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Investigating the Quadratic Function.
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Introduction
Investigating the Quadratic Function
Type 1
Dan Plant
Mr. Maly
11 IB Mathematics
Thursday, May 10, 2007
Investigating the Quadratic Function Type 1
- Based on three resulting graphs, it can be determined that they are in fact all the same shape, of a parabola, however have varied locations. Furthermore, the locations of the graphs are specifically translated positively or negatively vertical according to the constant added or subtracted to the variable (x2). These three graphs may be generalized by the following statement; adding a positive or negative constant h will shift the graph vertically h units in the function y = (x2)+h. (Refer to attached graphs)
- Based on the three resulting graphs, it can be determined that they are in fact all the same shape, of a parabola, however they have varied locations. Furthermore, the locations of the graphs are specifically translated
Middle
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.
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(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
Conclusion
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.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.
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