1b 2009 Straight Line Graph

The Linear Law In this chapter you must be able to use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept. Reduction of laws to straight line (linear) form Non-linear functions in variables x and y can be reduced to linear functions in the form of Y = mX + c, where m is the gradient, c is the Yintercept and X and Y are expressions in x and/or y. The table below shows how the logarithmic functions can be reduced to the linear form: functions y = ab x lg y = lg a + x lg b lg y = x lg b + lg a

Y

X

m

c

lg y

x

lg b

lg a

y = ax b lg y = lg a + b lg x lg y = b (lg x) + lg a

lg y

lg x

b

lg a

Questions 1. Variables x and y are related by the equation y 2 = px q. When the graph of lg y against lg x is drawn, the resulting straight line has a gradient of - 2 and an intercept of 0.5 on the axis of lg y. Calculate the values of p and q. p = 10, q = - 4 2.

The variables x and y are related by the equation y = Ax n . A straight line of the graph of lg y against lg x passes through the points (1, 3) and (2, 1). Find the values of A and n. A = 105 , n = - 2

3. The variables x and y are connected by the equation y = ax b where a and b are constants. A straight line of lg y against lg x is drawn. The line passes through the points (2, 0) and (0, -1). Calculate the values of a and b and hence find the value of y when x = 5. a = 0.1, b = 0.5, 0.2236 4. A straight line graph which passes through the points (1, 2) and (2, 5) is obtained by plotting lg y against lg (x + 1). Find (i) y in terms of x y=

( x + 1)3 10

(ii) 6.4

the value of y when x = 3.

1

5. The variables x and y are related in such a way that when lg y is plotted against lg x, a straight line is obtained. Given that this line passes through (0, 1) and (6, 4), find (i) the value of x when lg y = 3. 10,000 (ii) The value of a and n when the relationship between x and y is expressed in the form y = axn. a = 100, b = √10 6. A straight line of the graph of lg y against x is drawn and passes through the points (4, 5) and (2, 2). Find the values of the constants A and b for which the straight line represent the equation y = Abx. A = 0.1, b = 31.6 7. The table shows experimental values of two quantities x and y which are known to be connected by a law of the form y = kbx. x y

1 30

2 75

3 190

4 470

Plot lg y against x and use your graph to estimate the values of k and b. b = 2.45, k = 11.9 8. The following values of x and y are believed to obey a law of the form y = abx, where a and b are constants. x y

1.00 2.00 3.00 4.00 5.00 3.80 9.20 22.1 53.1 127.4

Show graphically that this is so and estimate the values of a and b, correct to 1 decimal place. y = 1.6 (2.4)x 9. The following readings were obtained in an experiment. x y

20 55

40 150

60 280

80 430

100 600

Show graphically that, allowing for small errors of observation, there is a relation between y and x of the form y = ax k. Find the

2

approximate values for a and k.

y = 0.60 x

1.5

10. The table below shows experimental values of two variables, x and y. x y

5 10 15 20 25 57.7 37.0 23.7 15.2 9.7

It is known that x and y are related by the equation y = Ae- k x, where A and k are constants. Using graph paper, plot ln y against x for the above data and use your graph to estimate the value of A and of k. (J96/4) 11. The table below shows experimental values of two variables, x and y. x y

0.5 6.2

1.0 4.7

1.5 3.7

2.0 2.9

2.5 2.2

3.0 1.7

It is known that x and y are related by the equation y = Ab x, where A and k are constants. (i) Express the equation in a form suitable for drawing a straight line graph (ii) Draw the graph and use it to estimate the value of A and of b. (iii) Find the values of y when x = 0.8. (A.M J97/4)

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