CAIE A-Level Mathematics 3.2.4 Linearising Relationships

Practise taking logarithms of power and exponential models, matching the result to a straight-line equation and using plotted gradients and intercepts to recover constants.

Exam points

take logarithms and rearrange the relationship into Y = mX + c for the plotted axes
calculate the graph gradient and intercept from two points without confusing logged variables
match m and c to model parameters, then exponentiate an intercept to recover its constant

[Maximum number: 4]

Figure for Question 3.2.4—Linearising relationships question 1 — CAIE A-Level Mathematics A2

The variables x and y are related by the equation $y=a b^{x}$ , where a and b are constants. The diagram shows the result of plotting $\ln y$ against x for two pairs of values of x and y. The coordinates of these points are ( 1,3.7 ) and ( 2.2,6.46 ).

Use this information to find the values of a and b.

State or imply that $\ln y=\ln a+x \ln b$

B1

Carry out a completely correct method for finding $\ln a$ or $\ln b$

M1

$3.7=\ln a+\ln b$ and $6.46=\ln a+2.2 \ln b$
leading to $\ln a=1.4, \ln b=2.3$ .

Obtain value a=4.06

A1

Obtain value b=9.97

A1

SC B1 for $a=\mathrm{e}^{1.4}$ and $b=\mathrm{e}^{2.3}$ .

Alternative Method for Question 3

$e^{3.7}=a b^{1}$ and $e^{6.46}=a b^{2.2}$

B1

Divide to obtain $\mathrm{e}^{2.76}=b^{1.2}$ and state or imply $2.76=1.2 \ln b$

M1

Obtain value a=4.06

A1

Obtain value b=9.97

A1

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3.2.4—Linearising relationships question 1