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

5

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } \sqrt { 8 x ^ { 3 } + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
2. Describe the single transformation that maps the graph of \(y = \sqrt { 8 x ^ { 3 } + 1 }\) onto the graph of \(y = \sqrt { x ^ { 3 } + 1 }\).
3. The curve with equation \(y = \sqrt { x ^ { 3 } + 1 }\) is translated by \(\left[ \begin{array} { c } 2 \\ - 0.7 \end{array} \right]\) to give the curve with equation \(y = \mathrm { g } ( x )\). Find the value of \(\mathrm { g } ( 4 )\).
   (3 marks)

6

1. Sketch, on the axes given below, the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Describe the geometrical transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = \sin 5 x\).
3. Describe the single geometrical transformation that maps the graph of \(y = \sin 5 x\) onto the graph of \(y = \sin \left( 5 x + 10 ^ { \circ } \right)\).
   [0pt] [2 marks]
   1. \includegraphics[max width=\textwidth, alt={}, center]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-12_675_1417_906_370}

9 A curve has equation \(y = 3 \times 12 ^ { x }\).

1. The point ( \(k , 6\) ) lies on the curve \(y = 3 \times 12 ^ { x }\). Use logarithms to find the value of \(k\), giving your answer to three significant figures.
2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } 3 \times 12 ^ { x } \mathrm {~d} x\), giving your answer to two significant figures.
3. The curve \(y = 3 \times 12 ^ { x }\) is translated by the vector \(\left[ \begin{array} { l } 1 \\ p \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the curve \(y = \mathrm { f } ( x )\) passes through the origin ( 0,0 ), find the value of the constant \(p\).
4. The curve with equation \(y = 2 ^ { 2 - x }\) intersects the curve \(y = 3 \times 12 ^ { x }\) at the point \(T\). Show that the \(x\)-coordinate of \(T\) can be written in the form \(\frac { 2 - \log _ { 2 } 3 } { q + \log _ { 2 } 3 }\), where \(q\) is an integer. State the value of \(q\).
   [0pt] [5 marks]
   \includegraphics[max width=\textwidth, alt={}]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-20_2288_1707_221_153}

7 The diagram shows a sketch of two curves. \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-14_448_527_370_762} The equations of the two curves are \(y = 1 + \sqrt { x }\) and \(y = 4 ^ { \frac { x } { 9 } }\).
The curves meet at the points \(P ( 0,1 )\) and \(Q ( 9,4 )\).

1. 1. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x }\) onto the graph of \(y = 1 + \sqrt { x }\).
   2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { \frac { x } { 9 } }\).
2. 1. Given that \(\int _ { 0 } ^ { 9 } \sqrt { x } \mathrm {~d} x = 18\), find the value of \(\int _ { 0 } ^ { 9 } ( 1 + \sqrt { x } ) \mathrm { d } x\).
   2. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 9 } 4 ^ { \frac { x } { 9 } } \mathrm {~d} x\). Give your answer to one decimal place.
   3. Hence find an approximate value for the area of the shaded region bounded by the two curves and state, with an explanation, whether your approximation will be an overestimate or an underestimate of the true value for the area of the shaded region.
      [0pt] [3 marks]

2

1. Sketch the graph of \(y = ( 0.2 ) ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
2. Use logarithms to solve the equation \(( 0.2 ) ^ { x } = 4\), giving your answer to three significant figures.
3. Describe the geometrical transformation that maps the graph of \(y = ( 0.2 ) ^ { x }\) onto the graph of \(y = 5 ^ { x }\).
   [0pt] [1 mark]

3 The diagram shows a curve with a maximum point \(M\). \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-06_512_867_354_589} The curve is defined for \(x > 0\) by the equation $$y = 6 x ^ { \frac { 1 } { 2 } } - x - 3$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the \(y\)-coordinate of the maximum point \(M\).
3. Find an equation of the normal to the curve at the point \(P ( 4,5 )\).
4. It is given that the normal to the curve at \(P\), when translated by the vector \(\left[ \begin{array} { l } k \\ 0 \end{array} \right]\), passes through the point \(M\). Find the value of the constant \(k\).
   [0pt] [3 marks]

5

1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 2 } ^ { 11 } \sqrt { x ^ { 2 } + 9 } \mathrm {~d} x\). Give your answer to one decimal place.
2. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x ^ { 2 } + 9 }\) onto the graph of :
   1. \(y = 5 + \sqrt { x ^ { 2 } + 9 }\);
   2. \(y = 3 \sqrt { x ^ { 2 } + 1 }\).

2 Describe a sequence of two geometrical transformations that maps the graph of \(y = \sec x\) onto the graph of \(y = 1 + \sec 3 x\).

7

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 4 x ^ { 2 } - 5\).
2. Sketch the graph of \(y = \left| 4 x ^ { 2 } - 5 \right|\), indicating the coordinates of the point where the curve crosses the \(y\)-axis.
3. 1. Solve the equation \(\left| 4 x ^ { 2 } - 5 \right| = 4\).
   2. Hence, or otherwise, solve the inequality \(\left| 4 x ^ { 2 } - 5 \right| \geqslant 4\).

4 The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = 3 \cos \frac { 1 } { 2 } x , & \text { for } 0 \leqslant x \leqslant 2 \pi \\ \mathrm {~g} ( x ) = | x | , & \text { for all real values of } x \end{array}$$

1. Find the range of f .
2. The inverse of f is \(\mathrm { f } ^ { - 1 }\).
   1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
   2. Solve the equation \(\mathrm { f } ^ { - 1 } ( x ) = 1\), giving your answer in an exact form.
3. 1. Write down an expression for \(\mathrm { gf } ( x )\).
   2. Sketch the graph of \(y = \operatorname { gf } ( x )\) for \(0 \leqslant x \leqslant 2 \pi\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos \frac { 1 } { 2 } x\).

5

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x - \mathrm { e } )\).
2. Sketch, on the axes given below, the graph of \(y = | 4 \ln ( x - \mathrm { e } ) |\), indicating the exact value of the \(x\)-coordinate where the curve meets the \(x\)-axis.
3. 1. Solve the equation \(| 4 \ln ( x - e ) | = 4\).
   2. Hence, or otherwise, solve the inequality \(| 4 \ln ( x - e ) | \geqslant 4\). \includegraphics[max width=\textwidth, alt={}, center]{7aa76d26-e3c4-4374-ae4f-8bb61e61b135-3_655_1428_2023_315}

4 The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-5_629_1113_370_461}

1. On the axes below, sketch the curve with equation \(y = | \mathrm { f } ( x ) |\).
2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( 2 x - 1 )\).

8 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } + 3\). \includegraphics[max width=\textwidth, alt={}, center]{d5b78fa6-ea3c-497b-94d8-1d5f61288aa5-4_833_1034_1027_513}

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
2. Use the mid-ordinate rule with four strips of equal width to find an estimate for the area of the shaded region \(A\), giving your answer to three significant figures.
3. Find the exact value of the area of the shaded region \(A\).
4. The region \(B\) is indicated on the diagram. Find the area of the region \(B\), giving your answer in the form \(p \mathrm { e } ^ { 8 } + q \mathrm { e } ^ { 4 }\), where \(p\) and \(q\) are numbers to be determined.

5

1. The diagram shows part of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the point \(( a , 0 )\) and the \(y\)-axis at the point \(( 0 , - b )\). \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-4_569_853_1206_589} On separate diagrams, sketch the curves with the following equations. On each diagram, indicate, in terms of \(a\) or \(b\), the coordinates of the points where the curve crosses the coordinate axes.
   1. \(y = | \mathrm { f } ( x ) |\).
   2. \(\quad y = 2 \mathrm { f } ( x )\).
2. 1. Describe a sequence of geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x + 1 ) - 2\).
   2. Find the exact values of the coordinates of the points where the graph of \(y = 4 \ln ( x + 1 ) - 2\) crosses the coordinate axes.

4

1. Sketch the graph of \(y = \left| 50 - x ^ { 2 } \right|\), indicating the coordinates of the point where the graph crosses the \(y\)-axis.
2. Solve the equation \(\left| 50 - x ^ { 2 } \right| = 14\).
3. Hence, or otherwise, solve the inequality \(\left| 50 - x ^ { 2 } \right| > 14\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 50 - x ^ { 2 }\).

2

1. The diagram shows the graph of \(y = \sec x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   
   [diagram]
   1. The point \(A\) on the curve is where \(x = 0\). State the \(y\)-coordinate of \(A\).
   2. Sketch, on the axes given on page 3, the graph of \(y = | \sec 2 x |\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(\sec x = 2\), giving all values of \(x\) in degrees in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
3. Solve the equation \(\left| \sec \left( 2 x - 10 ^ { \circ } \right) \right| = 2\), giving all values of \(x\) in degrees in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

8 The diagram shows the curves \(y = \mathrm { e } ^ { 2 x } - 1\) and \(y = 4 \mathrm { e } ^ { - 2 x } + 2\). \includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-6_958_1492_372_242} The curve \(y = 4 \mathrm { e } ^ { - 2 x } + 2\) crosses the \(y\)-axis at the point \(A\) and the curves intersect at the point \(B\).

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } - 1\).
2. Write down the coordinates of the point \(A\).
3. 1. Show that the \(x\)-coordinate of the point \(B\) satisfies the equation $$\left( \mathrm { e } ^ { 2 x } \right) ^ { 2 } - 3 \mathrm { e } ^ { 2 x } - 4 = 0$$
   2. Hence find the exact value of the \(x\)-coordinate of the point \(B\).
4. Find the exact value of the area of the shaded region bounded by the curves \(y = \mathrm { e } ^ { 2 x } - 1\) and \(y = 4 \mathrm { e } ^ { - 2 x } + 2\) and the \(y\)-axis.

7 The sketch shows part of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-5_632_1029_712_541}

1. On Figure 2 on page 6, sketch the curve with equation \(y = | \mathrm { f } ( x ) |\).
2. On Figure 3 on page 6, sketch the curve with equation \(y = \mathrm { f } ( | x | )\).
3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )\).
4. The maximum point of the curve with equation \(y = \mathrm { f } ( x )\) has coordinates \(( - 1,10 )\). Find the coordinates of the maximum point of the curve with equation \(y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )\).
   (2 marks)
   1. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-6_785_1022_358_548}
      \end{figure}
   2. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-6_776_1022_1395_548}
      \end{figure}

6

1. Sketch the graph of \(y = \cos ^ { - 1 } x\), where \(y\) is in radians. State the coordinates of the end points of the graph.
2. Sketch the graph of \(y = \pi - \cos ^ { - 1 } x\), where \(y\) is in radians. State the coordinates of the end points of the graph.
   (2 marks) \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-05_759_1258_678_431} \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-05_751_1241_1564_443}

7 The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-06_620_1216_356_422}

1. On Figure 1, below, sketch the curve with equation \(y = - \mathrm { f } ( 3 x )\), indicating the values where the curve cuts the coordinate axes.
2. On Figure 2, on page 7, sketch the curve with equation \(y = \mathrm { f } ( | x | )\), indicating the values where the curve cuts the coordinate axes.
3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } \left( - \frac { 1 } { 2 } x \right)\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-06_732_1237_1649_443}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-07_727_1211_340_466}
   \end{figure}

4 The sketch shows part of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-08_536_1054_367_539}

1. On Figure 2 below, sketch the curve with equation \(y = - | \mathrm { f } ( x ) |\).
2. On Figure 3 on the page opposite, sketch the curve with equation \(y = \mathrm { f } ( | 2 x | )\).
3. 1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( 2 x + 2 )\).
   2. Find the coordinates of the image of the point \(P ( 4 , - 3 )\) under the sequence of transformations given in part (c)(i). \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-09_778_1032_424_529}
      \end{figure}

4

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x - 5 }\).
2. The normal to the curve \(y = \mathrm { e } ^ { 2 x - 5 }\) at the point \(P \left( 2 , \mathrm { e } ^ { - 1 } \right)\) intersects the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Show that the area of the triangle \(O A B\) is \(\frac { \left( \mathrm { e } ^ { 2 } + 1 \right) ^ { m } } { \mathrm { e } ^ { n } }\), where \(m\) and \(n\) are integers.
   [0pt] [6 marks]

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the axes at \(( p , 0 )\) and \(( 0 , q )\) and the lines \(x = 1\) and \(y = 2\) are asymptotes of the curve.

1. Showing the coordinates of any points of intersection with the axes and the equations of any asymptotes, sketch on separate diagrams the graphs of
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 2 \mathrm { f } ( x + 1 )\). Given also that $$\mathrm { f } ( x ) \equiv \frac { 2 x - 1 } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq 1$$
2. find the values of \(p\) and \(q\),
3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

5. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1 .$$

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.

5. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 3,2\) ) and a minimum point at \(( 2 , - 4 )\).

1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
   1. \(y = \mathrm { f } ( | x | )\),
   2. \(y = 3 \mathrm { f } ( 2 x )\).
2. Write down the values of the constants \(a\) and \(b\) such that the curve with equation \(y = a + \mathrm { f } ( x + b )\) has a minimum point at the origin \(O\).