# Graphing Techniques

###### 5 Essential Questions

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##### Graphing Techniques Q1: Prelims Revision

The curve $C$ has equation $y=\frac{{{x}^{2}}+ax+b}{x+c}$, where $a$, $b$ and $c$ are constants. The line $x=3$ is an asymptote to $C$ and the range of values that $y$ can take is given by $y\le -2$ or $y\ge 4$. Find the values of $a$, $b$ and $c$.



##### Suggested Handwritten and Video Solutions  ##### ACJC Tutorial 2 Q8

It is given that $y=\frac{2{{x}^{2}}+3}{x-2},x\in \mathbb{R},x\ne 2$.

(i)

Sketch the graph of $y=\frac{2{{x}^{2}}+3}{x-2}$ and label clearly the equations of the asymptotes and intercepts if any.

(i) Sketch the graph of $y=\frac{2{{x}^{2}}+3}{x-2}$ and label clearly the equations of the asymptotes and intercepts if any.

(ii)

The graph of $y=\frac{2{{x}^{2}}+3}{x-2}$ cannot lie between values $p$ and $m$, where $p>m$ . State the value of $p$ and $m$ .

(ii) The graph of $y=\frac{2{{x}^{2}}+3}{x-2}$ cannot lie between values $p$ and $m$, where $p>m$ . State the value of $p$ and $m$ .

(iii)

By means of a graphical argument, state the maximum number of real roots the equation $kx\left( x-2 \right)-2{{x}^{2}}=3$ have.

(iii) By means of a graphical argument, state the maximum number of real roots the equation $kx\left( x-2 \right)-2{{x}^{2}}=3$ have.

##### Suggested Handwritten and Video Solutions
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• (iii)    ##### 2013 YJC P1 Q12

The curve $C$ has equation $y=\frac{{{x}^{2}}}{x-2}$.

(i)

Find the equation(s) of the asymptote(s) of $C$.



(i) Find the equation(s) of the asymptote(s) of $C$.



(ii)

Sketch the curve $C$, labelling the equation(s) of its asymptote(s) and coordinates of any axial intercepts and turning points.



(ii) Sketch the curve $C$, labelling the equation(s) of its asymptote(s) and coordinates of any axial intercepts and turning points.



(iii)

Hence find the range of values of $k$ for which the equation ${{x}^{2}}=k\left( {{x}^{2}}-4 \right)$ has no real roots.

(iii) Hence find the range of values of $k$ for which the equation ${{x}^{2}}=k\left( {{x}^{2}}-4 \right)$ has no real roots.

##### Suggested Handwritten and Video Solutions
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• (iii)   ##### 2013 MJC Promo Q6

The curve $C$ has equation $y=\sqrt{5{{x}^{2}}+4}$.

(i)

Sketch curve $C$, indicating clearly the axial intercepts, the equations of the asymptotes and the coordinates of the stationary points.

(i) Sketch curve $C$, indicating clearly the axial intercepts, the equations of the asymptotes and the coordinates of the stationary points.

(ii)

Hence, by inserting a suitable graph, determine the range of values of $h$, where $h$ is a positive constant, such that the equation $\sqrt{5{{x}^{2}}+4}=h\sqrt{(1-{{x}^{2}})}$ has no real roots.

(ii) Hence, by inserting a suitable graph, determine the range of values of $h$, where $h$ is a positive constant, such that the equation $\sqrt{5{{x}^{2}}+4}=h\sqrt{(1-{{x}^{2}})}$ has no real roots.

##### Suggested Handwritten and Video Solutions
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• (ii)  ##### 2007 SAJC P1 Q7 [Modified]

The curve $C$ has equation $y=\frac{a{{x}^{2}}+2}{x-1}$ where $x\ne 1$ and $a$ is a non-zero constant.

(i)

Show that if the curve $C$ has no stationary points, then $-2<a<0$.

(i) Show that if the curve $C$ has no stationary points, then $-2<a<0$.

(ii)

Sketch the curve for $a=1$, showing clearly the asymptotes and coordinates of any intersections with the coordinate axes.

(ii) Sketch the curve for $a=1$, showing clearly the asymptotes and coordinates of any intersections with the coordinate axes.

(iii)

Verify that $y=k(x-1)+2$ passes through $(1,2)$ for all real values of $k$.

(iii) Verify that $y=k(x-1)+2$ passes through $(1,2)$ for all real values of $k$.

(iv)

By considering the equation of an appropriate line drawn on the same diagram with the curve $C$, find the range of values of $k$ for which the equation ${{x}^{2}}+2=k{{(x-1)}^{2}}+2(x-1)$ has no real roots.

(iv) By considering the equation of an appropriate line drawn on the same diagram with the curve $C$, find the range of values of $k$ for which the equation ${{x}^{2}}+2=k{{(x-1)}^{2}}+2(x-1)$ has no real roots.

##### Suggested Handwritten and Video Solutions
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