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

The diagram below shows the graph of \(y = \mathrm{f}(x)\) (\(-4 \leq x \leq 4\)) The graph meets the \(x\)-axis at \(x = 1\) and \(x = 3\) The graph meets the \(y\)-axis at \(y = 2\) \includegraphics{figure_7}

1. Sketch the graph of \(y = |\mathrm{f}(x)|\) on the axes below. Show any axis intercepts. [2 marks] \includegraphics{figure_7a}
2. Sketch the graph of \(y = \frac{1}{\mathrm{f}(x)}\) on the axes below. Show any axis intercepts and asymptotes. [3 marks] \includegraphics{figure_7b}
3. Sketch the graph of \(y = \mathrm{f}(|x|)\) on the axes below. Show any axis intercepts. [2 marks] \includegraphics{figure_7c}

The transformation S is represented by the matrix \(\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}\) The transformation T is a translation by the vector \(\begin{pmatrix} 0 \\ -5 \end{pmatrix}\) 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. [3 marks]

The function f is defined by $$f(x) = 4x^3 - 8x^2 - 51x - 45 \quad (x \in \mathbb{R})$$

1. 1. Fully factorise \(f(x)\) [2 marks]
   2. Hence, solve the inequality \(f(x) < 0\) [2 marks]
2. The graph of \(y = f(x)\) is translated by the vector \(\begin{pmatrix} 7 \\ 0 \end{pmatrix}\) The new graph is then reflected in the \(x\)-axis, to give the graph of \(y = g(x)\) Solve the inequality \(g(x) \leq 0\) [3 marks]

The function \(f\) is defined by \(f(x) = \frac{8}{x^2}\).

1. Sketch the graph of \(y = f(x)\). [2]
2. On a separate set of axes, sketch the graph of \(y = f(x - 2)\). Indicate the vertical asymptote and the point where the curve crosses the \(y\)-axis. [3]
3. Sketch the graphs of \(y = \frac{8}{x}\) and \(y = \frac{8}{(x-2)^2}\) on the same set of axes. Hence state the number of roots of the equation \(\frac{8}{(x-2)^2} = \frac{8}{x}\). [2]

Figure 1 shows a sketch of the graph of \(y = f(x)\). The graph has a minimum point at \((-3, -4)\) and intersects the \(x\)-axis at the points \((-8, 0)\) and \((2, 0)\). \includegraphics{figure_1}

1. Sketch the graph of \(y = f(x + 3)\), indicating the coordinates of the stationary point and the coordinates of the points of intersection of the graph with the \(x\)-axis. [3]
2. Figure 2 shows a sketch of the graph having one of the following equations with an appropriate value of either \(p\), \(q\) or \(r\). \(y = f(px)\), where \(p\) is a constant \(y = f(x) + q\), where \(q\) is a constant \(y = rf(x)\), where \(r\) is a constant \includegraphics{figure_2} Write down the equation of the graph sketched in Figure 2, together with the value of the corresponding constant. [2]

The diagram below shows a sketch of the graph of \(y = f(x)\). The graph passes through the points \((-2, 0)\), \((0, 8)\), \((4, 0)\) and has a maximum point at \((1, 9)\). \includegraphics{figure_3}

1. Sketch the graph of \(y = 2f(x + 3)\). Indicate the coordinates of the stationary point and the points where the graph crosses the \(x\)-axis. [3]
2. Sketch the graph of \(y = 5 - f(x)\). Indicate the coordinates of the stationary point and the point where the graph crosses the \(y\)-axis. [3]

A function \(f\) is given by \(f(x) = |3x + 4|\).

1. Sketch the graph of \(y = f(x)\). Clearly label the coordinates of the point \(A\), where the graph meets the \(x\)-axis, and the coordinates of the point \(B\), where the graph cuts the \(y\)-axis. [3]
2. On a separate set of axes, sketch the graph of \(y = \frac{1}{2}f(x) - 6\), where the points \(A\) and \(B\) are transformed to the points \(A'\) and \(B'\). Clearly label the coordinates of the points \(A'\) and \(B'\). [3]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{g}(x)\). The curve has a single turning point, a minimum, at the point \(M(4, -1.5)\). The curve crosses the \(x\)-axis at two points, \(P(2, 0)\) and \(Q(7, 0)\). The curve crosses the \(y\)-axis at a single point \(R(0, 5)\).

1. State the coordinates of the turning point on the curve with equation \(y = 2\text{g}(x)\). [1]
2. State the largest root of the equation $$\text{g}(x + 1) = 0$$ [1]
3. State the range of values of \(x\) for which \(\text{g}'(x) \leqslant 0\) [1]

Given that the equation \(\text{g}(x) + k = 0\), where \(k\) is a constant, has no real roots,

1. state the range of possible values for \(k\). [1]

\includegraphics{figure_3} Figure 3 shows part of the curve with equation \(y = 3\cos x^2\). The point \(P(c, d)\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.

1. State the value of \(c\) and the value of \(d\). [1]
2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3\cos x^2\) to the curve with equation
   1. \(y = 3\cos\left(\frac{x^2}{4}\right)\)
   2. \(y = 3\cos(x - 36)^2\)
   [2]
3. Solve, for \(450° \leq \theta < 720°\), $$3\cos\theta = 8\tan\theta$$ giving your solution to one decimal place. [4]

A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]

What transformations could be used, and in which order, to transform the curve \(y = \sin x\) into the curve \(y = 2 \sin(3x + 30°)\)? [3]

\includegraphics{figure_1} Figure 1 shows part of the graph of \(y = f(x)\), \(x \in \mathbb{R}\). The graph consists of two line segments that meet at the point \((1, a)\), \(a < 0\). One line meets the \(x\)-axis at \((3, 0)\). The other line meets the \(x\)-axis at \((-1, 0)\) and the \(y\)-axis at \((0, b)\), \(b < 0\). In separate diagrams, sketch the graph with equation

1. \(y = f(x + 1)\), [2]
2. \(y = f(|x|)\). [2]

Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(f(x) = |x - 1| - 2\), find

1. the value of \(a\) and the value of \(b\), [2]
2. the value of \(x\) for which \(f(x) = 5x\). [3]

The function \(f(x)\) is such that \(f(x) = -x^3 + 2x^2 + kx - 10\) The graph of \(y = f(x)\) crosses the \(x\)-axis at the points with coordinates \((a, 0)\), \((2, 0)\) and \((b, 0)\) where \(a < b\)

1. Show that \(k = 5\) [1 mark]
2. Find the exact value of \(a\) and the exact value of \(b\) [3 marks]
3. The functions \(g(x)\) and \(h(x)\) are such that $$g(x) = x^3 + 2x^2 - 5x - 10$$ $$h(x) = -8x^3 + 8x^2 + 10x - 10$$
   1. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = g(x)\) Fully justify your answer. [2 marks]
   2. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = h(x)\) Fully justify your answer. [2 marks]

\includegraphics{figure_4} Figure 4 Figure 4 shows a sketch of the graph with equation $$y = |2x - 3k|$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = f(x)\) where $$f(x) = k - |2x - 3k|$$ stating • the coordinates of the maximum point • the coordinates of any points where the graph cuts the coordinate axes [4]
2. Find, in terms of \(k\), the set of values of \(x\) for which $$k - |2x - 3k| > x - k$$ giving your answer in set notation. [4]
3. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5f\left(\frac{1}{2}x\right)$$ [2]

The transformations \(\mathbf{R}\), \(\mathbf{S}\) and \(\mathbf{T}\) are defined as follows. \begin{align} \mathbf{R} &: \quad \text{reflection in the } x\text{-axis}
\mathbf{S} &: \quad \text{stretch in the } x\text{-direction with scale factor } 3
\mathbf{T} &: \quad \text{translation in the positive } x\text{-direction by } 4 \text{ units} \end{align}

1. The curve \(y = \ln x\) is transformed by \(\mathbf{R}\) followed by \(\mathbf{T}\). Find the equation of the resulting curve. [2]
2. Find, in terms of \(\mathbf{S}\) and \(\mathbf{T}\), a sequence of transformations that transforms the curve \(y = x^3\) to the curve \(y = \left(\frac{1}{3}x - 4\right)^3\). You should make clear the order of the transformations. [2]

A sequence of transformations maps the curve \(y = e^x\) to the curve \(y = e^{2x+3}\). Give details of these transformations. [3]

A cubic curve \(C\) has equation $$y = (3-x)(4+x)^2.$$

1. Sketch the graph of \(C\). [3] The sketch must include any points where the graph meets the coordinate axes.
2. Sketch in separate diagrams the graph of \(\ldots\)
   1. \(\ldots y = (3-2x)(4+2x)^2\). [2]
   2. \(\ldots y = (3+x)(4-x)^2\). [2]
   3. \(\ldots y = (2-x)(5+x)^2\). [2]
   Each of the sketches must include any points where the graph meets the coordinate axes.

The curve \(y = \sqrt{2x - 1}\) is stretched by scale factor \(\frac{1}{4}\) parallel to the \(x\)-axis and by scale factor \(\frac{1}{2}\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt{ax - b}\) where \(a\) and \(b\) are rational numbers. [3]

Describe a sequence of transformations which maps the graph of $$y = |2x - 5|$$ onto the graph of $$y = |x|$$ [3 marks]