The height of a laser beam trajectory relative to a datum is modeled by the function H(x)H(x)H(x), where H(x)=2∣3x−9∣+4,x≥0H(x) = 2|3x - 9| + 4, \quad x \ge 0H(x)=2∣3x−9∣+4,x≥0 The minimum point on the beam's path is at the vertex VVV.
Determine the coordinates of VVV.
Find the values of x x\,x for which H(x)=x+7H(x) = x + 7H(x)=x+7.
Consider the family of linear paths defined by G(x)=kx+1G(x) = kx + 1G(x)=kx+1, where k k\,k is a constant. Given that the equation H(x)=G(x)H(x) = G(x)H(x)=G(x) has exactly two distinct solutions,
find the range of possible values for kkk.
Practise Edexcel A Level Maths Functions and Graphs with exam-style questions for A Level Maths. 100 questions covering 2.1 The Modulus Function, 2.2 Functions and Mappings, 2.3 Composite Functions, 2.4 Inverse Functions, 2.5 y=|f(x)| and y=f(|x|), 2.6 Combining Transformations, and 2.7 Solving Modulus Problems, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.