Math Reflection 2

1. The area of a rectangle can be illustrated through Distributive Property by breaking up the rectangle into smaller pieces. Say you have a rectangle that has a side of x units, plus another 3 units, and then another side length of x units plus 5 units. In factored form, the equation for getting the area of the rectangle would be (x+3)(x+5). That would mean you would multiply: x by 5, x by x, x by 3, and 3 by 5. You would have to distribute all of these terms to get the area of the rectangle, resulting in the expanded equation of x^2+8x+15.

2a. If an equation is in factored form, you can distribute to get it into expanded form. For example, if an equation were (x+4)(x+4), you would first distribute all the terms: x(x)+4x+4x+4(4). If you simplify that, then the expanded form would be x^2+8x+16.

2b. If an equation is in expanded form, you can do the opposite of distributing and factor to get a factored equation. You can take the above equation, x^2+8x+16, and factor. You need to find 2 numbers that add up to 8, but also multiply together to get 16. Those numbers would be 4 and 4. After finding those numbers, the equation can now be read as x^2+4x+4x+4(4). To factor this, split up the equations into sections. For this particular one, I'll just split it in half: (x^2+4x) and (4x+4(4)). You need to do the opposite of distributing and find out how a number can be multiplied to get what is already there. For the first section, x can be multiplied by x and 4 to get x(x+4). For the second section, 4 can be multiplied by x and 4 to get 4(x+4). To simplify this, the equation would now be factored into x(x+4)+4(x+4) and then further factored into (x+4)(x+4).

3. You can recognize a quadratic function from its equation if the expanded equation as a coefficient of x, raised by a power. In the factored form, you can recognize a quadratic function if the equation involves x being multiplied by x.

4. The shape of a graph of a quadratic equation is that of a parabola.
If the graph has an equation, you can find its y-intercept, x-intercepts, maximum/minimum point, and line of symmetry.
The y-intercept is the number that is the product of the two constants in the equation of the graph.
The x-intercepts are the opposites of those same two constants.
To find the maximum or minimum point , you find the average of the 2 x-intercepts and make that the x-coordinate. To find the remaining y-coordinate, you plug the x-coordinate you got into the equation and solve.
Finally, the line of symmetry of the graph is the x-coordinate of the maximum/minimum point.