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AHL 2.8—Transformations of graphs, composite transformations - IB Questionbank | RevisionDojo

Practice AHL 2.8—Transformations of graphs, composite transformations with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

The point $A(1,0.5)$ lies on the curve $y=f(x)$ . Write down the coordinates of the corresponding point under the following transformations.

$y=f(x)+5$

[1]

$y=5 f(x)$

[1]

$y=f(x+5)$

[1]

$y=f(5 x)$

[1]

Question 2

SLPaper 1

Let function $f(x)=\frac{x^{3}}{3}-x^{2}-3 x$ . Part of the graph of $f$ is shown below.

There is a maximum point at $A$ and a minimum point at $B(3,-9)$ . Write down the coordinates of:

The image of $B$ after reflection in the $y$ - axis.

[1]

The image of $B$ after translation by the vector $\binom{-2}{5}$

[1]

The image of $B$ after reflection in the $x$ -axis followed by a horizontal stretch with scale factor $\frac{1}{2}$ .

[2]

Question 3

SLPaper 1

Let function $f(x)=2 x+1$

Draw the graph of $f(x)$ for $0 \leq x \leq 2$

[3]

Let $g(x)=f(x+3)-2$ , draw the graph of $g(x)$ for $-3 \leq x \leq-1$

[3]

Question 4

SLPaper 1

Let $f(x)=x^{2}+4$ and $g(x)=x-1$

Find $(f \circ g)(x)[2]$

The vector $\binom{3}{-1}$ translates the graph of $(f \circ g)(x)$ to the graph of $h$

[17]

Find the coordinates of the vertex of the graph of $h$ .

[4]

Show that $h(x)=x^{2}-8 x+19$

[2]

The line $y=2 x-6$ is a tangent to the graph of $h$ at the point $P$ . Find the $x$ -coordinate of $P$ .

[4]

The line $y=2 x-5$ intersects the graph of $h$ at two points. Find the coordinates of the two points of intersection.

[5]

Question 5

HLPaper 2

A city planner is analyzing the function representing the cost of public transportation, given by $h(x) = \sqrt{x + 1}$

This function is transformed to produce various other functions.

Write down the equation of $j(x)$ if it represents $h(x)$ reflected in the $x$ -axis.

[2]

Write down the equation of $g(x)$ if it represents $h(x)$ reflected in the $y$ -axis.

[2]

Describe the transformations applied to $h(x)$ to obtain $f(x) = -\sqrt{-x + 1}$ .

[2]

Question 6

HLPaper 2

The point $A(-1,3)$ lies on the curve $y=f(x)$ .

Find the coordinates of the corresponding point under $y=2 f(x-3)+4$ .

[4]

Question 7

HLPaper 2

A local community is developing a new park and wants to model the height of a parabolic fountain using the quadratic function $f(x) = p + qx - x^2$ . The fountain reaches its maximum height of 5 meters when positioned at $x = 3$ meters from the base.

Determine the values of $p$ and $q$ that define this fountain's height.

[4]

If the fountain's design is modified to shift it 3 meters to the right, calculate the equation of the new height function.

[3]

Question 8

HLPaper 2

A financial analyst is examining the natural logarithm function to model investment growth, given by $f(x) = \ln(x)$ .

Write the equation of the function after a vertical stretch of $f(x)$ by a factor of 14.

[2]

Write the equation of the function after a horizontal shrink of $f(x)$ by a factor of 3.

[1]

Describe the domain and range of the transformed function $y = \ln(3x)$ .

[2]

Question 9

SLPaper 1

[Maximum Mark :12]

Express $y=2 x^{2}-12 x+23$ in the form $y=2(x-c)^{2}+d$ .

The graph of $y=x^{2}$ is transformed into the graph of $y=2 x^{2}-12 x+23$ by the transformations. a vertical stretch with scale factor $k$ followed by a horizontal translation of $p$ units followed by a vertical translation of $q$ . 2. Find the value of $k, p$ and $q$ .

Express $y=2 x^{2}-12 x+23$ in the form $y=2(x-c)^{2}+d$ .

[2]

Find the value of $k, p$ and $q$ .

[5]

Question 10

HLPaper 2

The quadratic function $f$ is defined by $f(x) = 3x^2 - 12x + 11$ .

On the following grid, draw the graph of $f$ for $0 \leq x \leq 4$ .

[1]

Write function $f$ in the form $f(x) = 3(x - h)^2 + k$ .

[2]

The graph of $f$ is translated 3 units in the positive $x$ -direction and 5 units in the positive $y$ -direction. Find the function $g$ for the translated graph, giving your answer in the form $g(x) = 3(x - p)^2 + q$ .

[2]

Paper

Level

Question 1

HLPaper 1

The point $A(1,0.5)$ lies on the curve $y=f(x)$ . Write down the coordinates of the corresponding point under the following transformations.

$y=f(x)+5$

[1]

$y=5 f(x)$

[1]

$y=f(x+5)$

[1]

$y=f(5 x)$

[1]

Question 2

SLPaper 1

Let function $f(x)=\frac{x^{3}}{3}-x^{2}-3 x$ . Part of the graph of $f$ is shown below.

There is a maximum point at $A$ and a minimum point at $B(3,-9)$ . Write down the coordinates of:

The image of $B$ after reflection in the $y$ - axis.

[1]

The image of $B$ after translation by the vector $\binom{-2}{5}$

[1]

The image of $B$ after reflection in the $x$ -axis followed by a horizontal stretch with scale factor $\frac{1}{2}$ .

[2]

Question 3

SLPaper 1

Let function $f(x)=2 x+1$

Draw the graph of $f(x)$ for $0 \leq x \leq 2$

[3]

Let $g(x)=f(x+3)-2$ , draw the graph of $g(x)$ for $-3 \leq x \leq-1$

[3]

Question 4

SLPaper 1

Let $f(x)=x^{2}+4$ and $g(x)=x-1$

Find $(f \circ g)(x)[2]$

The vector $\binom{3}{-1}$ translates the graph of $(f \circ g)(x)$ to the graph of $h$

[17]

Find the coordinates of the vertex of the graph of $h$ .

[4]

Show that $h(x)=x^{2}-8 x+19$

[2]

The line $y=2 x-6$ is a tangent to the graph of $h$ at the point $P$ . Find the $x$ -coordinate of $P$ .

[4]

The line $y=2 x-5$ intersects the graph of $h$ at two points. Find the coordinates of the two points of intersection.

[5]

Question 5

HLPaper 2

A city planner is analyzing the function representing the cost of public transportation, given by $h(x) = \sqrt{x + 1}$

This function is transformed to produce various other functions.

Write down the equation of $j(x)$ if it represents $h(x)$ reflected in the $x$ -axis.

[2]

Write down the equation of $g(x)$ if it represents $h(x)$ reflected in the $y$ -axis.

[2]

Describe the transformations applied to $h(x)$ to obtain $f(x) = -\sqrt{-x + 1}$ .

[2]

Question 6

HLPaper 2

The point $A(-1,3)$ lies on the curve $y=f(x)$ .

Find the coordinates of the corresponding point under $y=2 f(x-3)+4$ .

[4]

Question 7

HLPaper 2

Determine the values of $p$ and $q$ that define this fountain's height.

[4]

If the fountain's design is modified to shift it 3 meters to the right, calculate the equation of the new height function.

[3]

Question 8

HLPaper 2

A financial analyst is examining the natural logarithm function to model investment growth, given by $f(x) = \ln(x)$ .

Write the equation of the function after a vertical stretch of $f(x)$ by a factor of 14.

[2]

Write the equation of the function after a horizontal shrink of $f(x)$ by a factor of 3.

[1]

Describe the domain and range of the transformed function $y = \ln(3x)$ .

[2]

Question 9

SLPaper 1

[Maximum Mark :12]

Express $y=2 x^{2}-12 x+23$ in the form $y=2(x-c)^{2}+d$ .

Express $y=2 x^{2}-12 x+23$ in the form $y=2(x-c)^{2}+d$ .

[2]

Find the value of $k, p$ and $q$ .

[5]

Question 10

HLPaper 2

The quadratic function $f$ is defined by $f(x) = 3x^2 - 12x + 11$ .

On the following grid, draw the graph of $f$ for $0 \leq x \leq 4$ .

[1]

Write function $f$ in the form $f(x) = 3(x - h)^2 + k$ .

[2]

Paper

Level

Question 1

HLPaper 1

The point $A(1,0.5)$ lies on the curve $y=f(x)$ . Write down the coordinates of the corresponding point under the following transformations.

$y=f(x)+5$

[1]

$y=5 f(x)$

[1]

$y=f(x+5)$

[1]

$y=f(5 x)$

[1]

Question 2

SLPaper 1

Let function $f(x)=\frac{x^{3}}{3}-x^{2}-3 x$ . Part of the graph of $f$ is shown below.

There is a maximum point at $A$ and a minimum point at $B(3,-9)$ . Write down the coordinates of:

The image of $B$ after reflection in the $y$ - axis.

[1]

The image of $B$ after translation by the vector $\binom{-2}{5}$

[1]

The image of $B$ after reflection in the $x$ -axis followed by a horizontal stretch with scale factor $\frac{1}{2}$ .

[2]

Question 3

SLPaper 1

Let function $f(x)=2 x+1$

Draw the graph of $f(x)$ for $0 \leq x \leq 2$

[3]

Let $g(x)=f(x+3)-2$ , draw the graph of $g(x)$ for $-3 \leq x \leq-1$

[3]

Question 4

SLPaper 1

Let $f(x)=x^{2}+4$ and $g(x)=x-1$

Find $(f \circ g)(x)[2]$

The vector $\binom{3}{-1}$ translates the graph of $(f \circ g)(x)$ to the graph of $h$

[17]

Find the coordinates of the vertex of the graph of $h$ .

[4]

Show that $h(x)=x^{2}-8 x+19$

[2]

The line $y=2 x-6$ is a tangent to the graph of $h$ at the point $P$ . Find the $x$ -coordinate of $P$ .

[4]

The line $y=2 x-5$ intersects the graph of $h$ at two points. Find the coordinates of the two points of intersection.

[5]

Question 5

HLPaper 2

A city planner is analyzing the function representing the cost of public transportation, given by $h(x) = \sqrt{x + 1}$

This function is transformed to produce various other functions.

Write down the equation of $j(x)$ if it represents $h(x)$ reflected in the $x$ -axis.

[2]

Write down the equation of $g(x)$ if it represents $h(x)$ reflected in the $y$ -axis.

[2]

Describe the transformations applied to $h(x)$ to obtain $f(x) = -\sqrt{-x + 1}$ .

[2]

Question 6

HLPaper 2

The point $A(-1,3)$ lies on the curve $y=f(x)$ .

Find the coordinates of the corresponding point under $y=2 f(x-3)+4$ .

[4]

Question 7

HLPaper 2

Determine the values of $p$ and $q$ that define this fountain's height.

[4]

If the fountain's design is modified to shift it 3 meters to the right, calculate the equation of the new height function.

[3]

Question 8

HLPaper 2

A financial analyst is examining the natural logarithm function to model investment growth, given by $f(x) = \ln(x)$ .

Write the equation of the function after a vertical stretch of $f(x)$ by a factor of 14.

[2]

Write the equation of the function after a horizontal shrink of $f(x)$ by a factor of 3.

[1]

Describe the domain and range of the transformed function $y = \ln(3x)$ .

[2]

Question 9

SLPaper 1

[Maximum Mark :12]

Express $y=2 x^{2}-12 x+23$ in the form $y=2(x-c)^{2}+d$ .

Express $y=2 x^{2}-12 x+23$ in the form $y=2(x-c)^{2}+d$ .

[2]

Find the value of $k, p$ and $q$ .

[5]

Question 10

HLPaper 2

The quadratic function $f$ is defined by $f(x) = 3x^2 - 12x + 11$ .

On the following grid, draw the graph of $f$ for $0 \leq x \leq 4$ .

[1]

Write function $f$ in the form $f(x) = 3(x - h)^2 + k$ .

[2]

AHL 2.8—Transformations of graphs, composite transformations Questionbank