Graph transformations

The function f is defined by

Fully factorise f (x)

Hence, solve the inequality f (x) 0

The new graph is then reflected in the x-axis, to give the graph of y g(x)

Show that r takes values in the range r k, where k is an integer.

Find the Cartesian equation of C in the form y² f (x)

The ellipse E has equation

Find the transformation that maps the graph of E onto C

The curve C₁ has equation

The curve C₂ has equation

Find a sequence of transformations that maps the graph of C₁ onto the graph of C₂

Find the equations of the asymptotes to C₂

Give your answers in the form ax by c 0 where a, b and c are integers.

Kamla transforms the graphs of various functions by applying first S, then T.

Leo says that, for some graphs, Kamla would get a different result if she applied first T, then S.

Kamla disagrees.

State who is correct.

Fully justify your answer.

The ellipse E₁ has equation

Write down the equation of E₂

The ellipse E₃ has equation

Describe the transformation that maps E₂ to E₃

Explain, without doing any calculations, why L_A has an equation of the form

where c is a constant.

H₁ is the locus of points such that the distance from the point (5, 0) is twice the distance from the line x 2

Show that the equation of H₁ can be written in the form

where q and r are integers.

H₂ is the hyperbola

Describe fully a sequence of two transformations which maps the graph of H₂ onto the graph of H₁

A parabola P₁ has equation y² 4ax where a 0

The line y mx is a tangent to P₂

Solutions using differentiation will be given no marks.