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

7

1. By sketching the curves \(y = x ( 2 x + 5 )\) and \(y = \cos ^ { - 1 } x\) (where \(y\) is in radians) in a single diagram, show that the equation \(x ( 2 x + 5 ) = \cos ^ { - 1 } x\) has exactly one real root.
2. Use the iterative formula $$x _ { n + 1 } = \frac { \cos ^ { - 1 } x _ { n } } { 2 x _ { n } + 5 } \text { with } x _ { 1 } = 0.25$$ to find the root correct to 3 significant figures. Show the result of each iteration correct to at least 4 significant figures.
3. Two new curves are obtained by transforming each of the curves \(y = x ( 2 x + 5 )\) and \(y = \cos ^ { - 1 } x\) by the pair of transformations:
   reflection in the \(x\)-axis followed by reflection in the \(y\)-axis.
   State an equation of each of the new curves and determine the coordinates of their point of intersection, giving each coordinate correct to 3 significant figures.

5

1. State the period of the function \(\mathrm { f } ( x ) = 1 + \cos 2 x\), where \(x\) is in degrees.
2. State a sequence of two geometrical transformations which maps the curve \(y = \cos x\) onto the curve \(y = \mathrm { f } ( x )\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\) for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\).

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { \sqrt { 2 x - x ^ { 2 } } }\).
The curve has asymptotes \(x = 0\) and \(x = a\). \begin{figure}[h] \end{figure}

1. Find \(a\). Hence write down the domain of the function.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x - 1 } { \left( 2 x - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }\). Hence find the coordinates of the turning point of the curve, and write down the range of the function. The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
3. (A) Show algebraically that \(\mathrm { g } ( x )\) is an even function.
   (B) Show that \(\mathrm { g } ( x - 1 ) = \mathrm { f } ( x )\).
   (C) Hence prove that the curve \(y = \mathrm { f } ( x )\) is symmetrical, and state its line of symmetry.

4 Fig. 4 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \sqrt { 1 - 9 x ^ { 2 } } , - a \leqslant x \leqslant a\). \begin{figure}[h] \end{figure}

1. Find the value of \(a\).
2. Write down the range of \(\mathrm { f } ( x )\).
3. Sketch the curve \(y = \mathrm { f } \left( \frac { 1 } { 3 } x \right) - 1\).

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), which has a \(y\)-intercept at \(\mathrm { P } ( 0,3 )\), a minimum point at \(\mathrm { Q } ( 1,2 )\), and an asymptote \(x = - 1\). \begin{figure}[h] \end{figure}

1. Find the coordinates of the images of the points P and Q when the curve \(y = \mathrm { f } ( x )\) is transformed to
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( x + 1 ) + 2\). You are now given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { x + 1 } , x \neq - 1\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence find the coordinates of the other turning point on the curve \(y = \mathrm { f } ( x )\).
3. Show that \(\mathrm { f } ( x - 1 ) = x - 2 + \frac { 4 } { x }\).
4. Find \(\int _ { a } ^ { b } \left( x - 2 + \frac { 4 } { x } \right) \mathrm { d } x\) in terms of \(a\) and \(b\). Hence, by choosing suitable values for \(a\) and \(b\), find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).

8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = ( 1 - x ) \mathrm { e } ^ { 2 x }\), with its turning point P . \begin{figure}[h] \end{figure}

1. Write down the coordinates of the intercepts of \(y = \mathrm { f } ( x )\) with the \(x\) - and \(y\)-axes.
2. Find the exact coordinates of the turning point P .
3. Show that the exact area of the region enclosed by the curve and the \(x\) - and \(y\)-axes is \(\frac { 1 } { 4 } \left( e ^ { 2 } - 3 \right)\). The function \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
4. Express \(\mathrm { g } ( x )\) in terms of \(x\). Sketch the curve \(y = \mathrm { g } ( x )\) on the copy of Fig. 8, indicating the coordinates of its intercepts with the \(x\) - and \(y\)-axes and of its turning point.
5. Write down the exact area of the region enclosed by the curve \(y = \mathrm { g } ( x )\) and the \(x\)-and \(y\)-axes.

8 Fig. 8 shows the line \(y = 1\) and the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { ( x - 2 ) ^ { 2 } } { x }\). The curve touches the \(x\)-axis at \(\mathrm { P } ( 2,0 )\) and has another turning point at the point Q . \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 1 - \frac { 4 } { x ^ { 2 } }\), and find \(\mathrm { f } ^ { \prime \prime } ( x )\). Hence find the coordinates of Q and, using \(\mathrm { f } ^ { \prime \prime } ( x )\), verify that it is a maximum point.
2. Verify that the line \(y = 1\) meets the curve \(y = \mathrm { f } ( x )\) at the points with \(x\)-coordinates 1 and 4 . Hence find the exact area of the shaded region enclosed by the line and the curve. The curve \(y = \mathrm { f } ( x )\) is now transformed by a translation with vector \(\binom { - 1 } { - 1 }\). The resulting curve has equation \(y = \mathrm { g } ( x )\).
3. Show that \(\mathrm { g } ( x ) = \frac { x ^ { 2 } - 3 x } { x + 1 }\).
4. Without further calculation, write down the value of \(\int _ { 0 } ^ { 3 } \mathrm {~g} ( x ) \mathrm { d } x\), justifying your answer.

4 \includegraphics[max width=\textwidth, alt={}, center]{074597e7-5bb1-4249-9cfa-784974a6fd2b-2_947_1305_986_420} The diagram shows the curve with equation $$y = \frac { a x + b } { x + c }$$ where \(a , b\) and \(c\) are constants.

1. Given that the asymptotes of the curve are \(x = - 1\) and \(y = - 2\) and that the curve passes through \(( 3,0 )\), find the values of \(a , b\) and \(c\).
2. Sketch the curve with equation $$y ^ { 2 } = \frac { a x + b } { x + c }$$ for the values of \(a , b\) and \(c\) found in part (i). State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes.

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 2 x + 3\).

1. Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\), express \(\mathrm { f } ( x )\) in a fully factorised form.
2. Sketch the graph of \(y = \mathrm { f } ( x )\), indicating the coordinates of any points of intersection with the axes.
3. Solve the inequality \(\mathrm { f } ( x ) < 0\), giving your answer in set notation.
4. The graph of \(y = \mathrm { f } ( x )\) is transformed by a stretch parallel to the \(x\)-axis, scale factor \(\frac { 1 } { 2 }\). Find the equation of the transformed graph.

9

1. Express \(3 \cos 3 x + 7 \sin 3 x\) in the form \(R \cos ( 3 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
2. Give full details of a sequence of three transformations needed to transform the curve \(y = \cos x\) to the curve \(y = 3 \cos 3 x + 7 \sin 3 x\).
3. Determine the greatest value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.
4. Determine the least value of \(3 \cos 3 x + 7 \sin 3 x\) as \(x\) varies and give the smallest positive value of \(x\) for which it occurs.

5

1. The graph of the function \(y = \mathrm { f } ( x )\) passes through the point \(P\) with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to \(P\) on each of the following curves.
   1. \(\quad y = \mathrm { f } ( x ) + 3\)
   2. \(\quad y = 2 \mathrm { f } ( 3 x - 1 )\)
   3. \(y = \mathrm { f } ^ { - 1 } ( x )\)
2. \includegraphics[max width=\textwidth, alt={}, center]{6b766f5c-8533-4e0c-bb10-0d9949dc777b-5_494_739_806_333} The diagram shows part of the graph of \(y = \mathrm { g } ^ { \prime } ( x )\). This is the graph of the gradient function of \(y = \mathrm { g } ( x )\). The graph intersects the \(x\)-axis at \(x = - 2\) and \(x = 4\).
   1. State the \(x\)-coordinate of any stationary points on the graph of \(y = \mathrm { g } ( x )\).
   2. State the set of values of \(x\) for which \(y = \mathrm { g } ( x )\) is a decreasing function.
   3. State the \(x\)-coordinate of any points of inflection on the graph of \(y = \mathrm { g } ( x )\).

5

1. The graph of \(y = 2 ^ { x }\) can be transformed to the graph of \(y = 2 ^ { x + 4 }\) either by a translation or by a stretch.
   1. Give full details of the translation.
   2. Give full details of the stretch.
2. In this question you must show detailed reasoning. Solve the equation \(\log _ { 2 } ( 8 x ) = 1 - \log _ { 2 } ( 1 - x )\).

7 The diagram shows a part \(A B C\) of the curve \(y = 3 - 2 x ^ { 2 }\), together with its reflections in the lines \(y = x\), \(y = - x\) and \(y = 0\). \includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-05_691_673_1957_678}

9. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve with equation \(y = 3 \cos x ^ { \circ }\).
The point \(P ( c , d )\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.

1. State the value of \(c\) and the value of \(d\).
2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3 \cos x ^ { \circ }\) to the curve with equation
   1. \(y = 3 \cos \left( \frac { x ^ { \circ } } { 4 } \right)\)
   2. \(y = 3 \cos ( x - 36 ) ^ { \circ }\)
3. Solve, for \(450 ^ { \circ } \leqslant \theta < 720 ^ { \circ }\), $$3 \cos \theta = 8 \tan \theta$$ giving your solution to one decimal place.
   In part (c) you must show all stages of your working.
   Solutions relying entirely on calculator technology are not acceptable.

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

$$f ( x ) = a x ^ { 3 } + 15 x ^ { 2 } - 39 x + b$$ and \(a\) and \(b\) are constants.
Given

• the point \(( 2,10 )\) lies on \(C\)
• the gradient of the curve at \(( 2,10 )\) is - 3
  1. (i) show that the value of \(a\) is - 2
     (ii) find the value of \(b\).
  2. Hence show that \(C\) has no stationary points.
  3. Write \(\mathrm { f } ( x )\) in the form \(( x - 4 ) \mathrm { Q } ( x )\) where \(\mathrm { Q } ( x )\) is a quadratic expression to be found.
  4. Hence deduce the coordinates of the points of intersection of the curve with equation

$$y = \mathrm { f } ( 0.2 x )$$ and the coordinate axes.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = g ( x )\).
The curve has a single turning point, a minimum, at the point \(M ( 4 , - 1.5 )\).
The curve crosses the \(x\)-axis at two points, \(P ( 2,0 )\) and \(Q ( 7,0 )\).
The curve crosses the \(y\)-axis at a single point \(R ( 0,5 )\).

1. State the coordinates of the turning point on the curve with equation \(y = 2 \mathrm {~g} ( x )\).
2. State the largest root of the equation $$g ( x + 1 ) = 0$$
3. State the range of values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) \leqslant 0\) Given that the equation \(\mathrm { g } ( x ) + k = 0\), where \(k\) is a constant, has no real roots,
4. state the range of possible values for \(k\).

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 9 x + 9 \right) e ^ { - x } , \quad x \in R$$ The curve has a minimum turning point at \(A\) and a maximum turning point at \(B\) as shown in the figure above.
a. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
b. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \left( 2 x ^ { 2 } - 13 x + 18 \right) e ^ { - x }\) c. Hence find the exact coordinates of the turning points of \(C\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation $$y = a \mathrm { f } ( x ) + b , \quad x \geq 0$$ The range of the graph with equation \(y = a \mathrm { f } ( x ) + b\) is \(0 \leq y \leq 9 e ^ { 2 } + 1\) Given that \(a\) and \(b\) are constants.
d. find the value of \(a\) and the value of \(b\).

1. The point \(P ( 2 , - 3 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).

State the coordinates of the image of \(P\) under the transformation represented by the curve
a. \(\quad y = | \mathrm { f } ( x ) |\) b. \(y = \mathrm { f } ( x - 2 )\) c. \(y = 3 \mathrm { f } ( 2 x ) + 2\)

5. $$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$

1. Write \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are integers to be found.
2. Sketch the curve with equation \(y = \mathrm { f } ( x )\) showing any points of intersection with the coordinate axes and the coordinates of any turning point.
3. 1. Describe fully the transformation that maps the curve with equation \(y = \mathrm { f } ( x )\) onto the curve with equation \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
   2. Find the range of the function $$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$

1. The point \(P ( - 2 , - 5 )\) lies on the curve with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\)

Find the point to which \(P\) is mapped, when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation

1. \(y = f ( x ) + 2\)
2. \(y = | f ( x ) |\)
3. \(y = 3 f ( x - 2 ) + 2\)

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { 4 \sin 2 x } { \mathrm { e } ^ { \sqrt { 2 } x - 1 } } , \quad 0 \leqslant x \leqslant \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.

1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2 x = \sqrt { 2 }$$
2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation
   1. \(y = \mathrm { f } ( 2 x )\).
   2. \(y = 3 - 2 \mathrm { f } ( x )\).

1. The point \(P ( 3 , - 2 )\) lies on the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\)

Find the coordinates of the point to which \(P\) is mapped when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation

1. \(y = \mathrm { f } ( x - 2 )\)
2. \(y = \mathrm { f } ( 2 x )\)
3. \(y = 3 \mathrm { f } ( - x ) + 5\)

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = k - | 2 x - 3 k |$$ stating
   $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 f \left( \frac { 1 } { 2 } x \right)$$

3 The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic polynomial in \(x\). This diagram is repeated in the Printed Answer Booklet. \includegraphics[max width=\textwidth, alt={}, center]{d44919ed-806d-48c0-9726-c5fd67764504-03_896_1467_1382_244}

1. State the values of \(x\) for which \(\mathrm { f } ( x ) < \frac { 1 } { 2 }\), giving your answer in set notation.
2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } ( - x )\).
3. Explain how you can tell that \(\mathrm { f } ( x )\) cannot be expressed as the product of three real linear factors.

2

1. The curve \(y = \frac { 2 } { 3 + x }\) is translated by four units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
2. Describe fully the single transformation that transforms the curve \(y = \frac { 2 } { 3 + x }\) to \(y = \frac { 5 } { 3 + x }\).