Transformations of modulus graphs

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x ) , \quad x \in \mathbb { R }\) The graph consists of two half lines that meet at the point \(P ( 2 , - 3 )\), the vertex of the graph.
The graph cuts the \(y\)-axis at the point \(( 0 , - 1 )\) and the \(x\)-axis at the points \(( - 1,0 )\) and \(( 5,0 )\).
Sketch, on separate diagrams, the graph of

1. \(y = \mathrm { f } ( | x | )\),
2. \(y = 2 \mathrm { f } ( x + 5 )\). In each case, give the coordinates of the points where the graph crosses or meets the coordinate axes. Also give the coordinates of any vertices corresponding to the point \(P\).

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 the points \(A ( 5,4 )\) and \(B ( - 5 , - 4 )\).
In separate diagrams, sketch the graph with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ( | x | )\),
3. \(y = 2 f ( x + 1 )\). On each sketch, show the coordinates of the points corresponding to \(A\) and \(B\).

6. \begin{figure}[h] \end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation

1. \(y = \mathrm { f } ( x + 1 )\),
2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
3. the value of \(a\) and the value of \(b\),
4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = f ( x ) , x \in \mathbb { R }\).
The graph consists of two line segments that meet at the point \(P\).
The graph cuts the \(y\)-axis at the point \(Q\) and the \(x\)-axis at the points \(( - 3,0 )\) and \(R\). Sketch, on separate diagrams, the graphs of

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ( - x )\). Given that \(\mathrm { f } ( x ) = 2 - | x + 1 |\),
3. find the coordinates of the points \(P , Q\) and \(R\),
4. solve \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\).

4. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve with equation \(y = \mathrm { f } ( x )\) The curve passes through the points \(P ( - 1.5,0 )\) and \(Q ( 0,5 )\) as shown.
On separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\)
2. \(y = \mathrm { f } ( | x | )\)
3. \(y = 2 f ( 3 x )\) Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the graph with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(Q ( 6 , - 1 )\). The graph crosses the \(y\)-axis at the point \(P ( 0,11 )\). Sketch, on separate diagrams, the graphs of

1. \(y = | f ( x ) |\)
2. \(y = 2 f ( - x ) + 3\) On each diagram, show the coordinates of the points corresponding to \(P\) and \(Q\).
   Given that \(\mathrm { f } ( x ) = a | x - b | - 1\), where \(a\) and \(b\) are constants,
3. state the value of \(a\) and the value of \(b\).

5. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve meets the \(x\)-axis at \(P ( p , 0 )\) and meets the \(y\)-axis at \(Q ( 0 , q )\).

1. On separate diagrams, sketch the curve with equation
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). In each case show, in terms of \(p\) or \(q\), the coordinates of points at which the curve meets the axes. Given that \(\mathrm { f } ( x ) = 3 \ln ( 2 x + 3 )\),
2. state the exact value of \(q\),
3. find the value of \(p\),
4. find an equation for the tangent to the curve at \(P\).

3. \includegraphics[max width=\textwidth, alt={}, center]{14ec6709-e1cb-42d7-af99-91365e50e4fc-1_535_810_877_406} The diagram 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 \(O\).

2 Given that \(\mathrm { f } ( x ) = | x |\) and \(\mathrm { g } ( x ) = x + 1\), sketch the graphs of the composite functions \(y = \mathrm { fg } ( x )\) and \(y = \operatorname { gf } ( x )\), indicating clearly which is which.

6 Given that \(\mathrm { f } ( x ) = | x |\) and \(\mathrm { g } ( x ) = x + 1\), sketch the graphs of the composite functions \(y = \mathrm { fg } ( x )\) and \(y = \operatorname { gf } ( x )\), indicating clearly which is which.

3．（a）On separate diagrams sketch the curves with the following equations．On each sketch you should mark the coordinates of the points where the curve crosses the coordinate axes．
（i）\(y = x ^ { 2 } - 2 x - 3\) （ii）\(y = x ^ { 2 } - 2 | x | - 3\) （iii）\(y = x ^ { 2 } - x - | x | - 3\) （b）Solve the equation $$x ^ { 2 } - x - | x | - 3 = x + | x |$$

2．（a）On separate diagrams，sketch the curves with the following equations．On each sketch you should label the exact coordinates of the points where the curve meets the coordinate axes．
（i）\(y = 8 + 2 x - x ^ { 2 }\) （ii）\(y = 8 + 2 | x | - x ^ { 2 }\) （iii）\(y = 8 + x + | x | - x ^ { 2 }\) （b）Find the values of \(x\) for which $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$

5

1. The diagram shows part of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the point \(( a , 0 )\) and the \(y\)-axis at the point \(( 0 , - b )\). \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-4_569_853_1206_589} On separate diagrams, sketch the curves with the following equations. On each diagram, indicate, in terms of \(a\) or \(b\), the coordinates of the points where the curve crosses the coordinate axes.
   1. \(y = | \mathrm { f } ( x ) |\).
   2. \(\quad y = 2 \mathrm { f } ( x )\).
2. 1. Describe a sequence of geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x + 1 ) - 2\).
   2. Find the exact values of the coordinates of the points where the graph of \(y = 4 \ln ( x + 1 ) - 2\) crosses the coordinate axes.

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 $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$

6. \begin{figure}[h] \end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation

1. \(y = \mathrm { f } ( x + 1 )\),
2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
   Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
3. the value of \(a\) and the value of \(b\),
4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).