When a quadratic expression is in its factored form, we can use the distributive property to find an equivalent expression in expanded form. If we have an expression, let's suppose (x+2)(x+3), we can distribute each term in the first set of parenthesis into the second, and come up with x(x)+2x+3x+6, or x^2+5x+6.
2. b) To find the equivalent factored form of a quadratic expression written in expanded form, we factor, or the opposite of distributing. Here's an example: The expanded form is x^2+5x+6. Often times we need to break up the second term, which in this case is 5x. We need to find 2 integers that add up to 5, and multiply up to 6. The result is 2 and 3. This expression can now be written as x^2+2x+3x+6. Next, break the expression into two groups, (x^2+2x) and (3x+6). Doing the opposite of distributing, we get x(x+2)+3(x+2). This can be further simplified to (x+2)(x+3), the factored form.
3. A quadratic expression, in its expanded form, follows the form y=ax^2+bx+c.
4. From the equation (x+2)(x+3), we can predict many things. The x-intercepts are the opposite of the integers, or -2 and -3. The y-intercept is those multiplied together, or 6. We can tell if the parabola is pointing up or down by looking at if ax^2 is positive or negative. In this case, it is positive, so it is pointing upwards. The line of symmetry is the x value between the x-intercepts.

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