Graphical equation solving with auxiliary line

14. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph of \(y = g ( x ) , - 3 \leqslant x \leqslant 4\) and part of the line \(l\) with equation \(y = \frac { 1 } { 2 } x\) The graph of \(y = \mathrm { g } ( x )\) consists of three line segments, from \(P ( - 3,4 )\) to \(Q ( 0,4 )\), from \(Q ( 0,4 )\) to \(R ( 2,0 )\) and from \(R ( 2,0 )\) to \(S ( 4,10 )\). The line \(l\) intersects \(y = \mathrm { g } ( x )\) at the points \(A\) and \(B\) as shown in Figure 4.

1. Use algebra to find the \(x\) coordinate of the point \(A\) and the \(x\) coordinate of the point \(B\). Show each step of your working and give your answers as exact fractions.
2. Sketch the graph with equation $$y = \frac { 3 } { 2 } g ( x ) , \quad - 3 \leqslant x \leqslant 4$$ On your sketch show the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.

12

1. Sketch the graph of \(y = x ( x - 3 ) ^ { 2 }\).
2. Show that the equation \(x ( x - 3 ) ^ { 2 } = 2\) can be expressed as \(x ^ { 3 } - 6 x ^ { 2 } + 9 x - 2 = 0\).
3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i).

3 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).

1. On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
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2. Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
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3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).
4. 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 \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
5. 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.
6. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).

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 \(y = \mathrm { f } ( x )\).

3 The diagram in the Printed Answer Booklet shows part of the graph of \(y = x ^ { 2 } - 4 x + 3\).

1. It is required to solve the equation \(x ^ { 2 } - 3 x + 1 = 0\) graphically by drawing a straight line with equation \(y = m x + c\) on the diagram, where \(m\) and \(c\) are constants. Find the values of \(m\) and \(c\).
2. Use the graph to find approximate values of the roots of the equation \(x ^ { 2 } - 3 x + 1 = 0\).
3. By shading, or otherwise, indicate clearly the regions where all of the following inequalities are satisfied. You should use the values of \(m\) and \(c\) found in part (a).
   \(x \geqslant 0\)
   \(x \leqslant 4\)
   \(y \leqslant x ^ { 2 } - 4 x + 3\)
   \(y \geqslant m x + c\)

6 Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).

4 Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).

13 Answer the whole of this question on the insert provided.
The insert shows the graph of \(y = \frac { 1 } { x } , x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac { 1 } { x } = 2 x + 3\), showing your method clearly.
2. Rearrange the equation \(\frac { 1 } { x } = 2 x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac { p \pm \sqrt { q } } { r }\).
3. Draw the graph of \(y = \frac { 1 } { x } + 2 , x \neq 0\), on the grid used for part (i).
4. Write down the values of \(x\) which satisfy the equation \(\frac { 1 } { x } + 2 = 2 x + 3\).