Solve |f(x)| compared to |g(x)| with parameters: sketch then solve

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1. Sketch, on the same diagram, the graphs of \(y = | x + 2 k |\) and \(y = | 2 x - 3 k |\), where \(k\) is a positive constant. Give, in terms of \(k\), the coordinates of the points where each graph meets the axes.
2. Find, in terms of \(k\), the coordinates of each of the two points where the graphs intersect.
3. Find, in terms of \(k\), the largest value of \(t\) satisfying the inequality $$\left| 2 ^ { t } + 2 k \right| \geqslant \left| 2 ^ { t + 1 } - 3 k \right| .$$

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = | 3 x + a | + a$$ and where \(a\) is a positive constant. The graph has a vertex at the point \(P\), as shown in Figure 2 .

1. Find, in terms of \(a\), the coordinates of \(P\).
2. Sketch the graph with equation \(y = g ( x )\), where $$g ( x ) = | x + 5 a |$$ On your sketch, show the coordinates, in terms of \(a\), of each point where the graph cuts or meets the coordinate axes. The graph with equation \(y = \mathrm { g } ( x )\) intersects the graph with equation \(y = \mathrm { f } ( x )\) at two points.
3. Find, in terms of \(a\), the coordinates of the two points. \includegraphics[max width=\textwidth, alt={}, center]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-11_2255_50_314_34}
   

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6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph with equation \(y = | 4 x + 10 a |\), where \(a\) is a positive constant. The graph cuts the \(y\)-axis at the point \(P\) and meets the \(x\)-axis at the point \(Q\) as shown.

1. 1. State the coordinates of \(P\).
   2. State the coordinates of \(Q\).
2. A copy of Figure 1 is shown on page 15. On this copy, sketch the graph with equation $$y = | x | - a$$ Show on the sketch the coordinates of each point where your graph cuts or meets the coordinate axes.
3. Hence, or otherwise, solve the equation $$| 4 x + 10 a | = | x | - a$$ giving your answers in terms of \(a\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-15_860_1128_447_392} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} \(\_\_\_\_\) 7

8. The functions f and g are defined for all real values of \(x\) by $$\begin{aligned} & \mathrm { f } : x \rightarrow | x - 3 a | \\ & \mathrm { g } : x \rightarrow | 2 x + a | \end{aligned}$$ where \(a\) is a positive constant.

1. Evaluate fg(-2a).
2. Sketch on the same diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), showing the coordinates of any points where each graph meets the coordinate axes.
3. Solve the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$

The function \(f\) is defined by $$f : x \mapsto |2x - a|, \quad x \in \mathbb{R},$$ where \(a\) is a positive constant.

1. Sketch the graph of \(y = f(x)\), showing the coordinates of the points where the graph cuts the axes. [2]
2. On a separate diagram, sketch the graph of \(y = f(2x)\), showing the coordinates of the points where the graph cuts the axes. [2]
3. Given that a solution of the equation \(f(x) = \frac{1}{2}x\) is \(x = 4\), find the two possible values of \(a\). [4]

The functions \(f\) and \(g\) are defined by $$f: x \mapsto |x - a| + a, \quad x \in \mathbb{R},$$ $$g: x \mapsto 4x + a, \quad x \in \mathbb{R},$$ where \(a\) is a positive constant.

1. On the same diagram, sketch the graphs of \(f\) and \(g\), showing clearly the coordinates of any points at which your graphs meet the axes. [5]
2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of \(f\) and \(g\) intersect. [3]
3. Find an expression for \(fg(x)\). [2]
4. Solve, for \(x\) in terms of \(a\), the equation $$fg(x) = 3a.$$ [3]

The function f is defined by $$f: x \mapsto |2x - a|, \quad x \in \mathbb{R}$$ where \(a\) is a positive constant.

1. Sketch the graph of \(y = f(x)\), showing the coordinates of the points where the graph cuts the axes. [2]
2. On a separate diagram, sketch the graph of \(y = f(2x)\), showing the coordinates of the points where the graph cuts the axes. [2]
3. Given that a solution of the equation f(x) = \(\frac{1}{2}x\) is \(x = 4\), find the two possible values of \(a\). [4]

The functions f and g are defined by $$\text{f}: x \alpha |x - a| + a, \quad x \in \mathbb{R},$$ $$\text{g}: x \alpha 4x + a, \quad x \in \mathbb{R}.$$ where \(a\) is a positive constant.

1. On the same diagram, sketch the graphs of f and g, showing clearly the coordinates of any points at which your graphs meet the axes. [5]
2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of f and g intersect. [3]
3. Find an expression for fg(x). [2]
4. Solve, for \(x\) in terms of \(a\), the equation $$\text{fg}(x) = 3a.$$ [3]