Graph y=a|bx+c|+d given: solve equation or inequality

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 \(k\).
   

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 \(b\).

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 \(k\).

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.