Investigating the Quadratic Function.

(b) This particular expression x2 – 10x + 32 is not a perfect square, however it can be broken down to a perfect square (x – h)2 with the addition of the constant g to provide the expression in the form of (x – h)2 + g. The following is the method required to obtain the desired form of the expression: x2 – 10x + 32 = x2 – 10x + (25 + 7) = (x2 – 10x + 25) + 7 = (x-5)2 + 7. Thus x2 – 10x + 32 = (x-5)2 + 7.

(c)  (i) x2 – 18x + 77 (ii) x2 – 14x + 57 (iii) x2 + 12x + 36 (iv) x2 –5x + 8.5

(i) x2 – 18x + 77 → x2 – 18x + (81 – 4) → (x2 – 18x + 81) – 4 → (x – 9)2 - 4

(ii)  x2 – 14x + 57 → x2 – 14x + (49 + 8) → (x2 – 14x + 49) + 8 → (x – 7)2 + 8

(iii) x2 + 12x + 36 → (x + 4)2 

(iv) The middle term of the following expression, x2 – 7x + 14.5, determines the constant h of (x – h)2. Therefore, h will equal 7/2. The resultant of (x – h)2, (x - 7/2)2, is x2 – 7x + 12.25. In order to achieve an expression equal to the original, it is required to add 2.25 in the place of the constant g. With the addition of the necessary 2.25, equalizing it to the original expression in the x2 + bx +c form, the result is (x - 7/2)2 + 2.25. The result in the (x-h)2 + g is therefore (x - 7/2)2 + 2.25.

(d) Considering the fact that the vast majority of real numbers are not perfect squares is essential for drawing a final conclusion to the question. As (a+b)2 is equal to ax2 + 2ab + bx2, a perfect square x2 + bx + c simplifies to (x+b/2)2. In order to account for the difference between c and b/2 in expression void of a perfect square, the constant (c - (b/2)2) is added to the expression (x + b/2)2. Thus, when formulating an expression x2 + bx + c in the form x2 + bx + c, one must use the formula (x + b/2)2 + (c - (b/2)2).

5. The expression y = (x-h)2 is a parabolic shape.  Based on the conclusions drawn from questions one and two, the graph y = (x-h)2 + g will be of identical shape with the graph of y = x2, but translated horizontally h units and vertically g units. Fundamentally, if f(x) = y, then f(x) + g = y + g, translating y in accordance with g. The entire premise on which this is built relies within the statement that the position of the y value is based entirely on that of x, while the position of the x value is based entirely on that of y.
6. The findings drawn from this investigation apply not only to these graphs; they may be employed in other circumstances as well. Other graphs that could employ these findings may be polynomial, trigonometric, exponential, and any graph where the x and y vales are based on each other. Generalizing, incorporating a constant k into any function f(x) will translate the function f(x) vertically k units. Replacing x in the function f(x) with the expression (x-h), h will translate any function f(x) horizontally h units.