Math Reflection

1. The area of a rectangle can demonstrate the distributive property. If a square's length was originally x inches, and you added 3 inches to both the length and width, the equation for its area would be A = (x + 3) (x + 3), A being the area, because length x width = A. Currently, the equation is in factor form. If you want to convert it to expanded form, you use the distributive property like so:

A = (x + 3) (x + 3)
A = x (x) + x(3) + 3(x) + 3(3)
A = x^2 + 3x + 3x + 9
A = x^2 + 6x + 9

2.a. If a quadratic expression is in factored form, like y = (x + 3) (x + 2), you can find an equivalent expression in expanded form by using the distributive property:

y = (x + 3) (x + 2)
y = x(x) + x(2) + 3(x) + 3(2)
y = x^2 + 2x + 3x + 6
y = x^2 + 5x + 6

b. If a quadratic function is in expanded form, like y = x^2 + 5x + 6, you can find its equivalent expression in factor form by finding two numbers that, when you multiply them, equal the constant in the equation, and when you add them, equal the coefficient in the equation. The equivalent expression of y = x^2 + 5x + 6 is y = (x + 5) (x + 1), because then you need to substitute these numbers into the equation y = (x + n) (x + a), where n is one constant and a is the other. y = x^2 + 5x + 6 is equal to y = (x + 5) (x + 1), because 5 + 1 = 6 and 5 x 1 = 5.

3. An expression in factored form is quadratic if it has two linear factors with their variables raised to the first power. An expression in expanded form is quadratic if the variable's highest power is 2. x (x + 2) and x^2 + 2x are quadratic expressions.

4. The graph of a quadratic function always forms a parabola. It has a high or low point, 2 x intercepts, and a line of symmetry. Here is how you find each of these features from a quadratic function's equation:

x-intercepts - the opposites of the two constants in the factored form of the equation. If there is only one, then the other is 0. Before you find the opposites of the constants, you have to simplify the equation to y = (x + n) (x + a), where n and a are the constants.

y-intercept - the constant in the expanded form of the equation.

minimum/maximum point - to find the x-coordinate, find the number exactly between the x-intercepts, then substitute this number into the equation to find the y-coordinate.

equation of the line of symmetry - the number exactly between the x-intercepts is a in the equation x = a.