Rational curve sketching with asymptotes and inequalities

Find asymptotes, sketch the rational curve, and solve related inequalities, without requiring proof of turning point existence or range restrictions.

9 questions · Standard +0.1

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Write down the value of $y$ when $x = 0$.
Write down the equations of the three asymptotes.
Sketch the curve.
Find the values of $x$ for which $\frac { 5 } { ( x + 2 ) ( 4 - x ) } = 1$ and hence solve the inequality $$\frac { 5 } { ( x + 2 ) ( 4 - x ) } < 1 .$$

Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
Describe the behaviour of the curve for large positive and large negative values of $x$, justifying your answers.
Sketch the curve.
Solve the inequality $\frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) } \leqslant 0$.

1. Write down the equations of the three asymptotes of this curve.
2. State the coordinates of the points at which the curve intersects the $x$-axis.
3. Sketch the curve.
  (You are given that the curve has no stationary points.)
Hence, or otherwise, solve the inequality $$\frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) } < 0$$

Find the equations of the asymptotes of this curve.
Given that the line $y = - 4 x + c$ intersects the curve, show that the $x$-coordinates of the points of intersection must satisfy the equation $$4 x ^ { 2 } - ( c + 3 ) x + c = 0$$
It is given that the line $y = - 4 x + c$ is a tangent to the curve.
1. Find the two possible values of $c$.
  (No credit will be given for methods involving differentiation.)
2. For each of the two values found in part (c)(i), find the coordinates of the point where the line touches the curve.

1. Write down the equations of the two asymptotes of the curve $y = \frac { 1 } { x - 3 }$.
2. Sketch the curve $y = \frac { 1 } { x - 3 }$, showing the coordinates of any points of intersection with the coordinate axes.
3. On the same axes, again showing the coordinates of any points of intersection with the coordinate axes, sketch the line $y = 2 x - 5$.
1. Solve the equation $$\frac { 1 } { x - 3 } = 2 x - 5$$
2. Find the solution of the inequality $$\frac { 1 } { x - 3 } < 2 x - 5$$ □ \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-08_367_197_2496_155}

1. Write down the equations of the three asymptotes of the curve.
2. Sketch the curve, showing the coordinates of any points of intersection with the coordinate axes.
Hence, or otherwise, solve the inequality $$\frac { 1 } { x ^ { 2 } - 4 } < - 2$$

Write down the equations of the asymptotes of $C$.
The line $L$ intersects the curve $C$ at two points. Find the $x$-coordinates of these two points.
Sketch $C$ and $L$ on the same axes.
(You are given that the curve $C$ has no stationary points.)
Solve the inequality $$\frac { x } { ( x + 1 ) ( x - 2 ) } \leqslant - \frac { 1 } { 2 }$$