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

Let $f\left(x\right)=2(x+4{)}^3$ 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 A on the graph of $h$ corresponds to the point $(2,8)$ on the graph of $g$.

Find the coordinates of A.

Let $f$ and $g$ be functions such that $g\left(x\right)=2f(x-1)+2$,  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 A$(3,4)$ on the graph of $f$ is mapped to point B on the graph of $g$.

Find the coordinates of B.

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 $(f\circ f)\left(2\right).$

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

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

Let $f\left(x\right)=1.1e^{x-1}-4$ 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 function $f$ is defined by

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

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 $g\left(x\right)$.

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

Write down the value of

(i)  $f(-2)$

(ii)  $f^{-1}\left(1\right)$.

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

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

Let $v\left(t\right)=4t^2+64$, 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 $\left(\begin{array}{c}8\\ 3\end{array}\right)$.

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 ${\mathrm{ms}}^{-1}$, at time $t$ seconds, is given by $g\left(t\right)$.

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

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 $g\left(x\right)=a(x-h{)}^2+k$.

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 $g\left(x\right)=a(x-h{)}^2+k$.

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

Write down the domain of $g$.

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.

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

Find the exact values of $a$ and $b$.

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.

On separate sets of axes, sketch the graphs of:

(i)

$y=2\left(f\right(x)+3)$.

(ii)

$y=f(2x+3)$.

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

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

(i)

Find the values of $a,b$ and $k$.

(ii)

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

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

Find  the coordinates of

(i)  the $x$-intercept

(ii)  the $y$-intercept

of the graph of  $y=f\left(x\right)$.

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

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

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

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

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.

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 translated to point A on the graph of h.

Find the coordinates of A.

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

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

Write down the values of

(i)  v

(ii)  a

(iii)  b.

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.

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 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.

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

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

The function f is defined by

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

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 $f\left(x\right)=-2x^3+54,$, 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 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 m s^-1.

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

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 $f\left(x\right)=\frac{(x-p)(x-q)}{2},x\in \mathrm{ℝ}$ where $p,q\in \mathrm{\mathbb{Z}}.$The graph of f has x- intercepts at $(p,0)$ and $\left(q,0\right)$, 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 $\left(0,-\frac{7}{2}\right)$  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, 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 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.

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

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

Write down an expression for the function $g$.

Consider the function $h$ defined by $h\left(x\right)=2^x.$

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 $h$.

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

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

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