Transformations of Graphs (DP IB Analysis & Approaches (AA): SL): Exam Questions

The following diagram shows the graph of $y=f\left(x\right)$. The points $\mathrm{A}\left(-1,5\right)$ and $\mathrm{B}\left(3,-3\right)$ lie on the graph.

On separate diagrams, sketch the graphs of

(i) $y=f\left(x-1\right)$

(ii) $y=f\left(x\right)+3$.

On each diagram, label the images of $\mathrm{A}$ and $\mathrm{B}$ and give their coordinates.

On the graph of $y=f\left(x+a\right)$, the image of one of the two points $\mathrm{A}$ and $\mathrm{B}$ has an $x$-coordinate of 2.

Find the two possible values of $a$.

The following diagram shows the graph of $y=f\left(x\right)$. The graph passes through the origin at the point $\mathrm{A}$, and the point $\mathrm{B}\left(4,8\right)$ lies on the graph.

On separate diagrams, sketch the graphs of

(i) $y=-f\left(x\right)$

(ii) $y=f\left(4x\right)$.

On each diagram, label the images of $\mathrm{A}$ and $\mathrm{B}$ and give their coordinates.

On the graph of $y=af\left(x\right)$, the image of one of the two points $\mathrm{A}$ and $\mathrm{B}$ has a $y$-coordinate of 4.

Find the value of $a$.

The following diagram shows the graph of $y=f\left(x\right)$. The graph meets the coordinate axes at the points $\mathrm{A}\left(0,6\right)$ and $\mathrm{B}\left(3,0\right)$. The graph has asymptotes with equations $y=\frac{20}{3}$ and $x=\frac{10}{3}$.

On separate diagrams, sketch the graphs of

(i) $y=f\left(x\right)-6$

(ii) $y=f\left(-x\right)$.

On each diagram, label the images of $\mathrm{A}$ and $\mathrm{B}$ and give their coordinates, and write down the equations of the asymptotes.

The graph of $y=f\left(x+a\right)$ has an asymptote along one of the coordinate axes.

Find the value of $a$.

Describe, in order, a sequence of transformations that maps the graph of $y=\mathrm{f}\left(x\right)$ onto the following graphs:

(i) $y=3\mathrm{f}(x+2)$

(ii) $y=\mathrm{f}(-x)-1$

Let $f\left(x\right)=3x^2-2x$.

The graph of $g$ is obtained from the graph of $f$ by a translation by the vector $\left(\begin{array}{c}2\\ 0\end{array}\right)$, followed by a vertical stretch with scale factor 4, followed by a translation by the vector $\left(\begin{array}{c}0\\ -3\end{array}\right)$.

Find $g\left(x\right)$, giving your answer in the form $a{x}^2+bx+c$.

(i) Sketch the graph of $y=p\left(x\right)$, where $p\left(x\right)=3x-4$.

(ii) On the same set of axes, sketch the graph of $y=p^{-1}\left(x\right)$.

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

(i) Find an expression for $p^{-1}\left(x\right)$.

(ii) Find an expression for $\frac{1}{9}\left[p\left(x\right)+16\right]$.

(iii) Hence describe a sequence of two transformations that maps the graph of $y=p\left(x\right)$ onto the graph of $y=p^{-1}\left(x\right)$.

The following diagram shows the graph of $y=f\left(x\right)$, where $f\left(x\right)={\left(x-a\right)}^2$, for $a>1$.

The graph meets the coordinate axes at the points $\mathrm{A}$ and $\mathrm{B}$.

Write down the coordinates of $\mathrm{A}$ and $\mathrm{B}$, in terms of $a$.

Sketch the graph of $y=-f\left(-x\right)$. On your sketch, label the images of $\mathrm{A}$ and $\mathrm{B}$ and give their coordinates in terms of $a$.

Write down the value of $a$ for which $\mathrm{A}$ is three times as far from the origin as $\mathrm{B}$.

The graph of $y=f\left(x\right)$ is transformed by a horizontal stretch with scale factor 2, followed by a reflection in the $x$-axis, followed by a translation by the vector $\left(\begin{array}{c}0\\ 2\end{array}\right)$, to give the graph of $y=g\left(x\right)$.

Write down an expression for $g\left(x\right)$ in terms of $f$.

The following diagram shows the graph of $y=f\left(t\right)$, where $f\left(t\right)=\sin 2t$, for $0°\le t\le 180°$.

(i) Write down the maximum value of $y$ when $y=3f\left(t\right)$.

(ii) Write down the value of $t$ at which this maximum occurs.

(i) Write down the minimum value of $y$ when $y=5f\left(t+30°\right)$.

(ii) Write down the value of $t$ at which this minimum occurs.

Describe two transformations that map the graph of $y=f\left(t\right)$ onto the graph of $y=2+\sin t$, for $0°\le t\le 360°$.

Let $f\left(x\right)=3x^2+18x+27$.

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

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

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

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 $\left(\begin{array}{c}0\\ 1\end{array}\right)$.

Find $g\left(x\right)$, giving your answer in the form $g\left(x\right)=r{x}^2+sx+t$.

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

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 $\left(\begin{array}{c}a\\ b\end{array}\right)$.

Write down the values of

(i) $v$

(ii) $a$

(iii) $b$.

The point $\mathrm{A}\left(3,4\right)$ on the graph of $f$ is mapped to the point $\mathrm{B}$ on the graph of $g$.

Find the coordinates of $\mathrm{B}$.

Let $f\left(x\right)=1.1{\mathrm{e}}^{x-1}-4$, for $-2\le x\le 3$.

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

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

Find an expression for $g\left(x\right)$.

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

Write down the value of

(i) $f\left(-2\right)$

(ii) $f^{-1}\left(1\right)$.

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

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

Let $f\left(x\right)=-2x^3+54$, 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 $g\left(x\right)$, giving your answer in the form $a{x}^3+b$.

(ii) Write down the domain of $g$.

A particle moves along a straight line so that its velocity, in ${\text{m s}}^{-1}$, at time $x$ seconds, is given by $g\left(x\right)$.

Find the value of $x$ when the particle’s velocity is 85 ${\text{m s}}^{-1}$.

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

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

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

Now consider the graph of $h$ as a transformation of the graph of $g$. The point $\mathrm{A}$ on the graph of $h$ corresponds to the point $\left(2,8\right)$ on the graph of $g$.

Find the coordinates of $\mathrm{A}$.

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

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

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

The graph of $g$ is obtained when the graph of $f$ is translated by the vector $\left(\begin{array}{c}2\\ 5\end{array}\right)$.

On the same grid, sketch the graph of $y=g\left(x\right)$ in the interval $0\le x\le 5$. Clearly label any intersections the graph makes with the coordinate axes.

Write down $g\left(x\right)$ in the form $a{x}^2+bx+c$, where $a$, $b$ and $c$ are constants to be determined.

The function $f$ is defined by

$f\left(x\right)=\{\begin{array}{cc}1+2x,& x\le 2\\ x^2-2x+5,& x>2\end{array}$

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

a translation by the vector $\left(\begin{array}{c}2\\ 0\end{array}\right)$,

followed by

a reflection in the $x$-axis.

Find an expression for $g\left(x\right)$.

Let $v\left(t\right)=4t^2+64$, where $t\ge 0$.

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 $\left(\begin{array}{c}8\\ 3\end{array}\right)$.

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

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

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

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

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

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

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

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

find the value of $b$.

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

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 $g\left(x\right)$ and state the coordinates of the $y$-intercept of the graph of $g$.

Let $f\left(x\right)=2x^2-12x+10$.

For the graph of $f$, find

(i) the $x$-intercepts

(ii) the $y$-intercept

(iii) the coordinates of the vertex.

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 $\left(\begin{array}{c}1\\ 6\end{array}\right)$.

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

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

Describe 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.

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

Now consider $h$ as a transformation of $g$. The point where $g\left(x\right)=-27$ is mapped to the point $\mathrm{A}$ on the graph of $h$.

Find the coordinates of $\mathrm{A}$.

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

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

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

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

Let $f\left(x\right)=a{x}^2-12x+c$, for $x\in \mathrm{ℝ}$, where $a$ and $c$ are negative integers.

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

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

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

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 $g\left(x\right)$, along with the coordinates of the $y$-intercept of the graph of $g$.

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.

Let $f\left(x\right)=x^2-4x$ and $g\left(x\right)=x^2+x-2$.

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

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

Find the area of triangle ABC.

The function $f$ is defined by

$f\left(x\right)=\{\begin{array}{cc}4-3x,& x\le 1\\ 2x^2-2x+1,& x>1\end{array}$

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 $\left(\begin{array}{c}-1\\ -4\end{array}\right)$.

Find $g\left(x\right)$.

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

(i) Find the values of $p$ and $q$.

(ii) Find the coordinates of the vertex, $\mathrm{V}$.

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

Find the coordinates of $\mathrm{B}$.