Linear Law “Transformation” of non-linear relationships into linear relationships

How it works Quadratic Curve: non-linear!

Transforming to linear relationship Linear ? General equation of linear relationship: −3 y x2x2 Plot y vs x 2

3 y 1/x

Plot (1/y) vs x 2 m = a, c = b Example 1

Plot xy vs x 2 m = a, c = b Example 2

m = b, c = a Example 3 Plot xy vs

Plot lg y vs x m = lg b, c = lg a Q9

Plot xy vs x Plot y vs Grad = b, xy-intercept = aGrad = a, y-intercept = b Q16

Plot lg y vs x m = lg b, c = lg a + 3 lg b Q17

Plot lg (y – 4) vs x m = lg b, c = lg a Q18

Plot lg y vs lg x m = b, c = - lg a Q19

m = p, c = - q Q20 Plot vs x

Express y in terms of x?y = ?? (0,1) (4,9) a) y x2x2

Express y in terms of x? (4,0) (0,2)

(5, 9) (2, 3) x + 1 lg y Q1

Q (3, 9) P (1, 3) ln (x – 1) ln y Q2

The following table gives values of y corresponding to some value of x. x12345 y It is known that x and y are related by the equation. (i)Explain how a straight-line graph of against can be drawn to represent the given equation and draw it for the given data. Use this graph to estimate the value of a and of b. (ii) Express the given equation in another form suitable for a straight-line graph to be drawn. State the variables whose values should be plotted.

. (i)Explain how a straight-line graph of against can be drawn to represent the given equation and draw it for the given data. Use this graph to estimate the value of a and of b. (i) In order to plot 1/y against 1/x, we need to arrange the equation into (1). b/a represents the gradient and 1/a represents the vertical intercept. (1)

x12345 y /x /y Choose appropriate scales

(0.6, 0.7) (0,0.25)

(ii) Express the given equation in another form suitable for a straight-line graph to be drawn. State the variables whose values should be plotted.

Q1 The data for x and y given in the table below are related by a law of the form, where p and q are constants. x12345 y By drawing a suitable straight line, find estimates for p and q. Plot (y ─ x) against x 2, p represents the gradient and q represents the (y-x) -intercept.

x12345 y x2x y ─ x

Q2 The table shows the experimental values of two variables x and y which are known to be related by an equation of the form p(x + y – q) = qx 3, where p and q are constants. x y Draw a suitable straight-line graph to represent the above data. Use your graph to estimate (i)the value of p and of q, (ii)the value of y when x = 2.2. Plot (x + y) against x 3, (q/p) represents the gradient and q represents the (x + y) - intercept.

x y x3x x+y

Q3 The table below shows experimental values of two variables, x and y. One value of y has been recorded incorrectly. x12345 y It is believed that x and y are related in the form y = x 2 – ax + b, where a and b are constants. Draw a suitable straight-line graph to represent the given data. Use your graph to estimate (i) the value of a and of b, (ii) a value of y to replace the incorrect value. Plot (y ─ x 2 ) against x, ─ a represents the gradient and b represents the (y ─ x 2 ) -intercept.

x12345 y x12345 y ─ x

Identify the incorrect readings/ outliers!!

x12345 y x y2y ? One of the values of y is subject to an abnormally large error Identify the abnormal reading and estimate its correct value. abnormal reading: y = 3.71 Correct value should be

Estimate the value of x when y = 2

Q4 The table below shows the experimental values of two variables x and y. It is known that one value of y has been recorded incorrectly x y It is known that x and y are related by an equation of the form, where a and b are constants. By plotting against x, obtain a straight-line graph to represent the above data. Use your graph to estimate the value of a and of b. (i) Use your graph to estimate a value of y to replace the incorrect value. (ii) Find the value of x when y =. (iii) By inserting another straight line to your graph, find the value of x and of y which satisfy the simultaneous equations and

x y x /y

abnormal reading: y = 0.70 Correct value should be x y x /y

Estimate the value of x when y =

and Need to draw this and find the point of intersection of the 2 lines Bear in mind: need to use the same axes as first line! Vertical intercept  (0, -1.2) Horizontal intercept  (0.8, 0)

Vertical intercept  (0, -1.2) Horizontal intercept  (0.8, 0)

Q5 The variables x and y are known to be connected by the equation An experiment gave pairs of values of x and y as shown in the table. One of the values of y is subject to an abnormally large error. x y Plot lg y against x and use the graph to (i) identify the abnormal reading and estimate its correct value. (ii) estimate the value of C and of a. (iii) estimate the value of x when y = 1.

x y lg y (i) abnormal reading: y = Correct value should be

estimate the value of x when y = 1.