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Exponential and Logarithmic Functions · Paper 1 — Free Maths AI HL Practice Questions

1FoundationSAQ-SExponential functions and their graphs5 marksPaper 1~8 min

The graph of an exponential function

f(x) = a \cdot 2^x

passes through the point

(3, 40)

(a)

Find the value of

a

. [2 marks]

(b)

State the equation of the horizontal asymptote of

f(x)

. [1 mark]

(c)

A second function is defined as

g(x) = f(x+2) - 20

. Determine the equation of the horizontal asymptote of

g(x)

and find the value of

x

for which

g(x) = 0

. [2 marks]

Solutions

2MasterySAQ-SExponential functions and their graphs6 marksPaper 1~9 min

The value of a rare painting is modelled by the function

V(t) = 8000 \times 10^{kt}

, where

V

is the value in USD,

t

is the number of years since purchase, and

k

is a constant. The painting was purchased for USD 8000. After

5

years, its value had increased to USD 11000.

(a)

Show that

k = \dfrac{1}{5}\log_{10}\!\left(\dfrac{11}{8}\right)

. [2 marks]

(b)

Calculate the value of

k

. Give your answer correct to

3

significant figures. [1 mark]

(c)

Find the number of years it takes for the painting to reach a value of USD 20000. Give your answer correct to the nearest year. [3 marks]

Solutions

3ChallengeSAQ-LExponential functions and their graphs8 marksPaper 1~12 min

A new smartphone app's number of downloads,

D

(in thousands), is modelled by

D(t) = 50 \times 2^{kt}

, where

t

is the time in weeks since the app was released. After 3 weeks, there were 200 thousand downloads.

(a)

Determine the value of

k

, giving your answer in exact form. [2 marks]

(b)

Determine the number of weeks it takes for the number of downloads to reach 1000 thousand. Give your answer correct to 3 significant figures. [2 marks]

(c)

A competing app has downloads modelled by

E(t) = 100 \times 1.5^{t}

. State which app has a faster initial rate of growth. Justify your answer by comparing the initial rates of change of

D

and

E

. [2 marks]

(d)

Determine the value of

t

at which both apps have the same number of downloads. Give your answer correct to 3 significant figures. [2 marks]

Solutions

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