1.02n Sketch curves: simple equations including polynomials

5 A curve has equation \(y = \frac { x ^ { 2 } + 5 x - 6 } { x + 3 }\) for \(x \neq - 3\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points on the curve.
2. Sketch the curve, justifying all significant features.

9

1. On the same axes, sketch the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
2. Find the exact area of the region contained between the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).

3 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } - 4 }\).

1. Show that the gradient of \(C\) is always negative.
2. Sketch \(C\), showing all significant features.

3

1. Express \(x ^ { 2 } + 2 x - 3\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
2. Sketch the graph of \(y = x ^ { 2 } + 2 x - 3\) giving the coordinates of the vertex and of any intersections with the coordinate axes.

10 Let \(\mathrm { f } ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 10 x ^ { 2 } + 28 x - 15\).

1. Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\).
2. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(x - 5\).
3. Factorise \(\mathrm { f } ( x )\) fully and hence sketch the graph of \(y = \mathrm { f } ( x )\).

5 The curve \(C\) has equation \(y = \frac { 12 ( x + 1 ) } { ( x - 2 ) ^ { 2 } }\).

1. Determine the coordinates of any stationary points of \(C\).
2. Sketch \(C\).

6

1. Find the coordinates of the stationary points of the curve with equation $$y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }$$ and determine their nature.
2. Sketch the graph of \(y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }\) and hence state the set of values of \(k\) for which the equation \(3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 } = k\) has exactly four distinct real roots.

6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).
3. Sketch \(C\).

6 The curve \(S\) has equation \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ^ { 2 } }\).

1. Write down the equations of the asymptotes of \(S\).
2. Determine \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any turning points of \(S\).
3. Sketch \(S\).

9

1. Find the coordinates of the stationary point of the curve with equation $$y = \ln x - k x , \text { where } k > 0 \text { and } x > 0$$ and determine its nature.
2. Hence show that the equation \(\ln x - k x = 0\) has real roots if \(0 < k \leqslant \frac { 1 } { \mathrm { e } }\).
3. In the particular case that \(k = \frac { 1 } { 3 }\), the equation \(\ln x - k x = 0\) has two roots, one of which is near \(x = 5\). Use the Newton-Raphson process to find, correct to 3 significant figures, the root of the equation \(\ln x - \frac { 1 } { 3 } x = 0\) which is near \(x = 5\).
4. Show that the equation \(\ln x - k x = 0\) has one real root if \(k \leqslant 0\).
5. Explain why the equation \(\ln x - k x = 0\) has two distinct real roots if \(0 < k < \frac { 1 } { \mathrm { e } }\).

6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).
3. Sketch \(C\).

6 Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3 \mathrm { e } ^ { - 0.02 t }\) units and the concentration of Coldcure is \(5 \mathrm { e } ^ { - 0.07 t }\) units. Each drug becomes ineffective when its concentration falls below 1 unit.

1. Show that Coldcure becomes ineffective before Antiflu.
2. Sketch, on the same diagram, the graphs of concentration against time for each drug.
3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later.

7 A curve has equation \(y = \frac { 4 x + 11 } { ( x + 3 ) ^ { 2 } }\).

1. Show that the curve meets the line \(y = k\) if and only if \(k \leq 4\), and deduce the coordinates of the turning point on the curve.
2. Sketch the curve, stating the coordinates of the points where it cuts the axes, and showing clearly its asymptotes and the turning point.

7 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x - 3 } { x + 2 } .$$

1. Find the equations of the asymptotes of \(C\).
2. Sketch \(C\), indicating clearly the asymptotes and any points where \(C\) meets the coordinate axes.

9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant.

1. Obtain the equation of each of the asymptotes of \(C\).
2. Find the coordinates of the turning points of \(C\).
3. In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\).

The variables \(x\) and \(y\) satisfy the differential equation $$(1 - \cos x)\frac{dy}{dx} = y \sin x.$$ It is given that \(y = 4\) when \(x = \pi\).

1. Solve the differential equation, obtaining an expression for \(y\) in terms of \(x\). [6]
2. Sketch the graph of \(y\) against \(x\) for \(0 < x < 2\pi\). [1]

The curve \(C\) has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).

1. Find the equations of the asymptotes of \(C\). [2]
2. Find the coordinates of any stationary points on \(C\). [4]
3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]

The curve \(C\) has equation $$y = \frac{x^2}{x + \lambda},$$ where \(\lambda\) is a non-zero constant. Obtain the equations of the asymptotes of \(C\). [3] In separate diagrams, sketch \(C\) for the cases where

1. \(\lambda > 0\),
2. \(\lambda < 0\).

[4]

The curve \(C\) has equation \(y = \frac{2x^2 + kx}{x + 1}\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. [5] For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes. [6]

The curve \(C\) has equation $$y = \frac{x^2 + ax - 1}{x + 1},$$ where \(a\) is constant and \(a > 1\).

1. Find the equations of the asymptotes of \(C\). [3]
2. Show that \(C\) intersects the \(x\)-axis twice. [1]
3. Justifying your answer, find the number of stationary points on \(C\). [2]
4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis. [3]

The curve \(C\) has equation $$y = \frac{5x^2 + 5x + 1}{x^2 + x + 1}.$$

1. Find the equation of the asymptote of \(C\). [2]
2. Show that, for all real values of \(x\), \(-\frac{1}{5} \leqslant y < 5\). [4]
3. Find the coordinates of any stationary points of \(C\). [2]
4. Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis. [2]

The line \(y = 2x + 1\) is an asymptote of the curve \(C\) with equation $$y = \frac{x^2 + 1}{ax + b}.$$

1. Find the values of the constants \(a\) and \(b\). [3]
2. State the equation of the other asymptote of \(C\). [1]
3. Sketch \(C\). [Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.] [3]

On separate diagrams, sketch the graphs of

1. \(y = (x + 3)^2\), [3]
2. \(y = (x + 3)^2 + k\), where \(k\) is a positive constant. [2]

Show on each sketch the coordinates of each point at which the graph meets the axes.