The diagram below shows the graphs of *f*(*x*)=-*x ^{2} *+ 2

*x*+15 and

*g*(

*x*)=-3

*x*+

*k*. Graph

*f*cuts the

*x*-axis at A(-3 ; 0) and B(5 ; 0), the

*y*-axis at C and has a turning point at D. Graph

*g*cuts the

*x*-axis at B and the

*y*-axis at C. E is a point on

*g*such DE is

parallel to the *y*-axis.

6.1 Show that *k *= 15 (1)

6.2 Determine the coordinates of D, the turning point of *f*. (3)

6.3 Determine the values of *x *for which *f *is increasing. (1)

6.4 Calculate the average gradient between points A and D. (3)

6.5 Calculate the length of DE. (2)

6.6 If *h*(*x*) = *f*(*x*-1)-2, determine the equation of *h *in the form *h*(*x*) = *a*(*x *+ *p*)^{2} + *q*. (4)

6.7 Determine the maximum value of *p*(*x*) = 3^{f(x)-12} (3)

6.8 Determine the values of *x *for which *f*(*x*) + *k *= 0 will have two distinct real roots. (2)

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