Graphs & curve sketching

A circle with centre C has equation

The graph with equation

y = │ 3 x – 36 │ – 8

intersects the circle at the points A, B and D as shown in the diagram.

Point D is vertically below point C

State the coordinates of C

State the coordinates of D

The coordinates of A are (a,10)

Find the value of a

Fully justify your answer.

Find, in radians, the angle ADB

Give your answer to three significant figures.

Hence or otherwise find the length of the minor arc AB

Give your answer to three significant figures.

Describe a sequence of two transformations which maps the graph with equation

onto the graph with equation

State the equation of the vertical asymptote of the graph with equation

It is given that

f (x) = x (x – a)(x – 6)

where 0 < a < 6

Sketch the graph of y = f(x) on the axes below.

Sketch the graph of y = f(–2x) on the axes below.

Sketch the curve with equation

The polynomial p (x) is given by

It is given that x 3 is a factor of p(x)

Use the factor theorem to show a 2

State the transformation which maps the curve with equation

onto the curve with equation

The polynomial x²(2x 2) 36 can be written as (x 3)(2x² bx c)

Without finding the values of b and c, use your answers to parts (a) and (b)(ii) to explain why

The polynomial p (x) is given by

where b is a constant.

Use the factor theorem to prove that (x 2) is a factor of p(x) for all values of b.

The graph of y p(x) meets the x-axis at exactly two points.

Sketch a possible graph of y p(x)

Given p (x) can be written as

find the value of b.

Fully justify your answer.

Figure 1 shows the graph of y 2x

On Figure 1 add a sketch of the graph of

Find the coordinates of the points of intersection of the two graphs.

Fully justify your answer.

Determine which one of these graphs does not represent y as a function of x.

Tick () one box.

Sketch the graph of any cubic function that has both three distinct real roots and

The function f (x) is defined by

Show that there is a turning point where the curve crosses the y-axis.

The equation f (x) 0 has three distinct real roots.

By considering the positions of the turning points find, in terms of p, the range of possible values of q.

Sketch the graph of y 2x a, where a is a positive constant.

Show clearly where the graph intersects the axes.