1.02w Graph transformations: simple transformations of f(x)

10

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{450c1c3a-9290-4afa-a051-112b60cf19c0-3_753_775_360_726} \captionsetup{labelformat=empty} \caption{Fig. 10}
   \end{figure} Fig. 10 shows a sketch of the graph of \(y = \frac { 1 } { x }\).
   Sketch the graph of \(y = \frac { 1 } { x - 2 }\), showing clearly the coordinates of any points where it crosses the axes.
2. Find the value of \(x\) for which \(\frac { 1 } { x - 2 } = 5\).
3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac { 1 } { x - 2 }\). Give your answers in the form \(a \pm \sqrt { b }\). Show the position of these points on your graph in part (i).

9

1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.

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

4 \includegraphics[max width=\textwidth, alt={}, center]{3b927f8b-ddf8-481d-a1ce-3b90bb1435f0-2_437_807_953_579} The graph shows a function \(y = \mathrm { f } ( x )\).
On separate graphs, sketch the graphs of the following functions:

1. \(\quad y = \mathrm { f } ( x ) + 1\),
2. \(y = \mathrm { f } ( x + 1 )\).

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.

10 The diagram shows the graph of \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-3_507_1085_933_383} A is the minimum point of the curve at \(( 3 , - 4 )\) and B is the point \(( 5,0 )\).
On separate diagrams on graph paper, draw the graphs of the following. In each case give the coordinates of the images of the points A and B .

1. \(\quad y = \mathrm { f } ( x ) + 2\),
2. \(y = \mathrm { f } ( x + 2 )\).

8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$

1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
   1. \(\quad y = \mathrm { f } ( x + 3 )\),
   2. \(y = \mathrm { f } ( - x )\).

6. \includegraphics[max width=\textwidth, alt={}, center]{00364339-8108-4031-8e67-6100810e8297-2_549_885_251_370} The diagram shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).

9. (i) Find an equation for the tangent to the curve \(y = x ^ { 2 } + 2\) at the point \(( 1,3 )\) in the form \(y = m x + c\).
(ii) Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers.
(iii) Describe fully the transformation that maps the graph of \(y = x ^ { 2 } + 2\) onto the graph of \(y = x ^ { 2 } - 6 x + 11\).
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve \(y = x ^ { 2 } - 6 x + 11\) at the point with \(x\)-coordinate 4.

8. (i) Sketch the graphs of \(y = 2 x ^ { 4 }\) and \(y = 2 \sqrt { x } , x \geq 0\) on the same diagram and write down the coordinates of the point where they intersect.
(ii) Describe fully the transformation that maps the graph of \(y = 2 \sqrt { x }\) onto the graph of \(y = 2 \sqrt { x - 3 }\).
(iii) Find and simplify the equation of the graph obtained when the graph of \(y = 2 x ^ { 4 }\) is stretched by a factor of 2 in the \(x\)-direction, about the \(y\)-axis.

9. \(f ( x ) = 2 x ^ { 2 } + 3 x - 2\).

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
3. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) crosses the coordinate axes. When the graph of \(y = \mathrm { f } ( x )\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.
4. Find the values of \(a , b\) and \(c\).

8. (i) Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
(ii) Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
(iii) Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).

1 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.

3 You are given that \(\mathrm { f } ( x ) = ( 2 x - 3 ) ( x + 2 ) ( x + 4 )\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. State the roots of \(\mathrm { f } ( x - 2 ) = 0\).
3. You are also given that \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + 15\).
   (A) Show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 2 x - 9\).
   (B) Show that \(\mathrm { g } ( 1 ) = 0\) and hence factorise \(\mathrm { g } ( x )\) completely.

4 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 \(\mathrm { f } ( x + 3 ) = 0\).

5 A cubic curve has equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis where \(x = - , \frac { 1 } { 2 }\) and 5 .

1. Write down three linear factors of \(\mathrm { f } ( x )\). Hence find the equation of the curve in the form \(y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 0 } { - 8 }\). State the coordinates of the point where the translated curve intersects the \(y\)-axis.
4. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression in factorised form for \(\mathrm { g } ( x )\) and state the coordinates of the point where the curve \(y = \mathrm { g } ( x )\) intersects the \(y\)-axis.

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 \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
   [0pt] [3]
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.
   [0pt] [6]
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 \(y = \mathrm { g } ( x )\).

1 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).

1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$

3

1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
   (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
   (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
   Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.

4 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 \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).

2 \begin{figure}[h] \end{figure} Fig. 13 shows a sketch of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2\).

1. Use the fact that \(x = 2\) is a root of \(\mathrm { f } ( x ) = 0\) to find the exact values of the other two roots of \(\mathrm { f } ( x ) = 0\), expressing your answers as simply as possible.
2. Show that \(\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10\).
3. Write down the roots of \(\mathrm { f } ( x - 3 ) = 0\).

4 \begin{figure}[h] \end{figure} Fig. 7 shows the graph of \(y = \mathrm { g } ( x )\). Draw the graphs of the following.

1. \(y = \mathrm { g } ( x ) + 3\)
2. \(y = \mathrm { g } ( x + 2 )\)

5 The point \(\mathrm { P } ( 5,4 )\) is on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P when the graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of

1. \(y = \mathrm { f } ( x - 5 )\),
2. \(y = \mathrm { f } ( x ) + 7\).

6

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