Sketch y=|f(x)| or y=f(|x|) for non-linear f(x) and solve

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.

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.

1. \(y = 8 + 2 x - x ^ { 2 }\)
2. \(y = 8 + 2 | x | - x ^ { 2 }\)
3. \(y = 8 + x + | x | - x ^ { 2 }\) (b)Find the values of \(x\) for which $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$

7

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 4 x ^ { 2 } - 5\).
2. Sketch the graph of \(y = \left| 4 x ^ { 2 } - 5 \right|\), indicating the coordinates of the point where the curve crosses the \(y\)-axis.
3. 1. Solve the equation \(\left| 4 x ^ { 2 } - 5 \right| = 4\).
   2. Hence, or otherwise, solve the inequality \(\left| 4 x ^ { 2 } - 5 \right| \geqslant 4\).

4

1. Sketch the graph of \(y = \left| 50 - x ^ { 2 } \right|\), indicating the coordinates of the point where the graph crosses the \(y\)-axis.
2. Solve the equation \(\left| 50 - x ^ { 2 } \right| = 14\).
3. Hence, or otherwise, solve the inequality \(\left| 50 - x ^ { 2 } \right| > 14\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 50 - x ^ { 2 }\).

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

3. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 2 } - 10 x \quad x \in \mathbb { R }$$

1. Solve the equation $$\mathrm { f } ( | x | ) = 48$$
2. Find the set of values of \(x\) for which $$| f ( x ) | \geqslant \frac { 5 } { 2 } x$$