Transformations of Graphs (Edexcel IGCSE Maths A (Modular): Higher Unit 2): Exam Questions

The diagram shows part of the curve with equation $y=\mathrm{f}\left(x\right)$. The coordinates of the maximum point of the curve are (3, 5).

Write down the coordinates of the maximum point of the curve with equation

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

[1]

(ii) $y\mathit{=}\mathit{-}\mathrm{f}\mathit{\left(}\mathit{x}\mathit{\right)}$

[1]

(iii) $y\mathit{=}\mathrm{f}\mathit{(}\mathit{-}\mathit{x}\mathit{)}$

[1]

The curve with equation $y=\mathrm{f}\left(x\right)$ is transformed to give the curve with equation $y=\mathrm{f}\left(x\right)-4$

Describe the transformation.

The diagram shows a sketch of the graph of $y=\cos x.$

Write down the coordinates of the point $A$.

On the same diagram, draw a sketch of the graph of $y=-\cos x.$

Here is the graph of $y=\sin x^{\circ }$ for  $-180⩽x⩽180$

On the grid, sketch the graph of $y=\sin x^{\circ }-2$ for $—180⩽x⩽180$

Here is the graph of $y=\sin x^{\circ }$ for  $-180⩽x⩽180$

On the grid above, sketch the graph of $y=\sin x^{\circ }+2$ for $—180⩽x⩽180$

Here is the graph of $y=\cos x^{\circ }$ for  $-180⩽x⩽180$

On the grid above, sketch the graph of $y=-\cos x^{\circ }$ for $—180⩽x⩽180$

The graph of $y=\mathrm{f}\left(x\right)$ is shown on the grids.

On this grid, sketch the graph of $y=\mathrm{f}\left(x-3\right)$

On this grid, sketch the graph of $y=—\mathrm{f}\left(x\right)$

$y=\mathrm{f}\left(x\right)$

The graph of $y=\mathrm{f}\left(x\right)$ is shown on the grid.

On the grid above, sketch the graph of $y=-\mathrm{f}\left(x\right)$.

The graph of  $y=\mathrm{f}\left(x\right)$is shown on the grid.

The graph G is a translation of the graph of $y=\mathrm{f}\left(x\right).$

Write down the equation of graph G.

The graph of  $y=a\cos (x+b)°$  for $0⩽x⩽360$ is drawn on the grid below.

Given that $a>0$ and that $0<b<360$

find the value of a and the value of $b$.

$\begin{array}{rcl}a& =& ...................................\\ b& =& ...................................\end{array}$

The graph of $y=\mathrm{g}\left(x\right)$ is shown on the grid below.

On the grid, sketch the graph of $y=2\mathrm{g}\left(x\right)$ for $–1⩽x⩽7$

The graph of $y=\mathrm{f}\left(x\right)$ is shown on the grid.

On the grid above, sketch the graph of $y=\mathrm{f}\left(\frac{1}{2}x\right)$

The graph of $y=\mathrm{f}(x+k)$ is shown on the grid below.

Write down the value of $k$

The diagram shows part of the curve with equation $y=\mathrm{f}\left(x\right)$

The coordinates of the minimum point on this curve are (3, –4)

Write down the coordinates of the minimum point on the curve with equation

(i) $y=\mathrm{f}(x–4)$

[1]

(ii) $y=3\mathrm{f}\left(x\right)$

[1]

(iii) $y=\mathrm{f}\left(\frac{1}{2}x\right)$

[1]

The curve with equation $y=\mathrm{f}\left(x\right)$  is translated to give curve $C$.

$C$ has a minimum at the point with coordinates (3, 5)

The equation of $C$ is $y=\mathrm{f}\left(x\right)+k$

Write down the value of $k$.

$k=$.......................

Here is the graph of $y=\cos x$for $0°⩽x⩽360°$

In parts (a) and (b) the graph of $y=\cos x$ is shown as a dashed line.

On the grid below, draw the graph of $y=\cos (x–90°)$ for $0°⩽x⩽360°$

On the grid below, draw the graph of $y=1+\cos x\mathrm{for}0°⩽x⩽360°$

Rita tries to draw the graph of$y=\cos (–x)$ for$0°⩽x⩽360°$ Here is her graph.

Give a reason why Rita’s graph is incorrect.

Here is a sketch of    $y=\mathrm{f}\left(x\right)$

The curve passes through the points    (–2, –10)  (–1, –3)  (0, –2)  (1, –1)  (2, 6)

On the grid, sketch the curve $y=\mathrm{f}(x+2)$

The curve with equation   $y=x^2-5x+2$ is reflected in the $x$-axis.

Circle the equation of the reflected curve.

The graph of $y=\cos \left(x-30\right)$ for  $0°⩽x⩽360°$ crosses the $x$-axis in two places.

Write down the values of $x$ where this occurs.

$x=$...........................

Sketch the graph of $y=-\sin x$ for $0°⩽x⩽360°$.

The graph of $y=\mathrm{f}\left(x\right)$ is shown on each of the grids.

On this grid, sketch the graph of $y=\mathrm{f}(x-3)$

On this grid, sketch the graph of $y=\mathrm{f}(-x)+2$

The graph of $y=\mathrm{f}\left(x\right)$ is shown on both grids below

On the grid above, sketch the graph of $y=\mathrm{f}\left(-x\right)$

On this grid, sketch the graph of $y=-\mathrm{f}\left(x\right)+3$

This is a sketch of the curve with the equation $y=\mathrm{f}\left(x\right).$

The only minimum point of the curve is at $P(3,—4)$.

Write down the coordinates of the minimum point of the curve with the equation $y=\mathrm{f}\left(x-2\right)$

Write down the coordinates of the minimum point of the curve with the equation  $y=\mathrm{f}\left(x+5\right)+6$

The graph of $y=\mathrm{f}\left(x\right)$ is transformed to give the graph of $y=—\mathrm{f}(x+3)$

The point $A$ on the graph of $y=\mathrm{f}\left(x\right)$ is mapped to the point $P$ on the graph of $y=—\mathrm{f}(x+3)$

The coordinates of point $A$ are $(9,1)$

Find the coordinates of point $P$.

The graph of the curve $\mathrm{C}$ with equation $y=\mathrm{f}\left(x\right)$ is transformed to give the graph of the curve $\mathrm{S}$ with equation $y=\mathrm{f}(-x)-3$

The point on $\mathrm{C}$ with coordinates $(7,2)$ is mapped to the point $Q$ on $\mathrm{S}$.

Find the coordinates of $Q$.

The graph of $y=h\left(x\right)$ intersects the $x$-axis at two points.

The coordinates of the two points are (–1, 0) and (6, 0)

The graph of $y=h(x+a)$ passes through the point with coordinates (2, 0), where $a$ is a constant.

Find the two possible values of $a$

The curve $C$ has equation $y=\mathrm{f}\left(x\right)$ where $\mathrm{f}\left(x\right)=9–3{(x+2)}^2$

The point $A$ is the maximum point on $\mathbf{C}$.

Write down the coordinates of $A$.

The curve $\mathbf{C}$ is transformed to the curve $\mathbf{S}$ by a translation of $\left(\begin{array}{c}4\\ 0\end{array}\right)$ Find an equation for the curve $\mathbf{S}$.

The curve $\mathbf{C}$ is transformed to the curve $\mathbf{T}$.
The curve $\mathbf{T}$ has equation $y=3{(x+2)}^2–9$

Describe fully the transformation that maps curve $\mathbf{C}$ onto curve $\mathbf{T}$.

The curve $\mathbf{S}$ has equation$y=\mathrm{f}\left(x\right)$where$\mathrm{f}\left(x\right)=x^2$

The curve $\mathbf{T}$ has equation $y=\mathrm{g}\left(x\right)$where $\mathrm{g}\left(x\right)=2x^2–12x+13$

By writing $\mathrm{g}\left(x\right)$ in the form $a{(x–b)}^2–c$, where $a,b$ and $c$ are constants, describe fully a series of transformations that map the curve $\mathbf{S}$ onto the curve $\mathbf{T}$.

The graph of$y=a\cos (x–b)°+c$ for $-180⩽x⩽360$is drawn on the grid below.

Find the value of $a$, the value of $b$ and the value of $c$.

a = .......................................................       
b = .......................................................       
c = ....................................................... 

The equation of a curve $\mathbf{C}$is $y=x^2+3x+4$ The curve $\mathbf{C}$is transformed to curve $\mathbf{S}$under the translation $\left(\begin{array}{c}4\\ 6\end{array}\right)$

Find an equation of curve $\mathbf{S}$.

You do need to simplify the equation.

The function f($x$) is defined as ƒ ($x$) = 3 − 8$x$ − 2$x^2$

Find the coordinates of the turning point on the graph of $y$ = $f\left(x\right)$

Using your result from part (a) write down the coordinates of the turning point on the following graphs:

(i) $y$ = $f\left(x\right)$ +2

[1]

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

[1]

(iii) $y=f\left(4x\right)$

[1]

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

[1]

(v) $y=3f\left(x+1\right)$

[2]

The graph of $y=x^3+6$ is translated 4 units to the right.

The translated graph has equation $y=\mathrm{f}\left(x\right)$

Work out $\mathrm{f}\left(x\right)$.

Give your answer in the form $x^3+a{x}^2+bx+c$ where $a,b$ and $c$ are integers.

Curve $\mathrm{P}$ has equation $y=2{(x–1)}^2–5$

Curve $\mathrm{Q}$ is a reflection in the $y$-axis of curve $\mathrm{P}$.

Work out the equation of curve $\mathrm{Q}$.

Give your answer in the form    $y=a{x}^2+bx+c$     where $a,b$ and $c$ are integers.

For all values of $x$

$$\begin{gathered} \mathrm{f}\left(x\right)=\sin x \\ \mathrm{g}\left(x\right)=x+90 \end{gathered}$$

On the grid, draw the graph of the composite function  $y=\mathrm{fg}\left(x\right)$  for  $0°⩽x⩽360°$

The diagram shows part of the curve with equation $y=\mathrm{f}\left(x\right)$

The coordinates of the minimum point on this curve are (3, –4)

Write down the coordinates of the minimum point on the curve with equation

(i) $y=\mathrm{f}(x–4)$

(ii) $y=3\mathrm{f}\left(x\right)$

(iii) $y=\mathrm{f}\left(\frac{1}{2}x\right)$

The curve with equation $y=\mathrm{f}\left(x\right)$ is translated to give curve $C$.

$C$ has a minimum at the point with coordinates (3, 5)
The equation of $C$ is $y=\mathrm{f}\left(x\right)+k$

Write down the value of $k$

A curve has equation $y=\mathrm{f}\left(x\right)$

There is only one turning point on the curve.
The coordinates of this turning point are (6, 5)

Write down the coordinates of the turning point on the curve with equation

$y=\mathrm{f}\left(x-4\right)$

$y=\mathrm{f}\left(3x\right)$

The curve with equation $\mathrm{f}\left(x\right)=5x^2+9x+2$ is transformed to the curve with equation $\mathrm{g}\left(x\right)=5{\left(x+4\right)}^2+9\left(x+4\right)+8$ by the translation $\left(\begin{array}{c}a\\ b\end{array}\right)$

Write down the value of $a$ and the value of $b$

The graph of $y=p\cos {\left(x+q\right)}^{°}$ for $0⩽x⩽360$ is drawn on the grid below

Given that $p>0$ and $0<q<360$, find the value of $p$and the value of $q$ .