1.02n Sketch curves: simple equations including polynomials

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 4 ( x - 1 ) } { x ( x - 3 ) }$$ The curve cuts the \(x\)-axis at \(( a , 0 )\). The lines \(y = 0 , x = 0\) and \(x = b\) are asymptotes to the curve.

1. Write down the value of \(a\) and the value of \(b\).
   (2)
2. On separate axes, sketch the curves with the following equations. On your sketches, you should mark the coordinates of any intersections with the coordinate axes and state the equations of any asymptotes.
   1. \(y = \mathrm { f } ( x + 2 ) - 4\)
   2. \(y = \mathrm { f } ( | x | ) - 3\)

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { x } + 1 , \quad x \neq 0$$ The curve \(C\) crosses the \(x\)-axis at the point \(A\).

1. State the \(x\) coordinate of the point \(A\). The curve \(D\) has equation \(y = x ^ { 2 } ( x - 2 )\), for all real values of \(x\).
2. A copy of Figure 1 is shown on page 7. On this copy, sketch a graph of curve \(D\).
   Show on the sketch the coordinates of each point where the curve \(D\) crosses the coordinate axes.
3. Using your sketch, state, giving a reason, the number of real solutions to the equation $$x ^ { 2 } ( x - 2 ) = \frac { 1 } { x } + 1$$ \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-06_942_1026_516_466} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure}

2.(a)On separate diagrams,sketch the curves with the following equations.On each sketch you should label the exact coordinates of the points where the curve meets the coordinate axes.

1. \(y = 8 + 2 x - x ^ { 2 }\)
2. \(y = 8 + 2 | x | - x ^ { 2 }\)
3. \(y = 8 + x + | x | - x ^ { 2 }\) (b)Find the values of \(x\) for which $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$

5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of \(m\) and \(n\).

1. On separate diagrams, sketch the curve in each of the following cases.
   (A) \(m = 1 , n = 1\),
   (B) \(m = 2 , n = 2\),
   (C) \(m = 2 , n = 4\),
   (D) \(m = 4 , n = 2\).
2. What feature does the curve have when \(m = n\) ? What is the effect on the curve of interchanging \(m\) and \(n\) when \(m \neq n\) ?
3. Describe how the \(x\)-coordinate of the maximum on the curve varies as \(m\) and \(n\) vary. Use calculus to determine the \(x\)-coordinate of the maximum.
4. Find the condition on \(m\) for the gradient to be zero when \(x = 0\). State a corresponding result for the gradient to be zero when \(x = 1\).
5. Use your calculator to investigate the shape of the curve for large values of \(m\) and \(n\). Hence conjecture what happens to the value of the integral \(\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x\) as \(m\) and \(n\) tend to infinity.
6. Use your calculator to investigate the shape of the curve for small values of \(m\) and \(n\). Hence conjecture what happens to the shape of the curve as \(m\) and \(n\) tend to zero. }{www.ocr.org.uk}) after the live examination series.
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8

1. Solve the equation \(5 - 8 x - x ^ { 2 } = 0\), giving your answers in simplified surd form.
2. Solve the inequality \(5 - 8 x - x ^ { 2 } \leqslant 0\).
3. Sketch the curve \(y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )\), giving the coordinates of the points where the curve crosses the coordinate axes.

7 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Each diagram shows a quadratic curve. State which diagram corresponds to the curve
   1. \(y = ( 3 - x ) ^ { 2 }\),
   2. \(y = x ^ { 2 } + 9\),
   3. \(y = ( 3 - x ) ( x + 3 )\).
   4. Give the equation of the curve which does not correspond to any of the equations in part (i).

5

1. Sketch the curve \(y = - x ^ { 3 }\).
2. The curve \(y = - x ^ { 3 }\) is translated by 3 units in the positive \(x\)-direction. Find the equation of the curve after it has been translated.
3. Describe a transformation that transforms the curve \(y = - x ^ { 3 }\) to the curve \(y = - 5 x ^ { 3 }\).

9

1. Sketch the curve \(y = 12 - x - x ^ { 2 }\), giving the coordinates of all intercepts with the axes.
2. Solve the inequality \(12 - x - x ^ { 2 } > 0\).
3. Find the coordinates of the points of intersection of the curve \(y = 12 - x - x ^ { 2 }\) and the line \(3 x + y = 4\).

5

1. Expand and simplify \(( 2 x + 1 ) ( x - 3 ) ( x + 4 )\).
2. Find the coefficient of \(x ^ { 4 }\) in the expansion of $$x \left( x ^ { 2 } + 2 x + 3 \right) \left( x ^ { 2 } + 7 x - 2 \right) .$$

10

1. Solve the equation \(9 x ^ { 2 } + 18 x - 7 = 0\).
2. Find the coordinates of the stationary point on the curve \(y = 9 x ^ { 2 } + 18 x - 7\).
3. Sketch the curve \(y = 9 x ^ { 2 } + 18 x - 7\), giving the coordinates of all intercepts with the axes.
4. For what values of \(x\) does \(9 x ^ { 2 } + 18 x - 7\) increase as \(x\) increases?

4

1. Expand \(( x - 2 ) ^ { 2 } ( x + 1 )\), simplifying your answer.
2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } ( x + 1 )\), indicating the coordinates of all intercepts with the axes.

10 A curve has equation \(y = ( 2 x - 1 ) ( x + 3 ) ( x - 1 )\).

1. Sketch the curve, indicating the coordinates of all points of intersection with the axes.
2. Show that the gradient of the curve at the point \(P ( 1,0 )\) is 4 .
3. The line \(l\) is parallel to the tangent to the curve at the point \(P\). The curve meets \(l\) at the point where \(x = - 2\). Find the equation of \(l\), giving your answer in the form \(y = m x + c\).
4. Determine whether \(l\) is a tangent to the curve at the point where \(x = - 2\).

8

1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\), giving the coordinates of all points of intersection with the axes.
2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - x - 3 > 0\).
3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of the constant \(k\).

7

1. Sketch the curve \(y = x ^ { 2 } ( 3 - x )\) stating the coordinates of points of intersection with the axes.
2. The curve \(y = x ^ { 2 } ( 3 - x )\) is translated by 2 units in the positive direction parallel to the \(x\)-axis. State the equation of the curve after it has been translated.
3. Describe fully a transformation that transforms the curve \(y = x ^ { 2 } ( 3 - x )\) to \(y = \frac { 1 } { 2 } x ^ { 2 } ( 3 - x )\).

2 Sketch the graph of \(y = 9 - x ^ { 2 }\).

4

1. Describe fully the transformation which maps the curve \(y = x ^ { 2 }\) onto the curve \(y = ( x + 4 ) ^ { 2 }\).
2. Sketch the graph of \(y = x ^ { 2 } - 4\).

6 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).

12 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface. \begin{figure}[h] \end{figure}

1. State the value of \(x\) at A , where the arch meets the road.
2. Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph. Hence find the height of the arch.
3. Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ede57eaa-2645-49df-aa09-68b6d5f35a9a-4_529_871_1489_678} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
   \end{figure}
4. Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.
   [0pt] [5]

12

1. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
   (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
2. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
   (A) Show that \(\mathrm { g } ( 5 ) = 0\).
   (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
   (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).

13 \begin{figure}[h] \end{figure} Fig. 13 shows the curve \(y = x ^ { 4 } - 2\).

1. Find the exact coordinates of the points of intersection of this curve with the axes.
2. Find the exact coordinates of the points of intersection of the curve \(y = x ^ { 4 } - 2\) with the curve \(y = x ^ { 2 }\).
3. Show that the curves \(y = x ^ { 4 } - 2\) and \(y = k x ^ { 2 }\) intersect for all values of \(k\).

11 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).

1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).

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

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. State the values of \(x\) which satisfy \(\mathrm { f } ( x + 3 ) = 0\).

11

1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes.

11

1. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) with the axes.
2. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) and the line \(y = x + 3\).
3. Find the set of values of \(k\) for which the line \(y = x + k\) does not intersect the curve \(y = 2 x ^ { 2 } - 5 x - 3\). \section*{END OF QUESTION PAPER}

11 The point A has \(x\)-coordinate 5 and lies on the curve \(y = x ^ { 2 } - 4 x + 3\).

1. Sketch the curve.
2. Use calculus to find the equation of the tangent to the curve at A .
3. Show that the equation of the normal to the curve at A is \(x + 6 y = 53\). Find also, using an algebraic method, the \(x\)-coordinate of the point at which this normal crosses the curve again.