Exam-Style Questions on Transformation of Graphs

The graph of the curve A with equation \(y=f(x)\) is transformed to give the graph of the curve B with equation \(y=5-f(x)\).

The point on A with coordinates (3, 9) is mapped to the point W on B.

Find the coordinates of W.

Worked Solution

The graph of the following equation is drawn and then reflected in the x-axis

(a) What is the equation of the reflected curve?

The original curve is reflected in the y-axis.

(b) What is the equation of this second reflected curve?

Worked Solution

\(f\) and \(g\) are two functions such that \(g(x)=3f(x+2)+7\).

The graph of \(f\) is mapped to the graph of \(g\) under the following transformations:

A vertical stretch by a factor of \(a\) , followed by a translation \(\begin{pmatrix}b \\c \\ \end{pmatrix}\)

Find the values of

(a) \(a\);

(b) \(b\);

(c) \(c\).

(d) Consider two other functions \(h\) and \(j\). Let \(h(x)=-j(2x)\). The point A(8, 7) on the graph of \(j\) is mapped to the point B on the graph of \(h\). Find the coordinates of B.

Worked Solution

Let \(f(x) = {x^2}\) and \(g(x) = 3{(x+2)^2}\) .

The graph of \(g\) can be obtained from the graph of \(f\) using two transformations.

(a) Give a full description of each of the two transformations.

(b) The graph of \(g\) is translated by the vector \( \begin{pmatrix}-4\\5\\ \end{pmatrix}\) to give the graph of \(h\).

The point \(( 2{\text{, }}-1)\) on the graph of \(f\) is translated to the point \(P\) on the graph of \(h\).

Find the coordinates of \(P\).

Worked Solution

Let \(f\) and \(g\) be functions such that \(g(x) = 3f(x - 2) + 7\) .

The graph of \(f\) is mapped to the graph of \(g\) under the following transformations: vertical stretch by a factor of \(k\) , followed by a translation \(\left( \begin{array}{l} p\\ q \end{array} \right)\) .

Write down the value of:

(a) \(k\)

(b) \(p\)

(c) \(q\)

(d) Let \(h(x) = - g(2x)\) . The point A(\(8\), \(7\)) on the graph of \(g\) is mapped to the point \({\rm{A}}'\) on the graph of \(h\) . Find \({\rm{A}}'\)

Worked Solution

Two functions are defined as follows: \(f(x) = 2\ln x\) and \(g(x) = \ln \frac{x^2}{3}\).

(a) Express \(g(x)\) in the form \(f(x) - \ln a\) , where \(a \in {{\mathbb{Z}}^ + }\) .

(b) The graph of \(g(x)\) is a transformation of the graph of \(f(x)\) . Give a full geometric description of this transformation.

Worked Solution