Graph transformations

The graph of y = e^x can be transformed to the graph of y = e^2x-1 by a stretch parallel to the x-axis followed by a translation.

State the scale factor of the stretch.

Give full details of the translation.

Alternatively the graph of y = e^x can be transformed to the graph of y = e^2x-1 by a stretch parallel to the x-axis and a stretch parallel to the y-axis.

State the scale factor of the stretch parallel to the y-axis.

The point P lies on the curve y = e^2x-1 and has x-coordinate of

Show that the tangent to the curve y = e^2x-1 at P has equation y = 2x.

Find the exact area enclosed by the curve y = e^2x-1, the tangent to the curve at P and the

y-axis.

The curve y = is translated by 4 units in the positive x-direction.

Write down the equation of the curve after it has been translated.

The curve y = 5x² is stretched parallel to the y-axis with scale factor 4.

The point on the curve y = 5x² with x-coordinate 3 is transformed to the point P.

Write down the coordinates of P.

The function f (x) is defined for all values of x as f (x) = | ax - b |, where a and b are positive constants.

The graph of y = f (x) + c, where c is a constant, has a vertex at (3, 1) and crosses the y-axis at (0, 7).

Find the values of a, b and c.

Explain why f^-1(x) does not exist.

The function g(x) is defined for x ⩾ as g(x) = | px - q|, where p and q are positive constants.

Find, in terms of p and q, an expression for g^-1(x), stating the domain of g^-1(x).

State the set of values of p for which the equation g(x) = g^-1(x) has no solutions.

The graph of y = 2^x can be transformed to the graph of y = 2^x+4 either by a translation or by a stretch.

Give full details of the translation.

Give full details of the stretch.

In this question you must show detailed reasoning.

Solve the equation log₂(8x) = 1 - log₂(1 - x).

Give full details of the single transformation that transforms the graph of y = x³ to the graph

The function f is defined by f(x) = x³ - 8.

Find an expression for f^-1(x).

State how the graphs of y = f(x) and y = f^-1(x) are related geometrically.

The graph of the function y = f (x) passes through the point P with coordinates (2, 6), and is a one-one function. State the coordinates of the point corresponding to P on each of the following curves.

y = f (x) + 3

y = 2f (3 x - 1)

y = f^-1(x)

The diagram shows part of the graph of y = g ′ (x). This is the graph of the gradient function of y = g (x). The graph intersects the x-axis at x =- 2 and x = 4.

State the x-coordinate of any stationary points on the graph of y = g (x).

State the set of values of x for which y = g (x) is a decreasing function.

State the x-coordinate of any points of inflection on the graph of y = g (x).

Express this property in the form of a differential equation.

It is given that C passes through the point (4, 3) and that x > 3 and y > 2 at all points on C.

Determine the equation of C giving your answer in the form y = f (x).

The curve C may be obtained by a transformation of part of the curve y = x³.

Describe fully this transformation.

A sequence of transformations maps the curve y = e^x to the curve y = e^2x+3.

Give details of these transformations.

Express 3 cos 3x + 7 sin 3x in the form R cos(3x - α), where R > 0 and 0 < α < .

Determine the greatest value of 3 cos 3 x + 7 sin 3 x as x varies and give the smallest positive

value of x for which it occurs.

Determine the least value of 3 cos 3 x + 7 sin 3 x as x varies and give the smallest positive value

of x for which it occurs.

In this question you must show detailed reasoning.

Solve the inequality | x - 2 | ⩽ | 2 x - 6 |.

Give full details of a sequence of two transformations needed to transform the graph of

y = | x - 2 | to the graph of y = | 2 x - 6 |.