Graph with equation y = a|bx+c| + d

6. \begin{figure}[h] \end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 2 | 2 x - 5 | + 3 \quad x \geqslant 0$$ The vertex of the graph is at point \(P\) as shown.

1. State the coordinates of \(P\).
2. Solve the equation \(\mathrm { f } ( x ) = 3 x - 2\) Given that the equation $$f ( x ) = k x + 2$$ where \(k\) is a constant, has exactly two roots,
3. find the range of values of \(k\).
   

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { 1 } { 2 } | 2 x + 7 | - 10$$

1. State the coordinates of the vertex, V, of the graph.
2. Solve, using algebra, $$\frac { 1 } { 2 } | 2 x + 7 | - 10 \geqslant \frac { 1 } { 3 } x + 1$$
3. Sketch the graph with equation $$y = | \mathrm { f } ( x ) |$$ stating the coordinates of the local maximum point and each local minimum point.

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation $$y = | 3 x - 5 a | - 2 a$$ where \(a\) is a positive constant.
The graph

• cuts the \(y\)-axis at the point \(P\)
• cuts the \(x\)-axis at the points \(Q\) and \(R\)
• has a minimum point at \(S\)
  1. Find, in simplest form in terms of \(a\), the coordinates of
     1. point \(P\)
     2. points \(Q\) and \(R\)
     3. point \(S\)
  2. Find, in simplest form in terms of \(a\), the values of \(x\) for which

$$| 3 x - 5 a | - 2 a = | x - 2 a |$$

8. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The graph shown in Figure 2 has equation $$y = a - | 2 x - b |$$ where \(a\) and \(b\) are positive constants, \(a > b\)

1. Find, giving your answer in terms of \(a\) and \(b\),
   1. the coordinates of the maximum point of the graph,
   2. the coordinates of the point of intersection of the graph with the \(y\)-axis,
   3. the coordinates of the points of intersection of the graph with the \(x\)-axis. On page 24 there is a copy of Figure 2 called Diagram 1.
2. On Diagram 1, sketch the graph with equation $$y = | x | - 1$$ Given that the graphs \(y = | x | - 1\) and \(y = a - | 2 x - b |\) intersect at \(x = - 3\) and \(x = 5\)
3. find the value of \(a\) and the value of \(b\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{76989f19-2624-4e86-a8ee-4978dd1014c2-24_675_652_1959_712} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

5. \begin{figure}[h] \end{figure} Figure 2 shows part of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = | k x - 9 | - 2 \quad x \in \mathbb { R }$$ and \(k\) is a positive constant. The graph intersects the \(y\)-axis at the point \(A\) and has a minimum point at \(B\) as shown.

1. 1. Find the \(y\) coordinate of \(A\)
   2. Find, in terms of \(k\), the \(x\) coordinate of \(B\)
2. Find, in terms of \(k\), the range of values of \(x\) that satisfy the inequality $$| k x - 9 | - 2 < 0$$ Given that the line \(y = 3 - 2 x\) intersects the graph \(y = \mathrm { f } ( x )\) at two distinct points,
3. find the range of possible values of \(k\)

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = 3 | x - 2 | - 10$$ The vertex of the graph is at point \(P\), shown in Figure 2.

1. Find the coordinates of \(P\)
2. Find \(\mathrm { ff } ( 0 )\)
3. Solve the inequality $$3 | x - 2 | - 10 < 5 x + 10$$
4. Solve the equation $$\mathrm { f } ( | x | ) = 0$$

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 | x - 5 | + 10$$ The point \(P\), shown in Figure 1, is the vertex of the graph.

1. State the coordinates of \(P\)
2. Use algebra to solve $$2 | x - 5 | + 10 > 6 x$$ (Solutions relying on calculator technology are not acceptable.)
3. Find the point to which \(P\) is mapped, when the graph with equation \(y = \mathrm { f } ( x )\) is transformed to the graph with equation \(y = 3 \mathrm { f } ( x - 2 )\)

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 21 - 2 | 2 - x | \quad x \geqslant 0$$

1. Find ff(6)
2. Solve the equation \(\mathrm { f } ( x ) = 5 x\) Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
3. state the set of possible values of \(k\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = a \mathrm { f } ( x - b )\) The vertex of the graph with equation \(y = a \mathrm { f } ( x - b )\) is (6, 3). Given that \(a\) and \(b\) are constants,
4. find the value of \(a\) and the value of \(b\). \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-11_2255_50_314_34}

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = | 3 x - 13 | + 5 \quad x \in \mathbb { R }$$ The vertex of the graph is at point \(P\), as shown in Figure 1.

1. State the coordinates of \(P\).
2. 1. State the range of f .
   2. Find the value of ff(4)
3. Solve, using algebra and showing your working, $$16 - 2 x > | 3 x - 13 | + 5$$ The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation \(y = a \mathrm { f } ( x + b )\) The vertex of the graph with equation \(y = a \mathrm { f } ( x + b )\) is \(( 4,20 )\) Given that \(a\) and \(b\) are constants,
4. find the value of \(a\) and the value of \(b\).

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the graph of \(C _ { 1 }\) with equation $$y = 5 - | 3 x - 22 |$$

1. Write down the coordinates of
   1. the vertex of \(C _ { 1 }\)
   2. the intersection of \(C _ { 1 }\) with the \(y\)-axis.
2. Find the \(x\) coordinates of the intersections of \(C _ { 1 }\) with the \(x\)-axis. Diagram 1, shown on page 21, is a copy of Figure 3.
3. On Diagram 1, sketch the curve \(C _ { 2 }\) with equation $$y = \frac { 1 } { 9 } x ^ { 2 } - 9$$ Identify clearly the coordinates of any points of intersection of \(C _ { 2 }\) with the coordinate axes.
4. Find the coordinates of the points of intersection of \(C _ { 1 }\) and \(C _ { 2 }\) (Solutions relying entirely on calculator technology are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-21_629_803_1137_573} \section*{Diagram 1} Solutions relying entirely on calculator technology are not acceptable.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 2 | 3 - x | + 5 \quad x \geqslant 0$$

1. Solve the equation $$f ( x ) = \frac { 1 } { 2 } x + 30$$ Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has two distinct roots,
2. state the set of possible values for \(k\).
   

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4. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$ Figure 2 shows a sketch of part of the graph \(y = \mathrm { g } ( x )\), where $$\operatorname { g } ( x ) = \frac { x + 9 } { 2 x + 3 } , \quad x \geqslant 0$$

1. Find the value of \(\mathrm { fg } ( 1 )\)
2. State the range of g
3. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain. Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
4. state the range of possible values of \(k\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The point \(P \left( \frac { 1 } { 3 } , 0 \right)\) is the vertex of the graph.
The point \(Q ( 0,5 )\) is the intercept with the \(y\)-axis. Given that \(\mathrm { f } ( x ) = | a x + b |\), where \(a\) and \(b\) are constants,

1. 1. find all possible values for \(a\) and \(b\),
   2. hence find an equation for the graph.
2. Sketch the graph with equation $$y = \mathrm { f } \left( \frac { 1 } { 2 } x \right) + 3$$ showing the coordinates of its vertex and its intercept with the \(y\)-axis.

12. Given that \(k\) is a positive constant,

1. sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis.
2. Find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$

8 Fig. 4 shows a sketch of the graph of \(y = 2 | x - 1 |\). It meets the \(x\) - and \(y\)-axes at ( \(a , 0\) ) and ( \(0 , b\) ) respectively. \begin{figure}[h] \end{figure} Find the values of \(a\) and \(b\).

4 Fig. 4 shows a sketch of the graph of \(y = 2 | x - 1 |\). It meets the \(x\) - and \(y\)-axes at ( \(a , 0\) ) and ( \(0 , b\) ) respectively. \begin{figure}[h] \end{figure} Find the values of \(a\) and \(b\).

5

1. The function \(\mathrm { f } ( x )\) is defined for all values of \(x\) as \(\mathrm { f } ( x ) = | a x - b |\), where \(a\) and \(b\) are positive constants.
   1. The graph of \(y = \mathrm { f } ( x ) + c\), where \(c\) is a constant, has a vertex at \(( 3,1 )\) and crosses the \(y\)-axis at \(( 0,7 )\). Find the values of \(a , b\) and \(c\).
   2. Explain why \(\mathrm { f } ^ { - 1 } ( x )\) does not exist.
2. The function \(\mathrm { g } ( x )\) is defined for \(x \geqslant \frac { q } { p }\) as \(\mathrm { g } ( x ) = | p x - q |\), where \(p\) and \(q\) are positive constants.
   1. Find, in terms of \(p\) and \(q\), an expression for \(\mathrm { g } ^ { - 1 } ( x )\), stating the domain of \(\mathrm { g } ^ { - 1 } ( x )\).
   2. State the set of values of \(p\) for which the equation \(\mathrm { g } ( x ) = \mathrm { g } ^ { - 1 } ( x )\) has no solutions.

1. a. Sketch the graph of the function with equation

$$y = 11 - 2 | 2 - x |$$ Stating the coordinates of the maximum point and any points where the graph cuts the \(y\)-axis.
b. Solve the equation $$4 x = 11 - 2 | 2 - x |$$ A straight line \(l\) has equation \(y = k x + 13\), where \(k\) is a constant.
Given that \(l\) does not meet or intersect \(y = 11 - 2 | 2 - x |\) c. find the range of possible value of \(k\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation $$y = 3 | x - 2 | + 5$$ The vertex of the graph is at the point \(P\), shown in Figure 1.

1. Find the coordinates of \(P\).
2. Solve the equation $$16 - 4 x = 3 | x - 2 | + 5$$ A line \(l\) has equation \(y = k x + 4\) where \(k\) is a constant.
   Given that \(l\) intersects \(y = 3 | x - 2 | + 5\) at 2 distinct points,
3. find the range of values of \(k\).

11. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation $$y = 2 | x + 4 | - 5$$ The vertex of the graph is at the point \(P\), shown in Figure 2.

1. Find the coordinates of \(P\).
2. Solve the equation $$3 x + 40 = 2 | x + 4 | - 5$$ A line \(l\) has equation \(y = a x\), where \(a\) is a constant.
   Given that \(l\) intersects \(y = 2 | x + 4 | - 5\) at least once,
3. find the range of possible values of \(a\), writing your answer in set notation.

4. A function \(f\) is given by \(f ( x ) = | 3 x + 4 |\).

1. Sketch the graph of \(y = f ( x )\). Clearly label the coordinates of the point \(A\), where the graph meets the \(x\)-axis, and the coordinates of the point \(B\), where the graph cuts the \(y\)-axis.
2. On a separate set of axes, sketch the graph of \(y = \frac { 1 } { 2 } f ( x ) - 6\), where the points \(A\) and \(B\) are transformed to the points \(A ^ { \prime }\) and \(B ^ { \prime }\).
   Clearly label the coordinates of the points \(A ^ { \prime }\) and \(B ^ { \prime }\).

1 Identify the graph of \(y = 1 - | x + 2 |\) from the options below.
Tick ( \(\checkmark\) ) one box. \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_389_526_845_500}
B \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_362_442_1279_525} \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_113_116_977_1107}
C \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_496_704_1688_523}
D \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_474_686_2211_534}

1 The graph of \(y = \mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-02_771_1324_676_447} One of the four equations listed below is the equation of the graph \(y = \mathrm { f } ( x )\) Identify which one is the correct equation of the graph.
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & y = | x + 2 | + 3 \\ & y = | x + 2 | - 3 \\ & y = | x - 2 | + 3 \\ & y = | x - 2 | - 3 \end{aligned}$$ □
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