1.02n Sketch curves: simple equations including polynomials

2. (a) Sketch, on the same axes,

1. \(y = | 2 x - 3 |\)
2. \(y = 4 - x ^ { 2 }\) (b) Find the set of values of \(x\) for which $$4 - x ^ { 2 } > | 2 x - 3 |$$

1. (a) Find the general solution of the differential equation
   (b) Find the particular solution for which \(y = 5\) at \(x = 1\), giving your answer in the form \(y = \mathrm { f } ( x )\).

$$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x ^ { 2 }$$ (c) (i) Find the exact values of the coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\), making your method clear.
(ii) Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of the turning points.

4

1. Sketch the graphs of \(y = 1 + \mathrm { e } ^ { 2 x }\) and \(y = | x - 4 |\) on the same diagram.
2. The two graphs meet at the point \(P\) .
   Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = \frac { 1 } { 2 } \ln ( 3 - x )\) . \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-06_2716_38_109_2012}
3. Use an iterative formula, based on the equation in part (b), to find the \(x\)-coordinate of \(P\) correct to 3 significant figures. Use an initial value of 0.45 and give the result of each iteration to 5 significant figures.

1. The hyperbola \(H\) has

• foci with coordinates \(\left( \pm \frac { 13 } { 2 } , 0 \right)\)
• directrices with equations \(x = \pm \frac { 72 } { 13 }\)
• eccentricity e

Determine

1. the value of \(e\)
2. an equation for \(H\), giving your answer in the form \(p x ^ { 2 } - q y ^ { 2 } = r\), where \(p , q\) and \(r\) are integers.

5 The curve with equation \(y = \frac { x ^ { 2 } - k x + 2 k } { x + k }\) is to be investigated for different values of \(k\).

1. Use your graphical calculator to obtain rough sketches of the curve in the cases \(k = - 2\), \(k = - 0.5\) and \(k = 1\).
2. Show that the equation of the curve may be written as \(y = x - 2 k + \frac { 2 k ( k + 1 ) } { x + k }\). Hence find the two values of \(k\) for which the curve is a straight line.
3. When the curve is not a straight line, it is a conic.
   (A) Name the type of conic.
   (B) Write down the equations of the asymptotes.
4. Draw a sketch to show the shape of the curve when \(1 < k < 8\). This sketch should show where the curve crosses the axes and how it approaches its asymptotes. Indicate the points A and B on the curve where \(x = 1\) and \(x = k\) respectively.

5 A curve has parametric equations \(x = \lambda \cos \theta - \frac { 1 } { \lambda } \sin \theta , y = \cos \theta + \sin \theta\), where \(\lambda\) is a positive constant.

1. Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = 0.5 , \quad \lambda = 3 \quad \text { and } \quad \lambda = 5 .$$
2. Given that the curve is a conic, name the type of conic.
3. Show that \(y\) has a maximum value of \(\sqrt { 2 }\) when \(\theta = \frac { 1 } { 4 } \pi\).
4. Show that \(x ^ { 2 } + y ^ { 2 } = \left( 1 + \lambda ^ { 2 } \right) + \left( \frac { 1 } { \lambda ^ { 2 } } - \lambda ^ { 2 } \right) \sin ^ { 2 } \theta\), and deduce that the distance from the origin of any point on the curve is between \(\sqrt { 1 + \frac { 1 } { \lambda ^ { 2 } } }\) and \(\sqrt { 1 + \lambda ^ { 2 } }\).
5. For the case \(\lambda = 1\), show that the curve is a circle, and find its radius.
6. For the case \(\lambda = 2\), draw a sketch of the curve, and label the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H }\) on the curve corresponding to \(\theta = 0 , \frac { 1 } { 4 } \pi , \frac { 1 } { 2 } \pi , \frac { 3 } { 4 } \pi , \pi , \frac { 5 } { 4 } \pi , \frac { 3 } { 2 } \pi , \frac { 7 } { 4 } \pi\) respectively. You should make clear what is special about each of these points.

5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of \(k\).

1. Firstly consider cases in which \(k\) is a positive even integer.
   (A) State the shape of the curve when \(k = 2\).
   (B) Sketch, on the same axes, the curves for \(k = 2\) and \(k = 4\).
   (C) Describe the shape that the curve tends to as \(k\) becomes very large.
   (D) State the range of possible values of \(x\) and \(y\).
2. Now consider cases in which \(k\) is a positive odd integer.
   (A) Explain why \(x\) and \(y\) may take any value.
   (B) State the shape of the curve when \(k = 1\).
   (C) Sketch the curve for \(k = 3\). State the equation of the asymptote of this curve.
   (D) Sketch the shape that the curve tends to as \(k\) becomes very large.
3. Now let \(k = \frac { 1 } { 2 }\). Sketch the curve, indicating the range of possible values of \(x\) and \(y\).
4. Now consider the modified equation \(| x | ^ { k } + | y | ^ { k } = 1\).
   (A) Sketch the curve for \(k = \frac { 1 } { 2 }\).
   (B) Investigate the shape of the curve for \(k = \frac { 1 } { n }\) as the positive integer \(n\) becomes very large.

5 On separate diagrams,

1. sketch the curve \(y = \frac { 1 } { x }\),
2. sketch the curve \(y = x \left( x ^ { 2 } - 1 \right)\), stating the coordinates of the points where it crosses the \(x\)-axis,
3. sketch the curve \(y = - \sqrt { } x\).

7

1. Solve the equation \(x ^ { 2 } - 8 x + 11 = 0\), giving your answers in simplified surd form.
2. Hence sketch the curve \(y = x ^ { 2 } - 8 x + 11\), labelling the points where the curve crosses the axes.
3. Solve the equation \(y - 8 y ^ { \frac { 1 } { 2 } } + 11 = 0\), giving your answers in the form \(p \pm q \sqrt { 5 }\).

5

1. Sketch the curve \(y = x ^ { 3 } + 2\).
2. Sketch the curve \(y = 2 \sqrt { x }\).
3. Describe a transformation that transforms the curve \(y = 2 \sqrt { x }\) to the curve \(y = 3 \sqrt { x }\).

6

1. Solve the equation \(x ^ { 2 } + 8 x + 10 = 0\), giving your answers in simplified surd form.
2. Sketch the curve \(y = x ^ { 2 } + 8 x + 10\), giving the coordinates of the point where the curve crosses the \(y\)-axis.
3. Solve the inequality \(x ^ { 2 } + 8 x + 10 \geqslant 0\).

3

1. Sketch the curve \(y = x ^ { 3 }\).
2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = - x ^ { 3 }\).
3. The curve \(y = x ^ { 3 }\) is translated by \(p\) units, parallel to the \(x\)-axis. State the equation of the curve after it has been transformed.

7

1. Calculate the discriminant of each of the following:
   1. \(x ^ { 2 } + 6 x + 9\),
   2. \(x ^ { 2 } - 10 x + 12\),
   3. \(x ^ { 2 } - 2 x + 5\).
   4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_391_446_628_397} \captionsetup{labelformat=empty} \caption{Fig. 1}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_449_625_888} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_389_442_630_1384} \captionsetup{labelformat=empty} \caption{Fig. 3}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_394_446_1119_644} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure} \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{e2a460a0-e411-4563-8f60-005189b6a3d9-3_396_447_1119_1137} \captionsetup{labelformat=empty} \caption{Fig. 5}
      \end{figure} State with reasons which of the diagrams corresponds to the curve
      (a) \(y = x ^ { 2 } + 6 x + 9\),
      (b) \(y = x ^ { 2 } - 10 x + 12\),
      (c) \(y = x ^ { 2 } - 2 x + 5\).

2

1. On separate diagrams, sketch the graphs of
   1. \(\mathrm { y } = \frac { 1 } { \mathrm { x } }\),
   2. \(y = x ^ { 4 }\).
2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = 8 x ^ { 3 }\).

10

1. Solve the equation \(3 x ^ { 2 } - 14 x - 5 = 0\). A curve has equation \(\mathrm { y } = 3 \mathrm { x } ^ { 2 } - 14 \mathrm { x } - 5\).
2. Sketch the curve, indicating the coordinates of all intercepts with the axes.
3. Find the value of C for which the line \(\mathrm { y } = 4 \mathrm { x } + \mathrm { C }\) is a tangent to the curve.

6

1. Expand and simplify \(( x - 5 ) ( x + 2 ) ( x + 5 )\).
2. Sketch the curve \(y = ( x - 5 ) ( x + 2 ) ( x + 5 )\), giving the coordinates of the points where the curve crosses the axes.

8

1. Find the coordinates of the stationary points on the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\).
2. Determine whether each stationary point is a maximum point or a minimum point.
3. By expanding the right-hand side, show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7 = ( x + 1 ) ^ { 2 } ( 2 x - 7 )$$
4. Sketch the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\), marking the coordinates of the stationary points and the points where the curve meets the axes.

11

1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.

12

1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).

10

1. Express \(x ^ { 2 } - 6 x + 2\) in the form \(( x - a ) ^ { 2 } - b\).
2. State the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 2\).
3. Sketch the graph of \(y = x ^ { 2 } - 6 x + 2\). You need not state the coordinates of the points where the graph intersects the \(x\)-axis.
4. Solve the simultaneous equations \(y = x ^ { 2 } - 6 x + 2\) and \(y = 2 x - 14\). Hence show that the line \(y = 2 x - 14\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 2\).

11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).

1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).

9 Explain why each of the following statements is false. State in each case which of the symbols ⇒, ⟸ or ⇔ would make the statement true.

1. ABCD is a square \(\Leftrightarrow\) the diagonals of quadrilateral ABCD intersect at \(90 ^ { \circ }\)
2. \(x ^ { 2 }\) is an integer \(\Rightarrow x\) is an integer

11 \begin{figure}[h] \end{figure} Fig. 11 shows a sketch of the circle with equation \(( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 125\) and centre C . The points \(\mathrm { A } , \mathrm { B }\), D and E are the intersections of the circle with the axes.

1. Write down the radius of the circle and the coordinates of C .
2. Verify that B is the point \(( 21,0 )\) and find the coordinates of \(\mathrm { A } , \mathrm { D }\) and E .
3. Find the equation of the perpendicular bisector of BE and verify that this line passes through C .

10

1. A quadratic function is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 8\).
   Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the points where it crosses the axes. Mark the lowest point on the curve, and give its coordinates.
2. Solve the inequality \(x ^ { 2 } - 6 x + 8 < 0\).
3. On the same graph, sketch \(y = \mathrm { f } ( x + 3 )\).
4. The graph of \(y = \mathrm { f } ( x + 3 ) - 2\) is obtained from the graph of \(y = \mathrm { f } ( x )\) by a transformation. Describe the transformation and sketch the curve on the same axes as in (i) and (iii) above. Label all these curves clearly.

11

1. Show algebraically that the equation \(x ^ { 2 } - 6 x + 10 = 0\) has no real roots.
2. Solve algebraically the simultaneous equations \(y = x ^ { 2 } - 6 x + 10\) and \(y + 2 x = 7\).
3. Plot the graph of the function \(y = x ^ { 2 } - 6 x + 10\) on graph paper, taking \(1 \mathrm {~cm} = 1\) unit on each axis, with the \(x\) axis from 0 to 6 and the \(y\) axis from - 2 to 10 .
   On the same axes plot the line with equation \(y + 2 x = 7\) showing clearly where the line cuts the quadratic curve.
4. Explain why these \(x\) coordinates satisfy the equation \(x ^ { 2 } - 4 x + 3 = 0\). Plot a graph of the function \(y = x ^ { 2 } - 4 x + 3\) on the same axes to illustrate your answer.