1.02n Sketch curves: simple equations including polynomials

10. (a) Factorise completely \(x ^ { 3 } - 6 x ^ { 2 } + 9 x\) (b) Sketch the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. Using your answer to part (b), or otherwise,
(c) sketch, on a separate diagram, the curve with equation $$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$ showing the coordinates of the points at which the curve meets the \(x\)-axis.

4. (a) Show that \(x ^ { 2 } + 6 x + 11\) can be written as $$( x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers to be found.
(b) In the space at the top of page 7 , sketch the curve with equation \(y = x ^ { 2 } + 6 x + 11\), showing clearly any intersections with the coordinate axes.
(c) Find the value of the discriminant of \(x ^ { 2 } + 6 x + 11\)

10. (a) On the axes below sketch the graphs of

1. \(y = x ( 4 - x )\)
2. \(y = x ^ { 2 } ( 7 - x )\) showing clearly the coordinates of the points where the curves cross the coordinate axes.
   (b) Show that the \(x\)-coordinates of the points of intersection of $$y = x ( 4 - x ) \text { and } y = x ^ { 2 } ( 7 - x )$$ are given by the solutions to the equation \(x \left( x ^ { 2 } - 8 x + 4 \right) = 0\) The point \(A\) lies on both of the curves and the \(x\) and \(y\) coordinates of \(A\) are both positive.
   (c) Find the exact coordinates of \(A\), leaving your answer in the form ( \(p + q \sqrt { } 3 , r + s \sqrt { } 3\) ), where \(p , q , r\) and \(s\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{65d61b2c-2e47-402e-b08f-2d46bb00c188-14_1178_1203_1407_379}

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

1. Sketch \(C\), showing the coordinates of the points at which \(C\) meets the axes.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + 14 x + 15\). The point \(A\), with \(x\)-coordinate - 5 , lies on \(C\).
3. Find the equation of the tangent to \(C\) at \(A\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. Another point \(B\) also lies on \(C\). The tangents to \(C\) at \(A\) and \(B\) are parallel.
4. Find the \(x\)-coordinate of \(B\).

8. $$4 x - 5 - x ^ { 2 } = q - ( x + p ) ^ { 2 }$$ where \(p\) and \(q\) are integers.

1. Find the value of \(p\) and the value of \(q\).
2. Calculate the discriminant of \(4 x - 5 - x ^ { 2 }\)
3. On the axes on page 17, sketch the curve with equation \(y = 4 x - 5 - x ^ { 2 }\) showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{089c3b5b-22ab-4fa2-8383-4f30cefa792a-11_1143_1143_260_388}

9. The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { 2 } + 8\) The line \(L\) has equation \(y = 3 x + k\), where \(k\) is a positive constant.

1. Sketch \(C\) and \(L\) on separate diagrams, showing the coordinates of the points at which \(C\) and \(L\) cut the axes. Given that line \(L\) is a tangent to \(C\),
2. find the value of \(k\).

1. (a) Factorise completely \(9 x - 4 x ^ { 3 }\) (b) Sketch the curve \(C\) with equation

$$y = 9 x - 4 x ^ { 3 }$$ Show on your sketch the coordinates at which the curve meets the \(x\)-axis. The points \(A\) and \(B\) lie on \(C\) and have \(x\) coordinates of - 2 and 1 respectively.
(c) Show that the length of \(A B\) is \(k \sqrt { } 10\) where \(k\) is a constant to be found.

5. $$f ( x ) = x ^ { 2 } - 8 x + 19$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) crosses the \(y\)-axis at the point \(P\) and has a minimum point at the point \(Q\).
2. Sketch the graph of \(C\) showing the coordinates of point \(P\) and the coordinates of point \(Q\).
3. Find the distance \(P Q\), writing your answer as a simplified surd.

9. (a) On separate axes sketch the graphs of

1. \(y = - 3 x + c\), where \(c\) is a positive constant,
2. \(y = \frac { 1 } { x } + 5\) On each sketch show the coordinates of any point at which the graph crosses the \(y\)-axis and the equation of any horizontal asymptote. Given that \(y = - 3 x + c\), where \(c\) is a positive constant, meets the curve \(y = \frac { 1 } { x } + 5\) at two distinct points,
   (b) show that \(( 5 - c ) ^ { 2 } > 12\) (c) Hence find the range of possible values for \(c\).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where

$$f ^ { \prime } ( x ) = ( x - 3 ) ( 3 x + 5 )$$ Given that the point \(P ( 1,20 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying each term.
2. Show that $$f ( x ) = ( x - 3 ) ^ { 2 } ( x + A )$$ where \(A\) is a constant to be found.
3. Sketch the graph of \(C\). Show clearly the coordinates of the points where \(C\) cuts or meets the \(x\)-axis and where \(C\) cuts the \(y\)-axis.

7. (a) Factorise completely \(x ^ { 3 } - 4 x\).
(3)
(b) Sketch the curve with equation \(y = x ^ { 3 } - 4 x\), showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
(c) On a separate diagram, sketch the curve with equation \(y = ( x - 1 ) ^ { 3 } - 4 ( x - 1 ) ,\) showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
\end{tabular} & Leave blank
\hline \end{tabular} \end{center} \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

1. In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} Figure 3 shows

• the curve \(C\) with equation \(y = x ^ { 2 } - 4 x + 5\)
• the line \(l\) with equation \(y = 2\)

The curve \(C\) intersects the \(y\)-axis at the point \(D\).

1. Write down the coordinates of \(D\). The curve \(C\) intersects the line \(l\) at the points \(E\) and \(F\), as shown in Figure 3.
2. Find the \(x\) coordinate of \(E\) and the \(x\) coordinate of \(F\). Shown shaded in Figure 3 is
   • the region \(R _ { 1 }\) which is bounded by \(C , l\) and the \(y\)-axis
   • the region \(R _ { 2 }\) which is bounded by \(C\) and the line segments \(E F\) and \(D F\)
   Given that \(\frac { \text { area of } R _ { 1 } } { \text { area of } R _ { 2 } } = k\), where \(k\) is a constant,
3. use algebraic integration to find the exact value of \(k\), giving your answer as a simplified fraction.

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = x ( x + 4 ) ( x - 2 )$$ The curve \(C\) crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).

1. Write down the \(x\)-coordinates of the points \(A\) and \(B\). The finite region, shown shaded in Figure 3, is bounded by the curve \(C\) and the \(x\)-axis.
2. Use integration to find the total area of the finite region shown shaded in Figure 3.

5. Sketch the graph of \(y = \ln | x |\), stating the coordinates of any points of intersection with the axes.

4. The function \(f\) is defined by $$f : x \mapsto | 2 x - 5 | , \quad x \in \mathbb { R }$$

1. Sketch the graph with equation \(y = \mathrm { f } ( x )\), showing the coordinates of the points where the graph cuts or meets the axes.
2. Solve \(\mathrm { f } ( x ) = 15 + x\). The function \(g\) is defined by $$g : x \mapsto x ^ { 2 } - 4 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 5$$
3. Find fg(2).
4. Find the range of g.

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the graph of \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left\{ \begin{array} { r r } 5 - 2 x , & x \leqslant 4 \\ \mathrm { e } ^ { 2 x - 8 } - 4 , & x > 4 \end{array} \right.$$

1. State the range of \(\mathrm { f } ( x )\).
2. Determine the exact value of ff(0).
3. Solve \(\mathrm { f } ( x ) = 21\) Give each answer as an exact answer.
4. Explain why the function f does not have an inverse.

2. A curve \(C\) has equation \(y = \mathrm { e } ^ { 4 x } + x ^ { 4 } + 8 x + 5\)

1. Show that the \(x\) coordinate of any turning point of \(C\) satisfies the equation $$x ^ { 3 } = - 2 - \mathrm { e } ^ { 4 x }$$
2. On the axes given on page 5, sketch, on a single diagram, the curves with equations
   1. \(y = x ^ { 3 }\),
   2. \(y = - 2 - e ^ { 4 x }\) On your diagram give the coordinates of the points where each curve crosses the \(y\)-axis and state the equation of any asymptotes.
3. Explain how your diagram illustrates that the equation \(x ^ { 3 } = - 2 - e ^ { 4 x }\) has only one root. The iteration formula $$x _ { n + 1 } = \left( - 2 - \mathrm { e } ^ { 4 x _ { n } } \right) ^ { \frac { 1 } { 3 } } , \quad x _ { 0 } = - 1$$ can be used to find an approximate value for this root.
4. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 5 decimal places.
5. Hence deduce the coordinates, to 2 decimal places, of the turning point of the curve \(C\). \includegraphics[max width=\textwidth, alt={}, center]{be00fdaa-2fe3-4f06-a710-08ec67fb911e-04_1285_1294_308_331}

2. Given that $$\mathrm { f } ( x ) = 2 \mathrm { e } ^ { x } - 5 , \quad x \in \mathbb { R }$$

1. sketch, on separate diagrams, the curve with equation
   1. \(y = \mathrm { f } ( x )\)
   2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of each point at which the curve meets or cuts the axes. On each diagram state the equation of the asymptote.
2. Deduce the set of values of \(x\) for which \(\mathrm { f } ( x ) = | \mathrm { f } ( x ) |\)
3. Find the exact solutions of the equation \(| \mathrm { f } ( x ) | = 2\)

8. The hyperbola \(H\) has cartesian equation \(x y = 16\) The parabola \(P\) has parametric equations \(x = 8 t ^ { 2 } , y = 16 t\).

1. Find, using algebra, the coordinates of the point \(A\) where \(H\) meets \(P\). Another point \(B ( 8,2 )\) lies on the hyperbola \(H\).
2. Find the equation of the normal to \(H\) at the point (8, 2), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
3. Find the coordinates of the points where this normal at \(B\) meets the parabola \(P\).

6. The rectangular hyperbola, \(H\), has cartesian equation $$x y = 36$$ The three points \(P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)\) and \(R \left( 6 r , \frac { 6 } { r } \right)\), where \(p , q\) and \(r\) are distinct, non-zero values, lie on the hyperbola \(H\).

1. Show that an equation of the line \(P Q\) is $$p q y + x = 6 ( p + q )$$ Given that \(P R\) is perpendicular to \(Q R\),
2. show that the normal to the curve \(H\) at the point \(R\) is parallel to the line \(P Q\).

6. The rectangular hyperbola \(H\) has equation \(x y = 25\)

1. Verify that, for \(t \neq 0\), the point \(P \left( 5 t , \frac { 5 } { t } \right)\) is a general point on \(H\). The point \(A\) on \(H\) has parameter \(t = \frac { 1 } { 2 }\)
2. Show that the normal to \(H\) at the point \(A\) has equation $$8 y - 2 x - 75 = 0$$ This normal at \(A\) meets \(H\) again at the point \(B\).
3. Find the coordinates of \(B\).
   

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\) The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
2. Determine the exact coordinates of \(Q\) Given that
   • the point \(R\) is the focus of \(C\)
   • the line \(l\) is the directrix of \(C\)
   • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
   • determine the exact length of \(Q S\)

6. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\).

1. Verify that the point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
2. Write down the coordinates of the focus \(S\) of \(C\).
3. Show that the normal to \(C\) at \(P\) has equation $$y + t x = 8 t + 4 t ^ { 3 }$$ The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(N\).
4. Find the area of triangle \(P S N\) in terms of \(t\), giving your answer in its simplest form.

7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant. The point \(\left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).

1. Find the value of \(a\).
2. Show that the equation for the tangent to \(C\) at the point \(\left( 4 t ^ { 2 } , 8 t \right)\) is $$y t = x + 4 t ^ { 2 } .$$ The tangent to \(C\) at the point \(A\) meets the tangent to \(C\) at the point \(B\) on the directrix of \(C\) when \(y = 15\).
3. Find the coordinates of \(A\) and the coordinates of \(B\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation $$y = \frac { 4 x } { 4 - | x | }$$ and the curve \(C _ { 2 }\) with equation $$y = x ^ { 2 } - 8 x$$ For \(x > 0 , C _ { 1 }\) has equation \(y = \frac { 4 x } { 4 - x }\)

1. Use algebra to show that \(C _ { 1 }\) touches \(C _ { 2 }\) at a point \(P\), stating the coordinates of \(P\)
2. Hence or otherwise, using algebra, solve the inequality $$x ^ { 2 } - 8 x > \frac { 4 x } { 4 - | x | }$$