Graphs of Logarithmic Functions

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Graphs of Logarithmic Functions — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionBasic log function graphsConcept Practice

2 marks~3 minCriterion A

The diagram shows the graph of

y = \log_2 x

for

0 < x \leq 8

State the coordinates of the point where the graph crosses the

x

-axis. [1]

State the equation of the vertical asymptote of the graph. [1]

Solutions

2QuestionComparing exponential and logarithmic graphsAssessment Practice

6 marks~9 minCriterion B

The diagram shows the first four figures of a pattern made from small squares arranged in two rows.

Figure 1 has 2 squares, Figure 2 has 6 squares, Figure 3 has 12 squares, and Figure 4 has 20 squares.

Write down the number of squares in Figure 5. [1]

Find a rule for the number of squares

S

in Figure

n

in the form

S = an^2 + bn

, where

a

and

b

are integers. [2]

A student claims that Figure

n

always has exactly

n

more squares than Figure

n - 1

. Justify whether this claim is correct, showing your working clearly. [3]

Solutions

3QuestionVertical asymptotesAssessment Practice

7 marks~11 minCriterion D

Data

x

(mL): 4, 7, 12

Observed pH:

-0.52

0.70

1.00

A chemist uses the function

f(x) = \log_{10}(x - 2)

to model the pH of a solution, where

x

is the total volume of acid added, in mL, and

x > 2

.

The diagram shows the graph of

f

for

2 < x \leq 12

, together with three observed pH readings recorded in the laboratory.

Observed

Write down the equation of the vertical asymptote of

f

and interpret its meaning in the context of the acid volume. [2]

Complete the table of predicted pH values, giving each answer to 2 decimal places.

x

(mL): 4, 7, 12

Predicted pH: ?, ?, ?

Hence calculate the percentage error at

x = 7

, using

\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{|\text{observed}|} \times 100\%.

Give your answer to 1 decimal place. [3]

The chemist considers using the model to predict the pH when

x = 2.05

mL. Justify whether the model is reliable for volumes this close to 2 mL, using the values of

f(2.05)

and

f(2.5)

in your reasoning. [2]

Solutions

4QuestionTransformations (shifts, reflections, stretches)Assessment Practice

10 marks~15 minCriterion C

The graph of

f(x) = \log_2 x

is transformed to obtain the graph of

g(x)

. The transformation consists of a horizontal translation of 3 units in the positive

x

-direction, followed by a vertical stretch with scale factor 4 from the

x

-axis.

Find the equation of

g(x)

and state its domain. [3]

The graph of

g(x)

passes through the point

(a, 12)

. Find the exact value of

a

. [3]

A student claims that

g(x) = f(x)

has no solution because the two graphs never intersect. Justify whether the student is correct, showing all algebraic reasoning. [4]

Solutions

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