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

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 \(q\).

5

1. The curve \(y = x ^ { 2 } + 3 x + 4\) is translated by \(\binom { 2 } { 0 }\).
   Find and simplify the equation of the translated curve.
2. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 3 \mathrm { f } ( - x )\). Describe fully the two single transformations which have been combined to give the resulting transformation.

10 The functions \(f\) and \(g\) are defined as follows: $$\begin{array} { l l } \mathrm { f } : x \mapsto x ^ { 2 } - 2 x , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto 2 x + 3 , & x \in \mathbb { R } . \end{array}$$

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 15\).
2. Find the range of f and state, with a reason, whether f has an inverse.
3. Show that the equation \(\operatorname { gf } ( x ) = 0\) has no real solutions.
4. Sketch, in a single diagram, the graphs of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\), making clear the relationship between the graphs.

11

1. The one-one function f is defined by \(\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 1\) for \(x < a\), where \(a\) is a constant.
   1. State the greatest possible value of \(a\).
   2. It is given that \(a\) takes this greatest possible value. State the range of f and find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
2. The function g is defined by \(\mathrm { g } ( x ) = ( x - 3 ) ^ { 2 }\) for \(x \geqslant 0\).
   1. Show that \(\operatorname { gg } ( 2 x )\) can be expressed in the form \(( 2 x - 3 ) ^ { 4 } + b ( 2 x - 3 ) ^ { 2 } + c\), where \(b\) and \(c\) are constants to be found.
   2. Hence expand \(\operatorname { gg } ( 2 x )\) completely, simplifying your answer.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

10. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 4 x - 3 ) ( x - 5 ) ^ { 2 }$$

1. Sketch \(C _ { 1 }\) showing the coordinates of any point where the curve touches or crosses the coordinate axes.
2. Hence or otherwise
   1. find the values of \(x\) for which \(\mathrm { f } \left( \frac { 1 } { 4 } x \right) = 0\)
   2. find the value of the constant \(p\) such that the curve with equation \(y = \mathrm { f } ( x ) + p\) passes through the origin. A second curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = \mathrm { f } ( x + 1 )\)
3. 1. Find, in simplest form, \(\mathrm { g } ( x )\). You may leave your answer in a factorised form.
   2. Hence, or otherwise, find the \(y\) intercept of curve \(C _ { 2 }\)

6. (a) Sketch the curve with equation $$y = - \frac { k } { x } \quad k > 0 \quad x \neq 0$$ (b) On a separate diagram, sketch the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ stating the coordinates of the point of intersection with the \(x\)-axis and, in terms of \(k\), the equation of the horizontal asymptote.
(c) Find the range of possible values of \(k\) for which the curve with equation $$y = - \frac { k } { x } + k \quad k > 0 \quad x \neq 0$$ does not touch or intersect the line with equation \(y = 3 x + 4\) \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-21_72_47_2615_1886}

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 4 ) ( x - 2 ) ( 2 x - 9 )$$ Given that the curve with equation \(y = \mathrm { f } ( x ) - p\) passes through the point with coordinates \(( 0,50 )\)

1. find the value of the constant \(p\). Given that the curve with equation \(y = \mathrm { f } ( x + q )\) passes through the origin,
2. write down the possible values of the constant \(q\).
3. Find \(\mathrm { f } ^ { \prime } ( x )\).
4. Hence find the range of values of \(x\) for which the gradient of the curve with equation \(y = \mathrm { f } ( x )\) is less than - 18 \includegraphics[max width=\textwidth, alt={}, center]{6c320b71-8793-461a-a078-e4f64c144a3a-23_68_37_2617_1914}

1. (a) On Diagram 1, sketch a graph of the curve \(C\) with equation

$$y = \frac { 6 } { x } \quad x \neq 0$$ The curve \(C\) is transformed onto the curve with equation \(y = \frac { 6 } { x - 2 } \quad x \neq 2\) (b) Fully describe this transformation. The curve with equation $$y = \frac { 6 } { x - 2 } \quad x \neq 2$$ and the line with equation $$y = k x + 7 \quad \text { where } k \text { is a constant }$$ intersect at exactly two points, \(P\) and \(Q\).
Given that the \(x\) coordinate of point \(P\) is - 4
(c) find the value of \(k\),
(d) find, using algebra, the coordinates of point \(Q\).
(Solutions relying entirely on calculator technology are not acceptable.) \section*{Diagram 1} Only use this copy of Diagram 1 if you need to redraw your graph. \includegraphics[max width=\textwidth, alt={}, center]{bb21001f-fe68-4776-992d-ede1aae233d7-19_709_739_1802_664} Copy of Diagram 1
(Total for Question 7 is 10 marks)

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

$$f ( x ) = ( x - 4 ) ( 2 x + 1 ) ^ { 2 }$$ The curve touches the \(x\)-axis at the point \(P\) and crosses the \(x\)-axis at the point \(Q\).

1. State the coordinates of the point \(P\).
2. Find \(f ^ { \prime } ( x )\).
3. Hence show that the equation of the tangent to the curve at the point where \(x = \frac { 5 } { 2 }\) can be expressed in the form \(y = k\), where \(k\) is a constant to be found. The curve with equation \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, passes through the origin \(O\).
4. State the possible values of \(a\).
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4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\) The curve has a minimum at \(P ( - 1,0 )\) and a maximum at \(Q \left( \frac { 3 } { 2 } , 2 \right)\) The line with equation \(y = 1\) is the only asymptote to the curve. On separate diagrams sketch the curves with equation

1. \(y = \mathrm { f } ( x ) - 2\)
2. \(y = \mathrm { f } ( - x )\) On each sketch you must clearly state
   • the coordinates of the maximum and minimum points
   • the equation of the asymptote

\begin{figure}[h] \end{figure} Figure 3 shows part of the graph of the trigonometric function with equation \(y = \mathrm { f } ( x )\)

1. Write down an expression for \(\mathrm { f } ( x )\) On a separate diagram,
2. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { f } \left( x + \frac { \pi } { 4 } \right)\) Show clearly the coordinates of all the points where the curve intersects the coordinate axes.
   (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_378_1251_1617_408} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows part of the graph of the trigonometric function with equation \(y = \mathrm { g } ( x )\)
3. Write down an expression for \(\mathrm { g } ( x )\) On a separate diagram,
4. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { g } ( x ) - 2\) Show clearly the coordinates of the \(y\) intercept.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the points \(( - 1,0 )\) and \(( 0,2 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( \mathrm { x } + 3 )\)
2. \(y = \mathrm { f } ( - 3 x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.

10. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\) The curve crosses the \(y\)-axis at the point \(P\), as shown.
2. Expand \(\mathrm { f } ( x )\) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found.
3. Hence, or otherwise, find
   1. the coordinates of \(P\),
   2. the gradient of the curve at \(P\). The curve with equation \(y = \mathrm { f } ( x )\) is translated two units in the positive \(x\) direction to a curve with equation \(y = \mathrm { g } ( x )\).
4. 1. Find \(\mathrm { g } ( x )\), giving your answer in a simplified factorised form.
   2. Hence state the \(y\) intercept of the curve with equation \(y = \mathrm { g } ( x )\).

5. (i) \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the points \(( - 5,0 )\) and \(( 0 , - 3 )\) and touches the \(x\)-axis at the point \(( 2,0 )\). On separate diagrams sketch the curve with equation

1. \(y = \mathrm { f } ( x + 2 )\)
2. \(y = \mathrm { f } ( - x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.
   (ii) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{dfb4b2bc-4bc8-4e5b-9b13-ffe4fbde1b4f-14_415_814_1548_571} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve with equation $$y = k \cos \left( x + \frac { \pi } { 6 } \right) \quad 0 \leqslant x \leqslant 2 \pi$$ where \(k\) is a constant.
   The curve meets the \(y\)-axis at the point \(( 0 , \sqrt { 3 } )\) and passes through the points \(( p , 0 )\) and ( \(q , 0\) ). Find
   1. the value of \(k\),
   2. the exact value of \(p\) and the exact value of \(q\).

8. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation $$y = 4 + 12 x - 3 x ^ { 2 }$$ The point \(M\) is the maximum turning point on \(C\).

1. 1. Write \(4 + 12 x - 3 x ^ { 2 }\) in the form $$a + b ( x + c ) ^ { 2 }$$ where \(a , b\) and \(c\) are constants to be found.
   2. Hence, or otherwise, state the coordinates of \(M\). The line \(l _ { 1 }\) passes through \(O\) and \(M\), as shown in Figure 4.
      A line \(l _ { 2 }\) touches \(C\) and is parallel to \(l _ { 1 }\)
2. Find an equation for \(l _ { 2 }\)

7. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 4,6 ) , Q ( - 1,6 ) , R ( 2,6 )\) and \(S ( 3,6 )\) lie on the curve.

1. Using Figure 1, find the range of values of \(x\) for which $$\mathrm { f } ( x ) < 6$$
2. State the largest solution of the equation $$f ( 2 x ) = 6$$
3. 1. Sketch the curve with equation \(y = \mathrm { f } ( - x )\). On your sketch, state the coordinates of the points to which \(P , Q , R\) and \(S\) are transformed.
   2. Hence find the set of values of \(x\) for which $$f ( - x ) \geqslant 6 \text { and } x < 0$$

3. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points ( \(2.5,0\) ) and ( 0,9 ), has a stationary point at ( 1,11 ), and has an asymptote \(y = 3\) On separate diagrams, sketch the curve with equation

1. \(y = 3 \mathrm { f } ( x )\)
2. \(y = \mathrm { f } ( - x )\) On each diagram show clearly the coordinates of the points of intersection of the curve with the two coordinate axes, the coordinates of the stationary point, and the equation of the asymptote.

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 8 } { x } + \frac { 1 } { 2 } x - 5 , \quad 0 < x \leqslant 12$$ The curve crosses the \(x\)-axis at \(( 2,0 )\) and \(( 8,0 )\) and has a minimum point at \(A\).

1. Use calculus to find the coordinates of point \(A\).
2. State
   1. the roots of the equation \(2 \mathrm { f } ( x ) = 0\)
   2. the coordinates of the turning point on the curve \(y = \mathrm { f } ( x ) + 2\)
   3. the roots of the equation \(\mathrm { f } ( 4 x ) = 0\)

13. \(\mathrm { f } ( x ) = 3 x ^ { 3 } + 3 x ^ { 2 } + c x + 12\), where \(c\) is a constant Given that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that \(c = - 14\)
2. Write \(\mathrm { f } ( x )\) in the form $$\mathrm { f } ( x ) = ( x + 3 ) \mathrm { Q } ( x )$$ where \(\mathrm { Q } ( x )\) is a quadratic function.
3. Use the answer to part (b) to prove that the equation \(\mathrm { f } ( x ) = 0\) has only one real solution. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{75d68987-2314-4c8f-8160-24977c5c4e34-32_595_915_1034_518} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). On separate diagrams sketch the curve with equation
4. 1. \(y = \mathrm { f } ( 3 x )\)
   2. \(y = - \mathrm { f } ( \mathrm { x } )\) On each diagram show clearly the coordinates of the points where the curve crosses the coordinate axes.

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \frac { 16 } { x } , \quad x > 0$$ The curve has a minimum turning point at \(A\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Hence find the coordinates of \(A\).
3. Use your answer to part (b) to write down the turning point of the curve with equation
   1. \(y = \mathrm { f } ( x + 1 )\),
   2. \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).

12. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the \(x\)-axis at the origin and at the point \(( 6,0 )\). The curve has maximum points at \(( 1,6 )\) and \(( 5,6 )\) and has a minimum point at \(( 3,2 )\). On separate diagrams sketch the curve with equation

1. \(y = - \mathrm { f } ( x )\)
2. \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\)
3. \(y = \mathrm { f } ( x + 4 )\) On each diagram show clearly the coordinates of the maximum and minimum points, and the coordinates of the points where the curve crosses the \(x\)-axis.

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(y\)-axis at the point \(( 0,8 )\). The line with equation \(y = 10\) is the only asymptote to the curve.
The curve has a single turning point, a minimum point at \(( 2,5 )\), as shown in Figure 3.

1. State the coordinates of the minimum point of the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 4 } x \right)\)
2. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( x ) - 3\) The curve with equation \(y = \mathrm { f } ( x )\) meets the line with equation \(y = k\), where \(k\) is a constant, at two distinct points.
3. State the set of possible values for \(k\).
4. Sketch the curve with equation \(y = - \mathrm { f } ( x )\). On your sketch, show clearly the coordinates of the turning point, the coordinates of the intersection with the \(y\)-axis and the equation of the asymptote. \section*{\textbackslash section*\{D\}}

14. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ^ { 2 } ( 2 x + 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(\left( - \frac { 1 } { 2 } , 0 \right)\), touches it at \(( 2,0 )\) and crosses the \(y\)-axis at ( 0,4 ). There is a maximum turning point at the point marked \(P\).

1. Use \(\mathrm { f } ^ { \prime } ( x )\) to find the exact coordinates of the turning point \(P\). A second curve \(C _ { 2 }\) has equation \(y = \mathrm { f } ( x + 1 )\).
2. Write down an equation of the curve \(C _ { 2 }\) You may leave your equation in a factorised form.
3. Use your answer to part (b) to find the coordinates of the point where the curve \(C _ { 2 }\) meets the \(y\)-axis.
4. Write down the coordinates of the two turning points on the curve \(C _ { 2 }\)
5. Sketch the curve \(C _ { 2 }\), with equation \(y = \mathrm { f } ( x + 1 )\), giving the coordinates of the points where the curve crosses or touches the \(x\)-axis.

1. (a) Sketch the graph of \(y = 3 ^ { x - 2 } , x \in \mathbb { R }\)

Give the exact values for the coordinates of the point where your graph crosses the \(y\)-axis. The table below gives corresponding values of \(x\) and \(y\), for \(y = 3 ^ { x - 2 }\) The values of \(y\) are rounded to 3 decimal places where necessary. (b) Use the trapezium rule with all the values of \(y\) from the table to find an approximate value for $$\int _ { 0.5 } ^ { 3 } 3 ^ { x - 2 } \mathrm {~d} x$$ Give your answer to 2 decimal places.