Transformations of Graphs (AQA GCSE Maths: Higher): 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 equation of a curve is $y=a^x$
$A$ is the point where the curve intersects the $y$-axis.

State the coordinates of $A$.

The equation of circle $\mathbf{C}$ is $x^2+y^2=16$
The circle $\mathbf{C}$ is translated by the vector $\left(\begin{array}{c}3\\ 0\end{array}\right)$ to give circle $\mathbf{B}$.

Draw a sketch of circle $\mathbf{B}$.

Label with coordinates    
the centre of circle $\mathbf{B}$    
and any points of intersection with the $x$-axis.

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.

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 a quadratic function has equation $y=\mathrm{f}\left(x\right)$, where $\mathrm{f}\left(x\right)=2x^2$ .

Describe the transformation of the graph when the equation is changed to $y=\mathrm{f}\left(x+2\right)$.

Find an expression for $\mathrm{f}\left(x+2\right)$ in terms of $x$.

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 $y=k^x$, where $k$ is a positive constant, is shown above.

Find the value of $k$.

The graph of $y=\sin x^{\circ }$ for values of $x$ from $—270\mathrm{to}+270$ is shown above.

On the same axes, draw the graph of $y=1-\sin x^{\circ }$ for values of $x$ from $—270\mathrm{to}+270$

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 $y=x^3+6$ 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 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 $\mathbf{S}$ 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 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 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.