1.02s Modulus graphs: sketch graph of |ax+b|

Identify the graph of \(y = 1 - |x + 2|\) from the options below. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_1}

Sketch the graph of \(y = |2x + a|\), where \(a\) is a positive constant. Show clearly where the graph intersects the axes. [3 marks] \includegraphics{figure_4}

The graph of \(y = f(x)\) is shown below. \includegraphics{figure_1} One of the four equations listed below is the equation of the graph \(y = f(x)\) Identify which one is the correct equation of the graph. Tick (\(\checkmark\)) one box. [1 mark] \(y = |x + 2| + 3\) \(y = |x + 2| - 3\) \(y = |x - 2| + 3\) \(y = |x - 2| - 3\)

The function f is defined by $$f(x) = \left|\sin x + \frac{1}{2}\right| \quad (0 \leq x \leq 2\pi)$$ Find the set of values of \(x\) for which $$f(x) \geq \frac{1}{2}$$ Give your answer in set notation. [5 marks]

The diagram shows the line \(y = 5 - x\) \includegraphics{figure_15}

1. On the diagram above, sketch the graph of \(y = |x^2 - 4x|\), including all parts of the graph where it intersects the line \(y = 5 - x\) (You do not need to show the coordinates of the points of intersection.) [3 marks]
2. Find the solution of the inequality $$|x^2 - 4x| > 5 - x$$ Give your answer in an exact form. [4 marks]

\includegraphics{figure_2} Figure 2 shows a sketch of part of the graph \(y = f(x)\), where $$f(x) = 2|3 - x| + 5, \quad x \geq 0$$

1. State the range of \(f\) [1]
2. Solve the equation $$f(x) = \frac{1}{2}x + 30$$ [3] Given that the equation \(f(x) = k\), where \(k\) is a constant, has two distinct roots,
3. state the set of possible values for \(k\). [2]

\includegraphics{figure_4} Figure 4 Figure 4 shows a sketch of the graph with equation $$y = |2x - 3k|$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = f(x)\) where $$f(x) = k - |2x - 3k|$$ stating • the coordinates of the maximum point • the coordinates of any points where the graph cuts the coordinate axes [4]
2. Find, in terms of \(k\), the set of values of \(x\) for which $$k - |2x - 3k| > x - k$$ giving your answer in set notation. [4]
3. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5f\left(\frac{1}{2}x\right)$$ [2]

Describe a sequence of transformations which maps the graph of $$y = |2x - 5|$$ onto the graph of $$y = |x|$$ [3 marks]