Laws of Exponents and Algebraic Manipulation

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Laws of Exponents and Algebraic Manipulation — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionNegative and Zero ExponentsConcept Practice

4 marks~6 minCriterion A

A biology researcher models bacterial decay using the function

y = 2^x

, where

x

represents time in hours (negative values indicate hours before a reference point) and

y

represents the relative bacterial population in millions.

The graph shows integer values of

x

from

-3

3

, but the

y

-values for

x = -1

x = -2

, and

x = -3

are missing.

Calculate the missing

y

-values for

x = -1

x = -2

, and

x = -3

using

a^{-n} = \dfrac{1}{a^n}

. [3]

The researcher claims the bacterial population was never zero in the hours before the reference point. Justify this claim using your results from part (a) and the behaviour of

y = 2^x

. [1]

Solutions

2QuestionProduct Quotient and Power LawsAssessment Practice

4 marks~6 minCriterion C

The functions

y = 2^x

and

y = 2^{x+3}

are defined for all real

x

State the transformation that maps the graph of

y = 2^x

onto the graph of

y = 2^{x+3}

. [1]

The point

(0, 1)

lies on

y = 2^x

. Show that the image of this point under the transformation confirms your answer to part (a). [1]

A student claims: "Adding 3 inside the exponent is equivalent to multiplying the original function by a constant, so the transformation could also be described as a vertical stretch." Justify whether this claim is correct, using algebraic reasoning. [2]

Solutions

3QuestionNegative and Zero ExponentsAssessment Practice

4 marks~6 minCriterion B

A population of bacteria doubles every hour. A scientist models the population using

y = 2^x

, where

x

is the number of hours elapsed (negative values represent time before observation began) and

y

is the population in millions.

Selected values from the model:

x

-2

-1

0

1

2

y

(millions):

\dfrac{1}{4}

\dfrac{1}{2}

?

2

4

Explain how the pattern in the table demonstrates that

2^0 = 1

. [2]

A second bacterial strain follows

y = 5^x

and a third follows

y = (-3)^x

. Evaluate

5^0

and

(-3)^0

, justifying your answers using the zero exponent rule. [1]

The scientist claims the population at

x = 0

is "effectively zero" because no growth has yet occurred. Assess whether the mathematical model supports or contradicts this claim, and interpret what

2^0 = 1

means in this context. [1]

Solutions

4QuestionExponents in Algebraic EquationsAssessment Practice

4 marks~6 minCriterion D

A doctor prescribes a 200 mg dose of a drug. The amount remaining in a patient's bloodstream after

t

hours is modelled by

A = 200(0.85)^t

where

A

is in mg.

Calculate the amount of drug remaining after 4 hours. [2]

The drug is considered safe to re-dose when less than 50 mg remains. Using your answer to part (a), explain whether a second dose could be given after 8 hours. [1]

Advise the doctor whether this model alone is sufficient to determine a safe re-dosing time for all patients. [1]

Solutions

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