2.6 Transformations of Graphs

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2 marks

Let $f (x) = 2 (x + 4)^{3}$ and $g (x) = x^{3}$ , for $x \in ℝ$ .

Give a full geometric description of the two individual transformations that can be combined to obtain the graph of $f$ from the graph of $g$ .

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4 marks

The graph of $f$ is translated by the vector $(\begin{matrix} 2 \\ - 5 \end{matrix})$ to give the graph of $h$ .

Now consider the graph of $h$ as a transformation of the graph of $g$ . The point A on the graph of $h$ corresponds to the point $(2, 8)$ on the graph of $g$ .

Find the coordinates of A.

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3 marks

Let $f$ and $g$ be functions such that $g (x) = 2 f (x - 1) + 2$ , for $x \in ℝ$ .

The transformation that maps the graph of $f$ onto the graph of $g$ may be represented as a combination of two simpler transformations:

a vertical stretch by a factor of $v$ ,

followed by

a translation by the vector $(\begin{matrix} a \\ b \end{matrix})$ .

Write down the values of

(i)

v

(ii)

a

(iii)

b .

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3 marks

The point A $(3, 4)$ on the graph of $f$ is mapped to point B on the graph of $g$ .

Find the coordinates of B.

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3 marks

Let $f (x) = x^{2} - 9$ , for $x \in ℝ$ .

Sketch the graph of $y = f (x)$ on the following grid in the interval $0 \leq x \leq 5$ . Use an appropriate scale and clearly label any intersections the graph makes with the coordinate axes.

q3a-2-6-hard-ib-aa-hl-maths

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2 marks

Find $(f \circ f) (2) .$

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5 marks

The function $g$ is obtained when the graph of $f$ is translated by the vector $(\begin{matrix} 2 \\ 5 \end{matrix})$ .

Sketch the graph of $g (x)$ on the same grid above, also for the interval $0 \leq x \leq 5$ . Clearly label any intersections the graph makes with the coordinate axes and label the graph in the form $g (x) = a x^{2} + b x + c$ where $a, b$ and $c$ are constants to be determined.

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4 marks

Let $f (x) = 1.1 e^{x - 1} - 4$ for $- 2 \leq x \leq 3$ .

Sketch the graph of $y = f (x)$ on the grid below, clearly labelling any intersections the graph makes with the coordinate axes.

q4a-2-6-hard-ib-aa-hl-maths

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2 marks

The graph of $f$ is reflected in the $x$ -axis and then translated by the vector $(\begin{matrix} 2 \\ - 3 \end{matrix})$ to obtain the graph of $y = g (x)$ .

Find an expression for $g (x)$ .

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4 marks

The function $f$ is defined by

$f (x) = \{\begin{cases} 1 + 2 x & if x \leq 2 \\ x^{2} - 2 x + 5 & if x > 2 \end{cases}$

The graph of the function $g$ is obtained by applying the following transformations to the graph of $f$ :

a translation by the vector $(\begin{matrix} 2 \\ 0 \end{matrix})$ ,

followed by

a reflection in the $x$ -axis.

Find an expression for $g (x)$ .

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2 marks

The following diagram shows the graph of $y = f (x), - 3 \leq x \leq 6 .$

q6a-2-6-hard-ib-aa-hl-maths

Write down the value of

(i)

f (- 2)

(ii)

f^{- 1} (1)

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1 mark

Find the value of $(f \circ f) (0)$ .

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2 marks

Given that $g (x) = f (x + 5) - 5$ , find the domain and range of $g$ .

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4 marks

Let $v (t) = 4 t^{2} + 64$ , where $t > 0$ .

The graph of a function $g$ is obtained when the graph of $v$ is transformed by

a vertical stretch by a factor of $\frac{1}{8}$ ,

followed by

a translation by the vector $(\begin{matrix} 8 \\ 3 \end{matrix})$ .

Find $g (t)$ , giving your answer in the form $a t^{2} + b t + c$ .

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2 marks

A particle moves along a straight line so that its velocity in ${ms}^{- 1}$ , at time $t$ seconds, is given by $g (t)$ .

Find the value of $t$ when the particle’s velocity is 11 ${ms}^{- 1}$ .

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4 marks

Let $f (x) = 2 x^{2} - 6 x$ , for $x \in ℝ .$

Sketch the graph of $y = f (x)$ on the grid below, clearly labelling the vertex as well as any intersections the graph makes with the coordinate axes.

q8a-2-6-hard-ib-aa-hl-maths

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5 marks

The graph of a function $g$ is obtained from the graph of $f$ by a reflection in the $y$ -axis, followed by a horizontal stretch with scale factor $\frac{1}{2}$ .

Find an expression for $g (x)$ , giving your answer in the form $g (x) = a (x - h)^{2} + k$ .

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3 marks

Let $f (x) = 2 x^{2} + b x + 8$ , for $x \in ℝ$ , where $b \in ℤ$ .

Given that the equation $f (x) = 0$ has two equal roots, and that $b < 0$ ,

find the value of $b$ .

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2 marks

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

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3 marks

The graph of a function $g$ is obtained from the graph of $f$ by a reflection in the $y$ -axis, followed by a horizontal stretch with scale factor 2.

Find an expression for $g (x)$ and state the coordinates of the $y$ -intercept of the graph of $g$ .

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10a

5 marks

Let $f (x) = 2 x^{2} - 12 x + 10$

For the graph of $f$ , find

(i)

the

x

-intercepts

(ii)

the

y

-intercept

(iii)

the coordinates of the vertex.

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10b

3 marks

The graph of a function $g$ is obtained from the graph of $f$ by a reflection in the $x$ -axis followed by a translation by the vector $(\begin{matrix} 1 \\ 6 \end{matrix})$ .

Find $g (x)$ , giving your answer in the form $g (x) = a (x - h)^{2} + k$ .

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11a

1 mark

Consider the functions $f$ and $g$ defined by $f (x) = \ln x$ and $g (x) = \ln (2 x + 5)$ , where each function has the largest possible valid domain.

Write down the domain of $g$ .

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11b

4 marks

The graph of $f$ can be transformed onto the graph of $g$ by a single translation and a single stretch, both of which are parallel to one of the coordinate axes.

Describe the sequence of transformations in the case where:

(i)

the translation occurs first.

(ii)

the stretch occurs first.

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11c

3 marks

The graph of $f$ can be also transformed onto the graph of $g$ by a single translation using the vector $(\begin{matrix} a \\ b \end{matrix})$ .

Find the exact values of $a$ and $b$ .

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12a

6 marks

The graph of a function $f$ is shown below. The points A $(1, 6)$ and B $(3, - 3)$ lie on the graph and are a local maximum and a local minimum respectively. The $x$ -axis is an asymptote to the graph.

q12a-2-6-hard-ib-aa-hl-maths

On separate sets of axes, sketch the graphs of:

(i)

y = 2 (f (x) + 3)

(ii)

y = f (2 x + 3)

In each case give the coordinates of the points onto which A and B are mapped, and state the equation of the asymptote.

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12b

5 marks

The graph of $y = f (x)$ is stretched horizontally by a scale factor of $k$ then translated by the vector $(\begin{matrix} a \\ b \end{matrix})$ to map it onto the graph of $y = f (5 x + 10) + 4$ .

(i)

Find the values of

a, b

and

k

(ii)

Find the coordinates of the points onto which A and B are mapped.

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13a

2 marks

Consider the function $f$ defined by $f (x) = 0.4 e^{x + 1} - 3, - 6 \leq x \leq 3 .$

Find the coordinates of

(i)

the

x

-intercept

(ii)

the

y

-intercept

of the graph of $y = f (x)$ .

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13b

2 marks

Sketch the graph of $y = f (x)$ .

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13c

2 marks

The graph of $f$ is first reflected in the $y$ -axis and then translated by the vector $(\begin{matrix} 2 \\ 1 \end{matrix})$ to obtain the graph of a function $g$ .

Find an expression for $g (x)$ .

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4 marks

The point $P (- 1, 4)$ lies on the curve with equation $y = f (x)$ .

State the coordinates of the image of point $P$ on the curves with the following equations:

(i)

y = f (x) + 3

(ii)

y = f (x + 3)

(iii)

y = 3 f (x)

(iv)

y = f (3 x)

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2 marks

The point $P (- 3, - 4)$ lies on the curve with equation $y = f (x)$ .

State the coordinates of the image of point $P$ on the curves with the following equations:

(i)

y = f (- x)

(ii)

y = - f (x)

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4 marks

The point $P (3, 2)$ lies on the curve with equation $y = f (x)$ .

(i)

On the graph of

y = f (x) + a

, where

a

is a constant, the point

P

is mapped to the point

(3, - 5)

. Determine the value of

a

(ii)

On the graph of

y = f (x + b)

, where

b

is a constant, the point

P

is mapped to the point

(- 1, 2)

. Determine the value of

b

(iii)

On the graph of

y = c f (x)

, where

c

is a constant, the point

P

is mapped to the point

(3, 1)

. Determine the value of

c

(iv)

On the graph of

y = f (d x)

, where

d

is a constant, the point

P

is mapped to the point

(1, 2)

. Determine the value of

d

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4 marks

The diagram below shows the graph of $y = f (x)$ . The two marked points $A (- 1, 5)$ and $B (3, - 3)$ lie on the graph.

q2a-2-6-medium-ib-aa-hl-maths

In separate diagrams, sketch the curves with equation

(i)

y = f (x - 1)

(ii)

y = f (x) + 3

On each diagram, give the coordinates of the images of points $A$ and $B$ under the given transformation.

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4 marks

The diagram below shows the graph of $y = f (x)$ . The marked point $B (4, 8)$ lies on the graph, and the graph meets the origin at the marked point $A$ .

q5a-2-6-medium-ib-aa-hl-maths

In separate diagrams, sketch the curves with equation

(i)

y = - f (x)

(ii)

y = f (4 x)

On each diagram, give the coordinates of the images of points $A$ and $B$ under the given transformation.

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2 marks

On the graph of $y = a f (x)$ the image of one of the two marked points has a $y$ coordinate of 4. Find the value of $a$ .

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6 marks

The diagram below shows the graph of $y = f (x)$ . The graph intersects the coordinate axes at the two marked points $A (0, 6)$ and $B (3, 0)$ . The graph has two asymptotes as shown, with equations $y = \frac{20}{3}$ and $x = \frac{10}{3}$ .

q6a-2-6-medium-ib-aa-hl-maths

In separate diagrams, sketch the curves with equation

(i)

y = f (x) - 6

(ii)

y = f (- x)

On each diagram, give the coordinates of the images of points $A$ and $B$ under the given transformation, as well as stating the equations of the transformed asymptotes.

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2 marks

The graph of $y = f (x + a)$ has an asymptote at one of the coordinate axes. Find the value of $a$ .

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3 marks

Describe, in order, a sequence of transformations that maps the graph of $y = f (x)$ onto the following graphs:

(i)

y = 3 f (x + 2)

(ii)

y = f (- x) - 1

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4 marks

Given that $f (x) = 3 x^{2} - 2 x$ find an expression for $g (x)$ , where $g (x)$ is obtained by applying the following sequence of transformations to $f (x)$ .

Translation by $(\begin{matrix} 2 \\ 0 \end{matrix})$
Vertical stretch of scale factor 4
Translation by $(\begin{matrix} 0 \\ - 3 \end{matrix})$

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4 marks

(i)

Sketch the graph of

y = p (x)

, where

p (x) = 3 x - 4

(ii)

On the same set of axes, sketch the graph of

y = p^{- 1} (x)

.
Label the coordinates of the points where each graph crosses the coordinate axes.

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4 marks

(i)

Find an expression for

p^{- 1} (x)

(ii)

Find an expression for

\frac{1}{9} [p (x) + 16]

(iii)

What can you deduce about the sequence of transformations given by

\frac{1}{9} [p (x) + 16]

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10a

2 marks

The equation $y = f (x)$ , where $f (x) = (x - a)^{2}$ , with $a > 1$ , is shown below.

q10a-2-6-medium-ib-aa-hl-maths

The points $A$ and $B$ are the points where the graph intercepts the coordinate axes.

Write down, in terms of $a$ , the coordinates of $A and B$ .

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10b

3 marks

Sketch the graph of $y = - f (- x)$ , labelling the images of the points $A$ and $B$ stating their coordinates in terms of $a$ .

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10c

1 mark

Write down the value of $a$ such that the point $A$ is three times as far from the origin as the point $B$ .

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3 marks

The function $f (x)$ is to be transformed by a sequence of functions, in the order detailed below:

A horizontal stretch by scale factor 2
A reflection in the $x$ -axis
A translation by $(\begin{matrix} 0 \\ 2 \end{matrix})$

Write down an expression for the combined transformation in terms of $f (x)$ .

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12a

2 marks

The diagram shows the graph of $y = f (t)$ , where $f (t) = \sin 2 t, 0 ° \leq x \leq 180 ° .$

q12a-2-6-medium-ib-aa-hl-maths

(i)

Write down the maximum value of

y

when

y = 3 f (t)

(ii)

Write down the first value of

t

for which this maximum occurs.

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12b

2 marks

(i)

Write down the minimum value of

y

when

y = 5 f (t + 30 °)

(ii)

Write down the first value of

t

for which this minimum occurs.

How did you do?

12c

2 marks

Find, in terms of $f (t)$ , the combination of transformations that would map the graph of $y = f (t)$ onto the graph of $y = 2 + \sin t, 0 ° \leq x \leq 180 ° .$

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13a

1 mark

Let $f (x) = 3 x^{2} + 18 x + 27 .$

Write down the value of $f (- 3) .$

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13b

3 marks

The function $f$ can be written in the form of $f (x) = a (x - h)^{2} + k .$

Find the values of $a, h$ and $k .$

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13c

4 marks

The graph of $g$ is obtained from the graph of $f$ by a reflection in the $x$ -axis followed by a translation by the vector $(\begin{matrix} 0 \\ 1 \end{matrix})$ .

Find $g (x)$ , giving your answer in the form of $g (x) = r x^{2} + s x + t$ .

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7 marks

The graph of $f$ is shown below. The points $A (- 2, 10)$ and $B (4, - 10)$ lie on the curve.

q14a-2-6-medium-ib-aa-hl-maths

Sketch the graph of:

(i)

y = f (2 x - 1)

(ii)

y = f (4 - x)

Clearly indicate the new coordinates of the images of the points A and B.

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4 marks

Describe a sequence of transformations that map the graph of $y = \ln x$ onto the graph of $y = 5 + \ln (\frac{1}{2} x + 4) .$

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16a

3 marks

The function $f$ is defined by

$f (x) = \{\begin{cases} a x + 1 & if x \leq 7, \\ x^{2} - 2 x + 1 & if x > 7 . \end{cases}$

Find the value of $a$ such that the graph of $f$ is continuous at $x = 7$ .

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16b

4 marks

The graph of the function $g$ is obtained by translating the graph of $f$ by the vector $(\begin{matrix} 1 \\ - 1 \end{matrix})$ , followed by a reflection in the $x$ -axis.

Find $g (x)$ .

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5 marks

Let $f (x) = \frac{1}{x}$ and $g (x) = \frac{x + 1}{x - 2} .$

Explain fully the transformations of the graph of $f$ to obtain the graph of $g$ .

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4 marks

Let $f (x) = \frac{4}{3} (x - 5)^{3} - 2$ and $g (x) = x^{3},$ for $x \in ℝ .$

Give a full geometric description of two individual transformations that can be combined to obtain the graph of f from the graph of g, given that:

(i) a stretch is to be applied first, followed by a translation

(ii) a translation is to be applied first, followed by a stretch.

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4 marks

The graph of f is translated by the vector $(\begin{matrix} - 3 \\ 6 \end{matrix})$ to give the graph of h.

Now consider h as a transformation of g. The point where $g (x) = - 27$ is translated to point A on the graph of h.

Find the coordinates of A.

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3 marks

Let f and g be functions defined for $x \in ℝ$ such that $g (x) = f (2 x + 6) - 4 .$

The graph of $g$ is obtained from the graph of f after the follow transformations:

a horizontal stretch by a factor of v,

followed by

a translation by the vector $(\begin{matrix} a \\ b \end{matrix})$ .

Write down the values of

(i)

(ii)

(iii)

How did you do?

5 marks

The point A (2 , 0) on the graph of $f$ is mapped to point B on the graph of g.

Find the distance between points A and B, giving your answer in the form $p \sqrt{q}$ where p and q are integers to be found, and where q has no square number factors.

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4 marks

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

Show that the graph of is a translation of the graph of f, and find the vector that translates the graph of f onto the graph of g.

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6 marks

The diagram below shows parts of the graphs of f and g. Point A is the y-intercept of f, point B is the intersection between the graphs of f and g and point C is the y-intercept of g.

q3-2-5-transformations-of-graphs-veryhard-ib-aa-sl-maths

Find the area of the triangle that has points A, B and C as its vertices.

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4 marks

Let $f (x) = \frac{1}{3} e^{- x - 2} - 1,$ for $- 4 \leq x \leq 4 .$

Sketch the graph of $y = f (x)$ on the grid below, clearly labelling any intersections the graph makes with the coordinate axes.

q4-2-5-transformations-of-graphs-veryhard-ib-aa-sl-maths

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3 marks

The graph of f is reflected in the x-axis and then translated by the vector $(\begin{matrix} - 1 \\ - 1 \end{matrix})$ to obtain the graph of $y = g (x)$ .

Show that the equation $g (x) = 0$ has no solutions.

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4 marks

The function f is defined by

$f (x) = \{\begin{cases} 4 - 3 x i f & x \leq 1 \\ 2 x^{2} - 2 x + 1 i f & x > 1 \end{cases}$

The graph of the function g is obtained by applying the following transformations to the graph of f:

a reflection in the y-axis.

followed by

a translation by the vector $(\begin{matrix} - 1 \\ - 4 \end{matrix})$ ,

Find $g (x)$ .

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4 marks

Let $f (x) = - 2 x^{3} + 54,$ , where $x < 0$ .

The graph of a function g is obtained when the graph of f is transformed by

a reflection in the y-axis,

followed by

a vertical stretch by a factor of $\frac{5}{4}$ .

(i)

Find

g (x)

, giving your answer in the form

a x^{3} + b

(ii)

Write down the domain of g.

How did you do?

2 marks

A particle moves along a straight line so that its velocity in m s^-1 at time x seconds is given by $g (x)$

Find the value of x when the particle’s velocity is 85 m s^-1.

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3 marks

Let $f (x) = a x^{2} - 12 x + c,$ $x \in ℝ,$ where $a, c \in ℤ^{-}$ .

The equation $f (x) = 0$ has two equal roots.

Given that $4 a = c$ , find the values of a and c.

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2 marks

Find the coordinates of the vertex of the graph of f.

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3 marks

The graph of a function g is obtained from the graph of f by a reflection in the y -axis, followed by a horizontal stretch by a factor of $\frac{2}{3}$ .

Find an expression for $g (x)$ , along with the coordinates of the y -intercept of the graph of g.

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2 marks

Using the geometric nature of the two transformations by which the graph of g was obtained from the graph of f, explain why the graphs of f and g must have the same y -intercept.

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6 marks

Let $f (x) = \frac{(x - p) (x - q)}{2}, x \in ℝ$ where $p, q \in ℤ .$ The graph of f has x- intercepts at $(p, 0)$ and $(q, 0),$ , with $(p, 0)$ lying on the negative x-axis and $(q, 0),$ lying on the positive x-axis. The y-intercept of the graph is at $(0, \frac{- 7}{2})$ and the vertex of the graph lies in the fourth quadrant. This information is represented on the diagram below.

q8-2-5-transformations-of-graphs-veryhard-ib-aa-sl-maths

(i)

Find the values of p and q.

(ii)

Find the coordinates of the vertex, V.

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4 marks

The graph of a function g is obtained from the graph of f by a translation by the vector $(\begin{matrix} 2 \\ - 1 \end{matrix})$ , followed by a reflection in the y-axis. Point A on the graph of f has an x-coordinate of 1 and is mapped to point B on the graph of g.

Find the coordinates of B.

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4 marks

Consider the function $f$ defined by $f (x) = 8^{x - 2}$ .

The graph of $f$ is stretched horizontally by scale factor of 0.75 and then translated by the vector $(\begin{matrix} - 6 \\ 0 \end{matrix})$ . The function corresponding to the transformed graph is denoted $g$ .

Write down an expression for the function $g$ .

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4 marks

Consider the function $h$ defined by $h (x) = 2^{x} .$

The graph of $f$ can be transformed onto the graph of $h$ by a horizontal stretch and a vertical stretch.

Describe the stretches needed to map the graph of $f$ onto the graph of $h$ .

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10a

5 marks

Let $f$ be a function.

For each of the functions below, describe two different sequences of transformations that map the graph of $f$ onto the graph of the function. Each sequence should consist of exactly two basic transformations (translations, stretches, or reflections).

(i)

f (5 - x)

(ii)

f (\frac{x + 3}{5}) .

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10b

6 marks

The graph of the function $g$ defined by $g (x) = 1 - f (2 x - 3)$ is shown below. There is a local minimum at A $(0.5, 1)$ and a local maximum at B $(1.5, 5)$ .

q10a-2-6-hard-ib-aa-hl-maths

Sketch the graph of $f$ . Clearly state the coordinates of the points corresponding to the points A and B.

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DP IB Maths: AA HL

Topic Questions

2.6 Transformations of Graphs