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

5. (a) Sketch the graph of \(y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\). Show on your sketch the coordinates of any turning points and the equations of any asymptotes.
(b) Find, in terms of \(\pi\), the \(x\)-coordinates of the points where the graph crosses the \(x\)-axis.

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

7. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\) which meets the coordinate axes at the points \(( a , 0 )\) and \(( 0 , b )\), where \(a\) and \(b\) are constants.

1. Showing, in terms of \(a\) and \(b\), the coordinates of any points of intersection with the axes, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ^ { - 1 } ( x )\),
   2. \(y = 2 \mathrm { f } ( 3 x )\). Given that $$\mathrm { f } ( x ) = 2 - \sqrt { x + 9 } , \quad x \in \mathbb { R } , \quad x \geq - 9 ,$$
2. find the values of \(a\) and \(b\),
3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.

7. \(\quad f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1\).

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. State the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
4. Describe fully two transformations that would map the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) onto the graph of \(y = \sqrt { x } , x \geq 0\).
5. Find an equation for the normal to the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point where \(x = 8\).

7. The function f is defined by $$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R } ,$$ where \(a\) is a positive constant.

1. Showing the coordinates of any points where each graph meets the axes, sketch on separate diagrams the graphs of
   1. \(\quad y = | \mathrm { f } ( x ) |\),
   2. \(y = \mathrm { f } ( | x | )\). The function g is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
2. Find fg(a) in terms of \(a\).
3. Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$

8 A curve has equation \(y ^ { 2 } = 12 x\).

1. Sketch the curve.
2. 1. The curve is translated by 2 units in the positive \(y\) direction. Write down the equation of the curve after this translation.
   2. The original curve is reflected in the line \(y = x\). Write down the equation of the curve after this reflection.
3. 1. Show that if the straight line \(y = x + c\), where \(c\) is a constant, intersects the curve \(y ^ { 2 } = 12 x\), then the \(x\)-coordinates of the points of intersection satisfy the equation $$x ^ { 2 } + ( 2 c - 12 ) x + c ^ { 2 } = 0$$
   2. Hence find the value of \(c\) for which the straight line is a tangent to the curve.
   3. Using this value of \(c\), find the coordinates of the point where the line touches the curve.
   4. In the case where \(c = 4\), determine whether the line intersects the curve or not.

6 The diagram shows a circle \(C\) and a line \(L\), which is the tangent to \(C\) at the point \(( 1,1 )\). The equations of \(C\) and \(L\) are $$x ^ { 2 } + y ^ { 2 } = 2 \text { and } x + y = 2$$ respectively. \includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_760_1301_552_395} The circle \(C\) is now transformed by a stretch with scale factor 2 parallel to the \(x\)-axis. The image of \(C\) under this stretch is an ellipse \(E\).

1. On the diagram below, sketch the ellipse \(E\), indicating the coordinates of the points where it intersects the coordinate axes.
2. Find equations of:
   1. the ellipse \(E\);
   2. the tangent to \(E\) at the point \(( 2,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{a4c5d61d-1af9-449e-b27a-d1e656dcd75a-4_743_1301_1921_420}

7 A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 9 } - y ^ { 2 } = 1$$

1. Find the equations of the asymptotes of \(H\).
2. The asymptotes of \(H\) are shown in the diagram opposite. On the same diagram, sketch the hyperbola \(H\). Indicate on your sketch the coordinates of the points of intersection of \(H\) with the coordinate axes.
3. The hyperbola \(H\) is now translated by the vector \(\left[ \begin{array} { r } - 3 \\ 0 \end{array} \right]\).
   1. Write down the equation of the translated curve.
   2. Calculate the coordinates of the two points of intersection of the translated curve with the line \(y = x\).
4. From your answers to part (c)(ii), deduce the coordinates of the points of intersection of the original hyperbola \(H\) with the line \(y = x - 3\). \includegraphics[max width=\textwidth, alt={}, center]{f9345653-d426-4350-bf1d-901506211078-4_675_1157_1932_495}

9 [Figure 3, printed on the insert, is provided for use in this question.]
The diagram shows the curve with equation $$\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$$ and the straight line with equation $$x + y = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{354cbeda-d84e-433a-8834-a6f20e7e9513-05_805_1499_863_267}

1. Write down the exact coordinates of the points where the curve with equation \(\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1\) intersects the coordinate axes.
2. The curve is translated \(k\) units in the positive \(x\) direction, where \(k\) is a constant. Write down, in terms of \(k\), the equation of the curve after this translation.
3. Show that, if the line \(x + y = 2\) intersects the translated curve, the \(x\)-coordinates of the points of intersection must satisfy the equation $$3 x ^ { 2 } - 2 ( k + 4 ) x + \left( k ^ { 2 } + 6 \right) = 0$$
4. Hence find the two values of \(k\) for which the line \(x + y = 2\) is a tangent to the translated curve. Give your answer in the form \(p \pm \sqrt { q }\), where \(p\) and \(q\) are integers.
5. On Figure 3, show the translated curves corresponding to these two values of \(k\). \end{table} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2 (for use in Question 5)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_677_1056_886_466}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3 (for use in Question 9)} \includegraphics[alt={},max width=\textwidth]{354cbeda-d84e-433a-8834-a6f20e7e9513-10_798_1488_1891_274}
   \end{figure}

7 A curve \(C\) has equation $$y = 7 + \frac { 1 } { x + 1 }$$

1. Define the translation which transforms the curve with equation \(y = \frac { 1 } { x }\) onto the curve \(C\).
2. 1. Write down the equations of the two asymptotes of \(C\).
   2. Find the coordinates of the points where the curve \(C\) intersects the coordinate axes.
3. Sketch the curve \(C\) and its two asymptotes.

6 An ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Sketch the ellipse \(E\), showing the coordinates of the points of intersection of the ellipse with the coordinate axes.
2. The ellipse \(E\) is stretched with scale factor 2 parallel to the \(y\)-axis. Find and simplify the equation of the curve after the stretch.
3. The original ellipse, \(E\), is translated by the vector \(\left[ \begin{array} { l } a \\ b \end{array} \right]\). The equation of the translated ellipse is $$4 x ^ { 2 } + 3 y ^ { 2 } - 8 x + 6 y = 5$$ Find the values of \(a\) and \(b\).

8 The diagram shows the ellipse \(E\) with equation $$\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$$ and the straight line \(L\) with equation $$y = x + 4$$ \includegraphics[max width=\textwidth, alt={}, center]{9f8cd5ed-f5cf-4cf6-8c92-9fd0819238ca-5_675_1120_708_468}

1. Write down the coordinates of the points where the ellipse \(E\) intersects the coordinate axes.
2. The ellipse \(E\) is translated by the vector \(\left[ \begin{array} { c } p \\ 0 \end{array} \right]\), where \(p\) is a constant. Write down the equation of the translated ellipse.
3. Show that, if the translated ellipse intersects the line \(L\), the \(x\)-coordinates of the points of intersection must satisfy the equation $$9 x ^ { 2 } - ( 8 p - 40 ) x + \left( 4 p ^ { 2 } + 60 \right) = 0$$
4. Given that the line \(L\) is a tangent to the translated ellipse, find the coordinates of the two possible points of contact.
   (No credit will be given for solutions based on differentiation.)

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter \(a\) is a real number. You may find it helpful to use a slider (for \(a\) ) to investigate the family of curves \(y = f ( x )\). \begin{enumerate}[label=(\alph*)] \item \begin{enumerate}[label=(\roman*)] \item On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.

• \(a = - 2\)
• \(a = - 1\)
• \(a = 0\)
• State a feature which is common to the curve in all three cases, \(a = - 2\), \(a = - 1\) and \(a = 0\).
• State a feature of the curve for the cases \(a = - 2 , a = - 1\) that is not a feature of the curve in the case \(a = 0\).
• 1. Determine the equation of the oblique asymptote to the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) in terms of \(a\).
  2. For \(b \neq - 1,0,1\) let \(A\) be the point with coordinates ( \(- b , \mathrm { f } ( - b )\) ) and let \(B\) be the point with coordinates ( \(b , \mathrm { f } ( b )\) ).

Show that the \(y\)-coordinate of the point at which the chord to the curve \(y = f ( x )\) between \(A\) and \(B\) meets the \(y\)-axis is independent of \(a\).

With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which

Find its coordinates and fully justify that it is a cusp.

1 A family of curves is given by the equation $$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$ where the parameter \(a\) is a real number.

1. 1. On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
      • \(a = - 0.5\)
      • \(a = - 0.1\)
      • \(a = 0.5\)
      • State one feature of the curve for the cases \(a = - 0.5\) and \(a = - 0.1\) that is not a feature of the curve in the case \(a = 0.5\).
      • By using a slider for \(a\), or otherwise, write down the non-zero value of \(a\) for which the points on the curve (\textit{) all lie on a straight line.
      • Write down the equation of the vertical asymptote of the curve (}).
      The equation of the curve (*) can be written in the form \(y = x + A + \frac { a ^ { 2 } - a } { x - 1 }\), where \(A\) is a constant.
   2. Show that \(A = 0\).
   3. Hence, or otherwise, find the value of $$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$
   4. Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).
   5. In this part of the question the value of the parameter \(a\) satisfies \(0 < a < 1\). For values of \(a\) in this range the curve intersects the \(x\)-axis at points X and Y . The point Z has coordinates \(( 0 , - 1 )\). These three points form a triangle XYZ.
      1. Determine, in terms of \(a\), the area of the triangle XYZ.
      2. Find the maximum area of the triangle XYZ.

5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure} On the insert, draw the graph of

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