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SL 2.11—Transformation of functions - IB Questionbank | RevisionDojo

Practice SL 2.11—Transformation of functions with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

Let $f(x)=\sqrt{x}$ , defined for $x \geq 0$ . The graph of $g$ is obtained from the graph of $f$ by a horizontal stretch by a factor of 2 , followed by a vertical translation of 3 units up.

Express $g(x)$ in terms of $x$ .

[1]

Write down the domain and range of $g$ .

[2]

The point $B(4,2)$ lies on the graph of $f$ . Find the coordinates of the corresponding point on the graph of $g$ .

[2]

Question 2

SLPaper 1

The graph of $y=f(x)=\ln (x+2)$ , for $x>-2$ , is shown below, with points at $(-1,0)$ and $(0, \ln 2)$ .

Let $g(x)=\frac{1}{x}$ , defined for $x \neq 0$ . The graph of $h$ is obtained from the graph of $f$ by a horizontal translation of 3 units to the left, a vertical stretch with scale factor 2 , and a reflection in the x-axis.

Express $h(x)$ in terms of $x$ .

[3]

Find $(h \circ g)(x)$ .

[3]

Sketch the graph of $y=h \circ g(x)$ for $-4 \leq x \leq 4, x \neq 0$ , indicating the vertical asymptote and the point corresponding to $x=1$ on $g$ .

[4]

Find the equation of the line tangent to the graph of $h \circ g$ at $x=1$ .

[4]

Describe a sequence of transformations that transforms the graph of $y=\ln x$ to the graph of $y=h(x)$ .

[2]

Question 3

SLPaper 1

The point $A(3,2)$ lies on the curve $y=f(x)$ . Write down the coordinates of the corresponding point under the following transformations:

$y=f(x)+4$

[1]

$y=f(x+2)$

[1]

$y=2 f(x)$

[1]

$y=f(3 x)$

[1]

Question 4

SLPaper 1

The function $f$ is defined for $-4 \leq x \leq 0$ , with range $1 \leq y \leq 5$ . Let $g(x)=f(x)+3$ .

Write down the range of $g$ .

[2]

Let $h(x)=f(x-2)$ . Write down the domain of $h$ .

[2]

Question 5

SLPaper 1

Let $f(x)=x^{2}+2$ . The graph of $g$ is obtained from the graph of $f$ by applying the following transformations in order: a horizontal translation of 3 units to the left, followed by a vertical stretch by a factor of 2 .

Express $g(x)$ in the form $g(x)=a(x-h)^{2}+k$ .

[3]

The point $A(1,3)$ lies on the graph of $f$ . Find the coordinates of the corresponding point on the graph of $g$ .

[3]

Question 6

SL & HLPaper 1

Consider the quadratic function $f(x) = 2(x - 2)^2 + 4$ .

Describe the transformations applied to the function $y = x^2$ to obtain the function $f(x)$ .

[3]

Given that $x^2\geq f(x)$ under the interval $[2,4]$ , find the area between the curves $y = x^2$ and $f(x) = 2(x - 2)^2 + 4$ within that interval.

[5]

Question 7

SLPaper 1

The point $A(3,2)$ lies on the curve $y=f(x)$ . Write down the coordinates of the corresponding point under the following transformations:

$y=f(x)+4$

[1]

$y=f(x+2)$

[1]

$y=2 f(x)$

[1]

$y=f(3 x)$

[1]

Question 8

SLPaper 2

A rational function has vertical asymptote $x=4$ and horizontal asymptote $y=1$ . It passes through $(0,-1)$ and $(5,9)$ .

Show $f(x)=1+\dfrac{k}{x-4}$ and determine $k$ . Hence express $f(x)=\dfrac{ax+b}{cx+d}$ with integers.

[4]

Determine domain, range, intercepts, asymptotes, and the sign of $f(x)-1$ on $(- \infty,4)$ , $(4,\infty)$ .

[5]

Solve $f(x)=g(x)$ to 3 d.p. and justify one solution in each of $(-1,4)$ and $(4,\infty)$ .

[6]

Describe the transformations from $y=1/x$ to $y=f(x)$ .

[2]

Question 9

SLPaper 2

A rational function has vertical asymptote $x=4$ and horizontal asymptote $y=7$ . It passes through the points $(0,5)$ and $(6,11)$ .

Show that the function can be written in the form $f(x)=7+\frac{k}{x-4}$ for some constant $k$ , and determine $k$ . Hence write $f(x)$ explicitly as $\dfrac{ax+b}{cx+d}$ with integer coefficients.

[4]

Determine all key features of $f$ : domain, range, intercepts, asymptotes, and the sign of $f(x)-7$ on $(- \infty,4)$ and $(4,\infty)$ .

[5]

Solve $f(x)=g(x)$ to three decimal places and justify exactly one solution in each of $(-1,4)$ and $(4,\infty)$ .

[6]

Describe transformations mapping $y=1/x$ to $y=f(x)$ .

[2]

Question 10

SLPaper 1

A quadratic has $x$ -intercepts $x=-6$ and $x=2$ , and its vertex lies on $y=-x+1$ .

Find $f(x)$ .

[5]

Determine the range of $f(x)$ .

[2]

Find $f^{-1}(x)$ with restriction.

[5]

Transformations from $y=x^2$ .

[3]

SL 2.11—Transformation of functions Questionbank