Stationary points after transformation

5 \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-06_743_750_269_687} The diagram shows a curve which has a maximum point at \(( 8,12 )\) and a minimum point at \(( 8,0 )\). The curve is the result of applying a combination of two transformations to a circle. The first transformation applied is a translation of \(\binom { 7 } { - 3 }\). The second transformation applied is a stretch in the \(y\)-direction.

1. State the scale factor of the stretch.
2. State the radius of the original circle.
3. State the coordinates of the centre of the circle after the translation has been completed but before the stretch is applied.
4. State the coordinates of the centre of the original circle.

8 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-10_515_936_274_577} The diagram shows part of the graph of \(y = \sin ( a ( x + b ) )\), where \(a\) and \(b\) are positive constants.

1. State the value of \(a\) and one possible value of \(b\).
   Another curve, with equation \(y = \mathrm { f } ( x )\), has a single stationary point at the point \(( p , q )\), where \(p\) and \(q\) are constants. This curve is transformed to a curve with equation $$y = - 3 f \left( \frac { 1 } { 4 } ( x + 8 ) \right) .$$
2. For the transformed curve, find the coordinates of the stationary point, giving your answer in terms of \(p\) and \(q\).

8. \begin{figure}[h] \end{figure} The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is shown in Figure 4. The curve \(C\)

• has a single turning point, a maximum at ( 4,9 )
• crosses the coordinate axes at only two places, \(( - 3,0 )\) and \(( 0,6 )\)
• has a single asymptote with equation \(y = 4\) as shown in Figure 4.
  1. state the possible values for \(k\). The curve \(C\) is transformed to a new curve that passes through the origin.
  2. 1. Given that the new curve has equation \(y = \mathrm { f } ( x ) - a\), state the value of the constant \(a\).
     2. Write down an equation for another single transformation of \(C\) that also passes through the origin.

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with the equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve has a turning point at \(A ( 3 , - 4 )\) and also passes through the point \(( 0,5 )\).

1. Write down the coordinates of the point to which \(A\) is transformed on the curve with equation
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 2 f \left( \frac { 1 } { 2 } x \right)\).
2. Sketch the curve with equation $$y = \mathrm { f } ( | x | )$$ On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the \(y\)-axis. The curve with equation \(y = \mathrm { f } ( x )\) is a translation of the curve with equation \(y = x ^ { 2 }\).
3. Find \(\mathrm { f } ( x )\).
4. Explain why the function f does not have an inverse.

10 The curve \(y = \mathrm { f } ( x )\) has a minimum point at \(( 3,5 )\).
State the coordinates of the corresponding minimum point on the graph of

1. \(y = 3 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( 2 x )\).

2 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 \(x = 1\).

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 \(O\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a minimum point at \(\left( - \frac { 3 } { 2 } , 0 \right)\), a maximum point at \(( 3,6 )\) and crosses the \(y\)-axis at \(( 0,4 )\). Sketch each of the following graphs on separate diagrams. In each case, show the coordinates of any turning points and of any points where the graph meets the coordinate axes.

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

The curve C has equation \(y = f(x)\) C has a maximum point at P with coordinates \((a, 2b)\) as shown in the diagram below. \includegraphics{figure_7}

1. C is mapped by a single transformation onto curve \(C_1\) with equation \(y = f(x + 2)\) State the coordinates of the maximum point on curve \(C_1\) [1 mark]
2. C is mapped by a single transformation onto curve \(C_2\) with equation \(y = 4f(x)\) State the coordinates of the maximum point on curve \(C_2\) [1 mark]
3. C is mapped by a stretch in the \(x\)-direction onto curve \(C_3\) with equation \(y = f(3x)\) State the scale factor of the stretch. [1 mark]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{g}(x)\). 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 \(y = 2\text{g}(x)\). [1]
2. State the largest root of the equation $$\text{g}(x + 1) = 0$$ [1]
3. State the range of values of \(x\) for which \(\text{g}'(x) \leqslant 0\) [1]

Given that the equation \(\text{g}(x) + k = 0\), where \(k\) is a constant, has no real roots,

1. state the range of possible values for \(k\). [1]

\includegraphics{figure_3} Figure 3 shows part of the curve with equation \(y = 3\cos x^2\). 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 \(d\). [1]
2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3\cos x^2\) to the curve with equation
   1. \(y = 3\cos\left(\frac{x^2}{4}\right)\)
   2. \(y = 3\cos(x - 36)^2\)
   [2]
3. Solve, for \(450° \leq \theta < 720°\), $$3\cos\theta = 8\tan\theta$$ giving your solution to one decimal place. [4]