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

Extracts from this document...

Introduction

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

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

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.

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

# Related AS and A Level Core & Pure Mathematics essays

1.  5 star(s)

Using this formula, the next 5 values of x and the gradient are: 4 256 1280 5 625 3125 6 1296 6480 7 2401 12005 8 4096 20480 9 6561 32805 General proof - (x+h)5 - x5 = x^5 + 5x4h + 10x�h� + 10x�h� + 5xh4 +h5 - x5

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

3 star(s)

This absolute value should be the higher the better. T- values in this case not as high as we want, they are 4.57 and 119 respectively. In order to estimate more precisely, DW (Durbin- Watson) can be considered. DW equal to 0 if there is extreme positive serial correlation.

1. ## Investigation into combined transoformations of 6 trigonometric functions

looking at the trigonometric function of cosine indicated in figure 1.0 by a green dotted line. A cosine graph has no asymptotes but has symmetry in the y-axis, once again I'm going to focus on looking at the maximums and minimums of the graph y=acos(x+c)

2. ## Investigation of the Phi Function

no factors other than one or itself and cannot have common factors. iii ?(7) = 6, again because it is a prime number and cannot share factors with any numbers smaller than it. iv ?(6) = {1, 5} = 2 v ?(25)

1. ## Math Portfolio Type II - Applications of Sinusoidal Functions

As a result, the domain is equal or in between 1 and 365. The range of the function represents the time of sunrise. Therefore, the minimum and maximum values are found out, equal or in between 5.516 and 7.208. The period of a sinusoidal function is , so the period equals to 365 (there is 365 days in a year).

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

This means that if a person was calculating their consumption for period t they would use all the information available to them up until that time. However this can be broken down into areas; information available at time t-1 and "new" information that came available between t-1 and t.

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

The amount that the community spends on consumption depends (i) partly on the amount of their income, (ii) partly on other objective attendant circumstances, and (iii) partly on the objective needs and the psychological propensities and habits of the individuals comparing it...

2. ## Triminoes Investigation

18a + 2b = 5 18 x 1/6 + 2b = 5 3 + 2b = 5 2b = 5 - 3 2b = 2 b = 2/2 b = 1 Substitute a = 1/6, b = 1 into equation 6. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 