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

12. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { x ^ { 3 } - 9 x ^ { 2 } - 81 x } { 27 }$$ The curve crosses the \(x\)-axis at the point \(A\), the point \(B\) and the origin \(O\). The curve has a maximum turning point at \(C\) and a minimum turning point at \(D\).

1. Use algebra to find exact values for the \(x\) coordinates of the points \(A\) and \(B\).
2. Use calculus to find the coordinates of the points \(C\) and \(D\). The graph of \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, has its minimum turning point on the \(y\)-axis.
3. Write down the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-35_29_37_182_1914}

2. 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 with equation

1. \(y = \mathrm { f } ( x + 2 )\)
2. \(y = - \mathrm { f } ( x )\)
3. \(2 y = f ( x )\)
4. \(y = \mathrm { f } ( x ) - 4\) State the coordinates of the image of \(P\) under the transformation represented by the curve
   with equation (a) \(y = \mathrm { f } ( x + 2 )\)

10. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 2 } ( 9 - 2 x ) .$$ There is a minimum at the origin, a maximum at the point \(( 3,27 )\) and \(C\) cuts the \(x\)-axis at the point \(A\).

1. Write down the coordinates of the point \(A\).
2. On separate diagrams sketch the curve with equation
   1. \(y = \mathrm { f } ( x + 3 )\),
   2. \(y = \mathrm { f } ( 3 x )\). On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant, has a maximum point at \(( 3,10 )\).
3. Write down the value of \(k\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the points \(( 2,0 )\) and \(( 4,0 )\). The minimum point on the curve is \(P ( 3 , - 2 )\). In separate diagrams sketch the curve with equation

1. \(y = - \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( 2 x )\). On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.

3. Given that \(\quad \mathrm { f } ( x ) = \frac { 1 } { x } , \quad x \neq 0\),

1. sketch the graph of \(y = \mathrm { f } ( x ) + 3\) and state the equations of the asymptotes.
2. Find the coordinates of the point where \(y = \mathrm { f } ( x ) + 3\) crosses a coordinate axis.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the points \(( 1,0 )\) and \(( 4,0 )\). The maximum point on the curve is \(( 2,5 )\).
In separate diagrams sketch the curves with the following equations.
On each diagram show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.

1. \(y = 2 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( - x )\). The maximum point on the curve with equation \(y = \mathrm { f } ( x + a )\) is on the \(y\)-axis.
3. Write down the value of the constant \(a\).

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). There is a maximum at \(( 0,0 )\), a minimum at \(( 2 , - 1 )\) and \(C\) passes through \(( 3,0 )\). On separate diagrams sketch the curve with equation

1. \(y = \mathrm { f } ( x + 3 )\),
2. \(y = \mathrm { f } ( - x )\). On each diagram show clearly the coordinates of the maximum point, the minimum point and any points of intersection with the \(x\)-axis.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve has a maximum point \(( - 2,5 )\) and an asymptote \(y = 1\), as shown in Figure 1. On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( x ) + 2\)
2. \(y = 4 \mathrm { f } ( x )\)
3. \(y = \mathrm { f } ( \mathrm { x } + 1 )\) On each diagram, show clearly the coordinates of the maximum point and the equation of the asymptote.

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { x - 2 } , \quad x \neq 2$$ The curve passes through the origin and has two asymptotes, with equations \(y = 1\) and \(x = 2\), as shown in Figure 1.

1. In the space below, sketch the curve with equation \(y = \mathrm { f } ( x - 1 )\) and state the equations of the asymptotes of this curve.
2. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } ( x - 1 )\) crosses the coordinate axes.

6. Figure 1 shows a sketch of the curve with equation \(y = \frac { 2 } { x } , x \neq 0\) The curve \(C\) has equation \(y = \frac { 2 } { x } - 5 , x \neq 0\), and the line \(l\) has equation \(y = 4 x + 2\)

1. Sketch and clearly label the graphs of \(C\) and \(l\) on a single diagram. On your diagram, show clearly the coordinates of the points where \(C\) and \(l\) cross the coordinate axes.
2. Write down the equations of the asymptotes of the curve \(C\).
3. Find the coordinates of the points of intersection of \(y = \frac { 2 } { x } - 5\) and \(y = 4 x + 2\)

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(y\)-axis at \(( 0,3 )\) and has a minimum at \(P ( 4,2 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( x + 4 )\),
2. \(y = 2 \mathrm { f } ( x )\). On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the \(y\)-axis.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve passes through the origin \(O\) and through the point \(( 6,0 )\). The maximum point on the curve is \(( 3,5 )\). On separate diagrams, sketch the curve with equation

1. \(y = 3 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( x + 2 )\). On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.

3. On separate diagrams, sketch the graphs of

1. \(y = ( x + 3 ) ^ { 2 }\),
2. \(y = ( x + 3 ) ^ { 2 } + k\), where \(k\) is a positive constant. Show on each sketch the coordinates of each point at which the graph meets the axes.

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 point ( 0,7 ) and has a minimum point at ( 7,0 ). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( x ) + 3\),
2. \(y = \mathrm { f } ( 2 x )\). On each diagram, show clearly the coordinates of the minimum point and the coordinates of the point at which the curve crosses the \(y\)-axis.

10. (a) Factorise completely \(x ^ { 3 } - 6 x ^ { 2 } + 9 x\) (b) Sketch the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x$$ showing the coordinates of the points at which the curve meets the \(x\)-axis. Using your answer to part (b), or otherwise,
(c) sketch, on a separate diagram, the curve with equation $$y = ( x - 2 ) ^ { 3 } - 6 ( x - 2 ) ^ { 2 } + 9 ( x - 2 )$$ showing the coordinates of the points at which the curve meets the \(x\)-axis.

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

1. \(y = \mathrm { f } ( x + 3 )\)
2. \(y = 2 \mathrm { f } ( x )\) On each diagram show clearly the coordinates of the maximum and minimum points.
   The graph of \(y = \mathrm { f } ( x ) + a\) has a minimum at (3, 0), where \(a\) is a constant.
3. Write down the value of \(a\).

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the origin and through \(( 6,0 )\).
The curve \(C\) has a minimum at the point \(( 3 , - 1 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( 2 x )\),
2. \(y = - \mathrm { f } ( x )\),
3. \(y = \mathrm { f } ( x + p )\), where \(p\) is a constant and \(0 < p < 3\). On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.

10. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } ( 9 - 2 x )$$ There is a minimum at the origin, a maximum at the point \(( 3,27 )\) and \(C\) cuts the \(x\)-axis at the point \(A\).

1. Write down the coordinates of the point \(A\).
2. On separate diagrams sketch the curve with equation
   1. \(y = \mathrm { f } ( x + 3 )\)
   2. \(y = \mathrm { f } ( 3 x )\) On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant, has a maximum point at \(( 3,10 )\).
3. Write down the value of \(k\).

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the point \(( - 1,0 )\) and touches the \(x\)-axis at the point \(( 2,0 )\).
The curve \(C\) has a maximum at the point ( 0,4 ).

1. The equation of the curve \(C\) can be written in the form $$y = x ^ { 3 } + a x ^ { 2 } + b x + c$$ where \(a\), \(b\) and \(c\) are integers.
   Calculate the values of \(a , b\) and \(c\).
2. Sketch the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) in the space provided on page 24 Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 3 ) ^ { 2 } ( x - 1 ) , \quad x \in \mathbb { R }$$ The curve crosses the \(x\)-axis at \(( 1,0 )\), touches it at \(( - 3,0 )\) and crosses the \(y\)-axis at \(( 0 , - 9 )\)

1. In the space below, sketch the curve \(C\) with equation \(y = \mathrm { f } ( x + 2 )\) and state the coordinates of the points where the curve \(C\) meets the \(x\)-axis.
2. Write down an equation of the curve \(C\).
3. Use your answer to part (b) to find the coordinates of the point where the curve \(C\) meets the \(y\)-axis.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum point \(A\) at \(( - 2,4 )\) and a minimum point \(B\) at \(( 3 , - 8 )\) and passes through the origin \(O\). On separate diagrams, sketch the curve with equation

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

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

1. Given that
   1. the curve with equation \(y = \mathrm { f } ( x ) - k , x \in \mathbb { R }\), passes through the origin, find the value of the constant \(k\),
   2. the curve with equation \(y = \mathrm { f } ( x + c ) , x \in \mathbb { R }\), has a minimum point at the origin, find the value of the constant \(c\).
2. Show that \(\mathrm { f } ^ { \prime } ( x ) = 12 x ^ { 2 } - 16 x - 35\) Points \(A\) and \(B\) are distinct points that lie on the curve \(y = \mathrm { f } ( x )\).
   The gradient of the curve at \(A\) is equal to the gradient of the curve at \(B\).
   Given that point \(A\) has \(x\) coordinate 3
3. find the \(x\) coordinate of point \(B\).
   \includegraphics[max width=\textwidth, alt={}]{c1b0a49d-9def-4289-a4cd-288991f67caf-28_2630_1826_121_121}

5. \begin{figure}[h] \end{figure} Figure 1 shows the sketch of a curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve crosses the \(y\)-axis at \(( 0,4 )\) and crosses the \(x\)-axis at \(( 5,0 )\). The curve has a single turning point, a maximum, at (2, 7). The line with equation \(y = 1\) is the only asymptote to the curve.

1. State the coordinates of the turning point on the curve with equation \(y = \mathrm { f } ( x - 2 )\).
2. State the solution of the equation f( \(2 x\) ) \(= 0\)
3. State the equation of the asymptote to the curve with equation \(y = \mathrm { f } ( - x )\). Given that the line with equation \(y = k\), where \(k\) is a constant, meets the curve \(y = \mathrm { f } ( x )\) at only one point,
4. state the set of possible values for \(k\).

7. (a) Factorise completely \(x ^ { 3 } - 4 x\).
(3)
(b) Sketch the curve with equation \(y = x ^ { 3 } - 4 x\), showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
(c) On a separate diagram, sketch the curve with equation \(y = ( x - 1 ) ^ { 3 } - 4 ( x - 1 ) ,\) showing the coordinates of the points where the curve crosses the \(x\)-axis.
(3)
\end{tabular} & Leave blank
\hline \end{tabular} \end{center} \begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = 4 ^ { x }\) A copy of Figure 1, labelled Diagram 1, is shown on the next page.

1. On Diagram 1, sketch the curve with equation
   1. \(y = 2 ^ { x }\)
   2. \(y = 4 ^ { x } - 6\) Label clearly the coordinates of any points of intersection with the coordinate axes. The curve with equation \(y = 2 ^ { x }\) meets the curve with equation \(y = 4 ^ { x } - 6\) at the point \(P\).
2. Using algebra, find the exact coordinates of \(P\).
   \includegraphics[max width=\textwidth, alt={}]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-05_1009_1490_264_219}
   \section*{Diagram 1}