Transformations of Graphs

5 Essential Questions

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• Q5
2021 NJC Lecture Test (b)

The diagram below shows a sketch of the curve with equation $y=\text{g}(x)$. The curve has asymptotes with equations$x=0$ and $y=2$. The curve crosses the $x$-axis at $A(2,0)$ and $C(7,0)$ and has a turning point at $B(4,-2)$.

On separate diagrams, sketch the curves with equations

(i)

$y=\left| \text{g}(-x) \right|$,

[3]

(i) $y=\left| \text{g}(-x) \right|$,

[3]

(ii)

$y=\frac{1}{\text{g}(x)}$.

[4]

(ii) $y=\frac{1}{\text{g}(x)}$.

[4]

For each curve, state clearly the equations of any asymptotes, the coordinates of any turning points and the coordinates of any points where the curves crosses the $x$- and $y$- axes.

• (i)
• (ii)
• (i)
• (ii)

ACJC Tutorial 3 Q9

A graph with the equation $y=\text{f}(x)$ undergoes, in succession, the following transformations:

$A$: A translation of $1$ unit in the direction of the positive $x$- axis.
$B$: A stretch parallel to the $x$- axis by factor $\frac{1}{2}$.
$C$: A reflection in the $y$- axis.

The equation of the resulting curve is $y=\frac{2}{2{{x}^{2}}+2x+1}$. Determine the equation $y=\text{f}(x)$.

2013 VJC P1 Q6 (i)

The diagram shows the graph of $y=\text{f}(x)$ which has a minimum point at $(0, 2)$ and asymptotes $x=2$ and $y=5.$

On separate diagrams, sketch the graphs of $y\ =\ \ \frac{1}{\text{f}(x)}$.

2020 VJC Promo Q8

(a)

The graph of $y=\text{f}(x)$ is shown in the diagram. It passes through the origin and has a minimum point at $\left( 2,\text{ }0 \right)$. The lines $x=1$ and $y=2$ are the asymptotes of the graph.Sketch the graph of $y=2-\text{f}(x)$, stating the coordinates of the point where the graph crosses the $y$-axis, the coordinates of any stationary point(s) and the equations of any asymptotes.

[2]

(a) The graph of $y=\text{f}(x)$ is shown in the diagram. It passes through the origin and has a minimum point at $\left( 2,\text{ }0 \right)$. The lines $x=1$ and $y=2$ are the asymptotes of the graph.Sketch the graph of $y=2-\text{f}(x)$, stating the coordinates of the point where the graph crosses the $y$-axis, the coordinates of any stationary point(s) and the equations of any asymptotes.

[2]

(b)

The curve $C$ has equation $y=\frac{2{{x}^{2}}+kx-2}{x+2}$, where $k$ is a constant.

(b) The curve $C$ has equation $y=\frac{2{{x}^{2}}+kx-2}{x+2}$, where $k$ is a constant.

(i)

Find the set of values of $k$ for which $C$ has $2$ stationary points.

[3]

(i) Find the set of values of $k$ for which $C$ has $2$ stationary points.

[3]

For the rest of the question, take $k=-3$.

For the rest of the question, take $k=-3$.

(ii)

By expressing the equation $y=\frac{2{{x}^{2}}-3x-2}{x+2}$ in the form $y=ax+b+\frac{c}{x+2}$, where $a$, $b$ and $c$ are constants, write down the equations of the asymptotes of $C$.

[3]

(ii) By expressing the equation $y=\frac{2{{x}^{2}}-3x-2}{x+2}$ in the form $y=ax+b+\frac{c}{x+2}$, where $a$, $b$ and $c$ are constants, write down the equations of the asymptotes of $C$.

[3]

(iii)

Hence sketch $C$, giving the equations of any asymptotes, the coordinates of any stationary points and of the points where $C$ crosses the axes.

[2]

(iii) Hence sketch $C$, giving the equations of any asymptotes, the coordinates of any stationary points and of the points where $C$ crosses the axes.

[2]

(iv)

Describe a pair of transformations which transform the graph of $y=x+\frac{6}{x+2}$ to $C$.

[2]

(iv) Describe a pair of transformations which transform the graph of $y=x+\frac{6}{x+2}$ to $C$.

[2]

• (a)
• (b)(i)
• (b)(ii)
• (b)(iii)
• (b)(iv)

• (a)
• (b)(i)
• (b)(ii)
• (b)(iii)
• (b)(iv)

2019 RI P1 Q5

The diagram shows the graph of Folium of Descartes with cartesian equation

${{x}^{3}}+{{y}^{3}}=3axy$,

where a is a positive constant. The curve passes through the origin, and has an oblique asymptote with equation $y=-x-a$.

(i)

Given that $\left( 0,0 \right)$ is a stationary point on the curve, find, in terms of a, the coordinates of the other stationary point.

[5]

(i) Given that $\left( 0,0 \right)$ is a stationary point on the curve, find, in terms of a, the coordinates of the other stationary point.

[5]

(ii)

Sketch the graph of

${{\left| x \right|}^{3}}+{{y}^{3}}=3a\left| x \right|y$,

including the equations of any asymptotes, coordinates of the stationary points and the point where the graph crosses the $x$-axis.

[3]

(ii) Sketch the graph of

${{\left| x \right|}^{3}}+{{y}^{3}}=3a\left| x \right|y$,

including the equations of any asymptotes, coordinates of the stationary points and the point where the graph crosses the $x$-axis.

[3]

• (i)
• (ii)
• (i)
• (ii)