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

3

1. 1. Express \(x ^ { 2 } - 7 x + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   2. Hence write down the minimum value of \(x ^ { 2 } - 7 x + 2\).
2. Describe the geometrical transformation which maps the graph of \(y = x ^ { 2 } - 7 x + 2\) onto the graph of \(y = ( x - 4 ) ^ { 2 }\).
   [0pt] [3 marks]

8. \begin{figure}[h] \end{figure} Figure 1 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. \(y = \mathrm { f } ( 2 x )\).

1. The curve \(C\) has the equation \(y = ( x - a ) ^ { 2 }\) where \(a\) is a constant.

Given that $$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 x - 6 ,$$

1. find the value of \(a\),
2. describe fully a single transformation that would map \(C\) onto the graph of \(y = x ^ { 2 }\).

6. \(f ( x ) = x ^ { 2 } - 10 x + 17\).

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. State the coordinates of the minimum point of the curve \(y = \mathrm { f } ( x )\).
3. Deduce the coordinates of the minimum point of each of the following curves:
   1. \(\quad y = \mathrm { f } ( x ) + 4\),
   2. \(y = \mathrm { f } ( 2 x )\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum at \(( - 3,4 )\) and a minimum at \(( 1 , - 2 )\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations

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

6

1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
   1. \(y = 2 \sin x\);
   2. \(y = - \sin x\);
   3. \(\quad y = \sin \left( x - 30 ^ { \circ } \right)\).
2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).

7 The diagram shows the graph of \(y = \cos 2 x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-4_518_906_1098_552}

1. Write down the coordinates of the points \(A , B\) and \(C\) marked on the diagram.
2. Describe the single geometrical transformation by which the curve with equation \(y = \cos 2 x\) can be obtained from the curve with equation \(y = \cos x\).
3. Solve the equation $$\cos 2 x = 0.37$$ giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\). (No credit will be given for simply reading values from a graph.)
   (5 marks)

8 The diagram shows a sketch of the curve with equation \(y = 3 ^ { x } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{4a4d4dcd-4137-427d-834f-ac2fe83f8aeb-5_535_1011_411_513} The curve intersects the \(y\)-axis at the point \(A\).

1. Write down the \(y\)-coordinate of point \(A\).
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximation for \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\), giving your answer to three significant figures.
      (4 marks)
   2. By considering the graph of \(y = 3 ^ { x } + 1\), explain with the aid of a diagram whether your approximation will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 1 } \left( 3 ^ { x } + 1 \right) \mathrm { d } x\).
      (2 marks)
3. The line \(y = 5\) intersects the curve \(y = 3 ^ { x } + 1\) at the point \(P\). By solving a suitable equation, find the \(x\)-coordinate of the point \(P\). Give your answer to four decimal places.
   (4 marks)
4. The curve \(y = 3 ^ { x } + 1\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
   (1 mark)

6

1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
   1. \(y = 2 \sin x\);
   2. \(y = - \sin x\);
   3. \(y = \sin \left( x - 30 ^ { \circ } \right)\).
2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).

8

1. Sketch the graph of \(y = 3 ^ { x }\), stating the coordinates of the point where the graph crosses the \(y\)-axis.
2. Describe a single geometrical transformation that maps the graph of \(y = 3 ^ { x }\) :
   1. onto the graph of \(y = 3 ^ { 2 x }\);
   2. onto the graph of \(y = 3 ^ { x + 1 }\).
3. 1. Using the substitution \(Y = 3 ^ { x }\), show that the equation $$9 ^ { x } - 3 ^ { x + 1 } + 2 = 0$$ can be written as $$( Y - 1 ) ( Y - 2 ) = 0$$
   2. Hence show that the equation \(9 ^ { x } - 3 ^ { x + 1 } + 2 = 0\) has a solution \(x = 0\) and, by using logarithms, find the other solution, giving your answer to four decimal places.
      (4 marks)

4 The diagram shows a sketch of the curves with equations \(y = 2 x ^ { \frac { 3 } { 2 } }\) and \(y = 8 x ^ { \frac { 1 } { 2 } }\). \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-3_433_720_1452_644} The curves intersect at the origin and at the point \(A\), where \(x = 4\).

1. 1. For the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\).
      (2 marks)
   2. Find an equation of the normal to the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\) at the point \(A\).
2. 1. Find \(\int 8 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
   2. Find the area of the shaded region bounded by the two curves.
3. Describe a single geometrical transformation that maps the graph of \(y = 2 x ^ { \frac { 3 } { 2 } }\) onto the graph of \(y = 2 ( x + 3 ) ^ { \frac { 3 } { 2 } }\).
   (2 marks)

7

1. Solve the equation \(\sin x = 0.8\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your answers in radians to three significant figures.
2. The diagram shows the graph of the curve \(y = \sin x , 0 \leqslant x \leqslant 2 \pi\) and the lines \(y = k\) and \(y = - k\). \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-5_497_780_552_689} The line \(y = k\) intersects the curve at the points \(P\) and \(Q\), and the line \(y = - k\) intersects the curve at the points \(R\) and \(S\). The point \(M\) is the minimum point of the curve.
   1. Write down the coordinates of the point \(M\).
   2. The \(x\)-coordinate of \(P\) is \(\alpha\). Write down the \(x\)-coordinate of the point \(Q\) in terms of \(\pi\) and \(\alpha\).
   3. Find the length of \(R S\) in terms of \(\pi\) and \(\alpha\), giving your answer in its simplest form.
3. Sketch the graph of \(y = \sin 2 x\) for \(0 \leqslant x \leqslant 2 \pi\), indicating the coordinates of points where the graph intersects the \(x\)-axis and the coordinates of any maximum points.

6

1. Sketch the curve with equation \(y = 2 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 2 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
3. Describe a geometrical transformation that maps the graph of \(y = 2 ^ { x }\) onto the graph of \(y = 2 ^ { x + 7 } + 3\).
4. The curve \(y = 2 ^ { x + k } + 3\) intersects the \(y\)-axis at the point \(A ( 0,8 )\). Show that \(k = \log _ { m } n\), where \(m\) and \(n\) are integers.

4

1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } \sqrt { 27 x ^ { 3 } + 4 } \mathrm {~d} x\), giving your answer to three significant figures.
2. The curve with equation \(y = \sqrt { 27 x ^ { 3 } + 4 }\) is stretched parallel to the \(x\)-axis with scale factor 3 to give the curve with equation \(y = \mathrm { g } ( x )\). Write down an expression for \(\mathrm { g } ( x )\).
   (2 marks)
   \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-05_1988_1717_719_150}

5

1. 1. Describe the geometrical transformation that maps the graph of \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) onto the graph of \(y = ( 1 + 2 x ) ^ { 6 }\).
   2. The curve \(y = \left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) is translated by the vector \(\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression for \(\mathrm { g } ( x )\), simplifying your answer.
2. The first four terms in the binomial expansion of \(\left( 1 + \frac { x } { 3 } \right) ^ { 6 }\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\), giving your answers in their simplest form.

7

1. Describe a geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 3 \times 4 ^ { x }\).
2. Sketch the curve with equation \(y = 3 \times 4 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
3. The curve with equation \(y = 4 ^ { - x }\) intersects the curve \(y = 3 \times 4 ^ { x }\) at the point \(P\). Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.

9

1. Write down the two solutions of the equation \(\tan \left( x + 30 ^ { \circ } \right) = \tan 79 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   (2 marks)
2. Describe a single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x + 30 ^ { \circ } \right)\).
3. 1. Given that \(5 + \sin ^ { 2 } \theta = ( 5 + 3 \cos \theta ) \cos \theta\), show that \(\cos \theta = \frac { 3 } { 4 }\).
   2. Hence solve the equation \(5 + \sin ^ { 2 } 2 x = ( 5 + 3 \cos 2 x ) \cos 2 x\) in the interval \(0 < x < 2 \pi\), giving your values of \(x\) in radians to three significant figures.

4 The diagram shows a curve \(C\) with equation \(y = \sqrt { x }\). The point \(O\) is the origin \(( 0,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-3_488_1136_1009_443} The region bounded by the curve \(C\), the \(x\)-axis and the vertical lines \(x = 1\) and \(x = 4\) is shown shaded in the diagram.

1. 1. Write \(\sqrt { x }\) in the form \(x ^ { p }\), where \(p\) is a constant.
   2. Find \(\int \sqrt { x } \mathrm {~d} x\).
   3. Hence find the area of the shaded region.
2. The point on \(C\) for which \(x = 4\) is \(P\). The tangent to \(C\) at the point \(P\) intersects the \(x\)-axis and the \(y\)-axis at the points \(A\) and \(B\) respectively.
   1. Find an equation for the tangent to the curve \(C\) at the point \(P\).
   2. Find the area of the triangle \(A O B\).
3. Describe the single geometrical transformation by which the curve with equation \(y = \sqrt { x - 1 }\) can be obtained from the curve \(C\).
4. Use the trapezium rule with four ordinates (three strips) to find an approximation for \(\int _ { 1 } ^ { 4 } \sqrt { x - 1 } \mathrm {~d} x\), giving your answer to three significant figures.

6 The diagram shows a sketch of the curve with equation \(y = 27 - 3 ^ { x }\). \includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-4_933_1074_376_484} The curve \(y = 27 - 3 ^ { x }\) intersects the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\).

1. 1. Find the \(y\)-coordinate of point \(A\).
   2. Verify that the \(x\)-coordinate of point \(B\) is 3 .
2. The region, \(R\), bounded by the curve \(y = 27 - 3 ^ { x }\) and the coordinate axes is shaded. Use the trapezium rule with four ordinates (three strips) to find an approximate value for the area of \(R\).
3. 1. Use logarithms to solve the equation \(3 ^ { x } = 13\), giving your answer to four decimal places.
   2. The line \(y = k\) intersects the curve \(y = 27 - 3 ^ { x }\) at the point where \(3 ^ { x } = 13\). Find the value of \(k\).
4. 1. Describe the single geometrical transformation by which the curve with equation \(y = - 3 ^ { x }\) can be obtained from the curve \(y = 27 - 3 ^ { x }\).
   2. Sketch the curve \(y = - 3 ^ { x }\).

8

1. Describe the single geometrical transformation by which the curve with equation \(y = \tan \frac { 1 } { 2 } x\) can be obtained from the curve \(y = \tan x\).
2. Solve the equation \(\tan \frac { 1 } { 2 } x = 3\) in the interval \(\mathbf { 0 } < \boldsymbol { x } < \mathbf { 4 } \boldsymbol { \pi }\), giving your answers in radians to three significant figures.
3. Solve the equation $$\cos \theta ( \sin \theta - 3 \cos \theta ) = 0$$ in the interval \(0 < \theta < 2 \pi\), giving your answers in radians to three significant figures.
   (5 marks)

8 The diagram shows a sketch of the curve with equation \(y = 6 ^ { x }\). \includegraphics[max width=\textwidth, alt={}, center]{a2525df8-dbd0-4b69-b6bb-f8ef6f96f7dc-5_403_506_370_769}

1. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. Explain, with the aid of a diagram, whether your approximate value will be an overestimate or an underestimate of the true value of \(\int _ { 0 } ^ { 2 } 6 ^ { x } \mathrm {~d} x\).
2. 1. Describe a single geometrical transformation that maps the graph of \(y = 6 ^ { x }\) onto the graph of \(y = 6 ^ { 3 x }\).
   2. The line \(y = 84\) intersects the curve \(y = 6 ^ { 3 x }\) at the point \(A\). By using logarithms, find the \(x\)-coordinate of \(A\), giving your answer to three decimal places.
      (4 marks)
3. The graph of \(y = 6 ^ { x }\) is translated by \(\left[ \begin{array} { c } 1 \\ - 2 \end{array} \right]\) to give the graph of the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

8 The diagram shows a sketch of the curve \(y = 2 ^ { 4 x }\). \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-9_435_814_374_623} The curve intersects the \(y\)-axis at the point \(A\).

1. Find the value of the \(y\)-coordinate of \(A\).
2. Use the trapezium rule with six ordinates (five strips) to find an approximate value for \(\int _ { 0 } ^ { 1 } 2 ^ { 4 x } \mathrm {~d} x\), giving your answer to two decimal places.
3. Describe the geometrical transformation that maps the graph of \(y = 2 ^ { 4 x }\) onto the graph of \(y = 2 ^ { 4 x - 3 }\).
4. The curve \(y = 2 ^ { 4 x }\) is translated by the vector \(\left[ \begin{array} { c } 1 \\ - \frac { 1 } { 2 } \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). The curve \(y = \mathrm { g } ( x )\) crosses the \(x\)-axis at the point \(Q\). Find the \(x\)-coordinate of \(Q\).
5. 1. Given that $$\log _ { a } k = 3 \log _ { a } 2 + \log _ { a } 5 - \log _ { a } 4$$ show that \(k = 10\).
   2. The line \(y = \frac { 5 } { 4 }\) crosses the curve \(y = 2 ^ { 4 x - 3 }\) at the point \(P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 1 } { 4 \log _ { 10 } 2 }\).

4

1. Sketch the curve with equation \(y = 4 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
   (2 marks)
2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { x } - 5\).
3. 1. Use the substitution \(Y = 2 ^ { x }\) to show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) can be written as \(Y ^ { 2 } - 4 Y - 5 = 0\).
   2. Hence show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) has only one real solution. Use logarithms to find this solution, giving your answer to three decimal places.
      (4 marks)

9 The diagram shows part of a curve whose equation is \(y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)\). \includegraphics[max width=\textwidth, alt={}, center]{a5fa3066-e330-46d0-98e3-92d438ed6f61-5_355_451_367_799}

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 0 } ^ { 1 } \log _ { 10 } \left( x ^ { 2 } + 1 \right) d x$$ giving your answer to three significant figures.
2. The graph of \(y = 2 \log _ { 10 } x\) can be transformed into the graph of \(y = 1 + 2 \log _ { 10 } x\) by means of a translation. Write down the vector of the translation.
3. 1. Show that \(\log _ { 10 } \left( 10 x ^ { 2 } \right) = 1 + 2 \log _ { 10 } x\).
   2. Show that the graph of \(y = 2 \log _ { 10 } x\) can also be transformed into the graph of \(y = 1 + 2 \log _ { 10 } x\) by means of a stretch, and describe the stretch.
   3. The curve with equation \(y = 1 + 2 \log _ { 10 } x\) intersects the curve \(y = \log _ { 10 } \left( x ^ { 2 } + 1 \right)\) at the point \(P\). Given that the \(x\)-coordinate of \(P\) is positive, find the gradient of the line \(O P\), where \(O\) is the origin. Give your answer in the form \(\log _ { 10 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

4

1. Sketch the graph of \(y = 9 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
   (2 marks)
2. Use logarithms to solve the equation \(9 ^ { x } = 15\), giving your value of \(x\) to three significant figures.
3. The curve \(y = 9 ^ { x }\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
   (l mark)