Mathematical Methods — Question 16

QCAA 2024, Paper 1 · 4 marks

Q16 · 2024 · Technology-freeComplex familiar4 marks

QUESTION 16 (4 marks)

The graph is of the form

y=\log_a(x+b)

. A point on the graph

(4,3)

is labelled. The line

x=-4

is an asymptote.

There is a point

P(x_P,y_P)

on the graph where

y_P

is twice the value of the

y

-intercept of the curve. Determine the value of

x_P

WORKED SOLUTION

4 marks

ANSWER

x_P=12

Worked solution

The vertical asymptote

x=-4

gives

b=4

, so

y=\log_a(x+4).

Substituting

(4,3)

gives

3=\log_a8,

hence

a^3=8

and

a=2

. Thus

y=\log_2(x+4).

The

y

-intercept is

\log_2(4)=2.

Therefore at

P

y_P=2\times2=4

. Then

4=\log_2(x_P+4),

x_P+4=2^4=16

and

x_P=12

correctly determines the value of b in the equation

[1 mark]

determines the value of a in the equation

[1 mark]

determines the y-coordinate of point P

[1 mark]

determines the x-coordinate of point P

[1 mark]

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