• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Investigating the Quadratic Function.

Extracts from this document...

Introduction

Investigating the Quadratic Function

Type 1

Dan Plant

Mr. Maly

11 IB Mathematics

Thursday, May 10, 2007

Investigating the Quadratic Function Type 1

  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)
  2. 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
...read more.

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.

(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

...read more.

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.

...read more.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related AS and A Level Core & Pure Mathematics essays

  1. Marked by a teacher

    The Gradient Function

    5 star(s)

    these data are very hard to work with and I cannot find a general pattern by looking at the gradients - they are not whole numbers. This will again, however, support the gradient function I will come up with using binomial expansion.

  2. Marked by a teacher

    Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    3 star(s)

    Again, in order to estimate this equation, we fits the real data from 1950 to 2002 to the equation, another consumption equation can be gained below: LC = + 0.6322*LC_1 + 0.2061 + 0.3519*LY Including the lagged consumption in the function has changed this permanent income.

  1. Math Portfolio Type II - Applications of Sinusoidal Functions

    the graph of function f into the graph of function g would be 0.520. Part B 1. Use the data in the list of sunrise times and the data in the list of sunset times for Toronto to create data for a new list that shows the number of hours of daylight for the 12 dates given.

  2. Estimate a consumption function for the UK economy explaining the economic theory and statistical ...

    + log(Yt-1) + log(Yt-2)] /3 If this value is put back into our equation we obtain the following result: ct = 0.6181 + 0.9507ypt This approach weights all the previous years' income at the same level, however it is rational to assume that consumers will weight the more recent years

  1. Investigating the Quadratic Function

    and graph C (y = x� - 2) show clear indications of vertical translations. B has three units upward and C two units downward because a second coefficient appeared in both equations ("+3", "-2"). Generally speaking if a number (second coefficient "+3" or "-2") is added to the parent function (y = x�), the result is a translation either up or down the y-axis.

  2. Estimate a consumption function for the UK economy explaining the statistical techniques you have ...

    However, during the 1950s there appeared to be a discrepancy between the consumption function estimated from long run, short run and cross-section series of data. It also failed to explain some of the more interesting features of aggregate consumer behaviour and failed to predict certain periods of sharp fall in

  1. Investigation of the Phi Function

    iii ?(11) = 10, because again, 11 is a prime number and, so, cannot be co-prime with any of the numbers smaller than it. This ?(11) = 11-1 =10 iv ?(24) = 8. This is because the prime factors of 24 are 2 and 3, and there are 8 numbers i.e.

  2. Triminoes Investigation

    = a x 4� + b x 4� + c x 4 + d = 64a + 16b + 4c + d = 35 - equation 4 Equation 4 - Equation 3 I am doing this to eliminate d from the two equations, to create another equation.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work