1.02w Graph transformations: simple transformations of f(x)

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

1. 1. State the scale factor of the stretch. [1]
   2. Give full details of the translation. [2]

Alternatively the graph of \(y = \text{e}^x\) can be transformed to the graph of \(y = \text{e}^{2x-1}\) by a stretch parallel to the \(x\)-axis and a stretch parallel to the \(y\)-axis.

1. State the scale factor of the stretch parallel to the \(y\)-axis. [1]

\(f(x) = 2x^2 + 4x + 9 \quad x \in \mathbb{R}\)

1. Write \(f(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are integers to be found. [3]
2. Sketch the curve with equation \(y = f(x)\) showing any points of intersection with the coordinate axes and the coordinates of any turning point. [3]
3. 1. Describe fully the transformation that maps the curve with equation \(y = f(x)\) onto the curve with equation \(y = g(x)\) where $$g(x) = 2(x - 2)^2 + 4x - 3 \quad x \in \mathbb{R}$$
   2. Find the range of the function $$h(x) = \frac{21}{2x^2 + 4x + 9} \quad x \in \mathbb{R}$$ [4]

A sequence of three transformations maps the curve \(y = \ln x\) to the curve \(y = \mathrm{e}^{3x} - 5\). Give details of these transformations. [4]

\includegraphics{figure_3} The diagram shows the graph of \(y = \text{g}(x)\). In the printed answer booklet, using the same scale as in this diagram, sketch the curves

1. \(y = \frac{3}{2}\text{g}(x)\), [2]
2. \(y = \text{g}\left(\frac{1}{2}x\right)\). [2]

The diagram below shows the graph of \(y = f(x)\). \includegraphics{figure_3}

1. On the diagram in the Printed Answer Booklet, draw the graph of \(y = f(\frac{1}{2}x)\). [1]
2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = f(x - 2) + 1\). [2]

The diagram below shows the graph of \(y = f(x)\). \includegraphics{figure_1}

1. On the diagram in the Printed Answer Booklet draw the graph of \(y = f(x + 3)\). [2]
2. Describe fully the transformation which transforms the graph of \(y = f(x)\) to the graph of \(y = -f(x)\). [1]

The point \((2, 3)\) lies on the graph of \(y = g(x)\). State the coordinates of its image when \(y = g(x)\) is transformed to

1. \(y = 4g(x)\) [1]
2. \(y = g(4x)\). [1]

Let \(f(x) = x^2(x - 2)\) and \(g(x) = 2x - 1\) for all real \(x\).

1. Sketch the graph of \(y = f(x)\) and explain briefly why the function f has no inverse. [2]
2. Write down \(g^{-1}(x)\). [1]
3. On the same diagram, sketch the graphs of \(y = f(x - 1) - 3\) and \(y = g^{-1}(x)\) and state the number of real roots of the equation \(f(x - 1) - 3 = g^{-1}(x)\). [3]

Functions f, g and h are defined for \(x \in \mathbb{R}\) by $$f : x \mapsto x^2 - 2x,$$ $$g : x \mapsto x^2,$$ $$h : x \mapsto \sin x.$$

1. 1. State whether or not f has an inverse, giving a reason. [2]
   2. Determine the range of the function f. [2]
2. 1. Show that gh(x) can be expressed as \(\frac{1}{2}(1 - \cos 2x)\). [2]
   2. Sketch the curve C defined by \(y = \text{gh}(x)\) for \(0 \leqslant x \leqslant 2\pi\). [3]

Sketch, on separate diagrams, the graphs of the following functions for \(0 \leqslant x \leqslant 2\pi\) giving the coordinates of all points of intersection with the axes.

1. \(y = \sin x\). [1]
2. \(y = \sin\left(x + \frac{1}{6}\pi\right)\). [2]

A circle, of radius \(\sqrt{5}\) and centre the origin \(O\), is divided into two segments by the line \(y = 1\).

1. Determine the area of the smaller segment. [4]

The line is rotated clockwise about \(O\) through \(45^{\circ}\) and then reflected in the \(x\)-axis.

1. Find the equation of the line in its final position. [3]

% Figure 4 shows curves with asymptotic behavior at x = 3 \includegraphics{figure_4} Figure 4

1. Figure 4 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{x^2 - 5}{3-x}, \quad x \in \mathbb{R}, x \neq 3$$ The curve has a minimum at the point \(A\), with \(x\)-coordinate \(\alpha\), and a maximum at the point \(B\), with \(x\)-coordinate \(\beta\). Find the value of \(\alpha\), the value of \(\beta\) and the \(y\)-coordinates of the points \(A\) and \(B\). [5]
2. The functions \(g\) and \(h\) are defined as follows $$g: x \to x + p \quad x \in \mathbb{R}$$ $$h: x \to |x| \quad x \in \mathbb{R}$$ where \(p\) is a constant. % Figure 5 shows curve with minimum points at C and D symmetric about y-axis \includegraphics{figure_5} Figure 5 Figure 5 shows a sketch of the curve with equation \(y = h(fg(x) + q)\), \(x \in \mathbb{R}\), \(x \neq 0\), where \(q\) is a constant. The curve is symmetric about the \(y\)-axis and has minimum points at \(C\) and \(D\).
   1. Find the value of \(p\) and the value of \(q\).
   2. Write down the coordinates of \(D\).
   [5]
3. The function \(\mathrm{m}\) is given by $$\mathrm{m}(x) = \frac{x^2 - 5}{3-x} \quad x \in \mathbb{R}, x < \alpha$$ where \(\alpha\) is the \(x\)-coordinate of \(A\) as found in part (a).
   1. Find \(\mathrm{m}^{-1}\)
   2. Write down the domain of \(\mathrm{m}^{-1}\)
   3. Find the value of \(t\) such that \(\mathrm{m}(t) = \mathrm{m}^{-1}(t)\)
   [10]

[Total 20 marks]