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

The curve \(y = x^2\) is transformed to the curve \(y = 4(x-3)^2 - 8\). Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations have been applied. [5]

The equation of a curve is \(y = x^2 - 8x + 5\).

1. Find the coordinates of the minimum point of the curve. [2]

The curve is stretched by a factor of 2 parallel to the \(y\)-axis and then translated by \(\begin{pmatrix} 4 \\ 1 \end{pmatrix}\).

1. Find the coordinates of the minimum point of the transformed curve. [2]
2. Find the equation of the transformed curve. Give the answer in the form \(y = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]

The function f is defined by f\((x) = 3 + 6x - 2x^2\) for \(x \in \mathbb{R}\).

1. Express f\((x)\) in the form \(a - b(x - c)^2\), where \(a\), \(b\) and \(c\) are constants, and state the range of f. [3]
2. The graph of \(y = \)f\((x)\) is transformed to the graph of \(y = \)h\((x)\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \)h\((x)\) has a minimum point at the origin. Give details of the reflection and translation involved. [2] The function g is defined by g\((x) = 3 + 6x - 2x^2\) for \(x \leqslant 0\).
3. Sketch the graph of \(y = \)g\((x)\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes. [2]
4. Sketch the graph of \(y = \)g\(^{-1}(x)\) on your diagram in (c), and find an expression for g\(^{-1}(x)\). You should label the two graphs in your diagram appropriately and show any relevant mirror line. [4]

\includegraphics{figure_5} In the diagram, the graph with equation \(y = \text{f}(x)\) is shown with solid lines and the graph with equation \(y = \text{g}(x)\) is shown with broken lines.

1. Describe fully a sequence of three transformations which transforms the graph of \(y = \text{f}(x)\) to the graph of \(y = \text{g}(x)\). [6]
2. Find an expression for g(x) in the form \(af(bx + c)\), where \(a\), \(b\) and \(c\) are integers. [2]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \text{f}(x)\) The curve \(C\) passes through the origin and through \((6, 0)\) The curve \(C\) has a minimum at the point \((3, -1)\) On separate diagrams, sketch the curve with equation

1. \(y = \text{f}(2x)\) [3]
2. \(y = \text{f}(x + p)\), where \(p\) is a constant and \(0 < p < 3\) [4]

On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve crosses the \(x\)-axis at the points \((2, 0)\) and \((4, 0)\). The minimum point on the curve is \(P(3, -2)\). In separate diagrams sketch the curve with equation

1. \(y = -f(x)\), [3]
2. \(y = f(2x)\). [3]

On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve passes through the origin \(O\) and through the point \((6, 0)\). The maximum point on the curve is \((3, 5)\). On separate diagrams, sketch the curve with equation

1. \(y = 3f(x)\), [2]
2. \(y = f(x + 2)\). [3]

On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the \(x\)-axis.

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x)\). The curve passes through the points \((0, 3)\) and \((4, 0)\) and touches the \(x\)-axis at the point \((1, 0)\). On separate diagrams, sketch the curve with equation

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

On each diagram show clearly the coordinates of all the points at which the curve meets the axes.

Given that \(f(x) = \frac{1}{x}\), \(x \neq 0\),

1. sketch the graph of \(y = f(x) + 3\) and state the equations of the asymptotes. [4]
2. Find the coordinates of the point where \(y = f(x) + 3\) crosses a coordinate axis. [2]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = \text{f}(x)\). The curve crosses the coordinate axes at the points \((0, 1)\) and \((3, 0)\). The maximum point on the curve is \((1, 2)\). On separate diagrams, sketch the curve with equation

1. \(y = \text{f}(x + 1)\), [3]
2. \(y = \text{f}(2x)\). [3]

On each diagram, show clearly the coordinates of the maximum point, and of each point at which the curve crosses the coordinate axes.

The function \(f\) is defined by $$f : x \mapsto |2x - a|, \quad x \in \mathbb{R},$$ where \(a\) is a positive constant.

1. Sketch the graph of \(y = f(x)\), showing the coordinates of the points where the graph cuts the axes. [2]
2. On a separate diagram, sketch the graph of \(y = f(2x)\), showing the coordinates of the points where the graph cuts the axes. [2]
3. Given that a solution of the equation \(f(x) = \frac{1}{2}x\) is \(x = 4\), find the two possible values of \(a\). [4]

The function \(f\) is defined by \(f: x \mapsto \frac{3x-1}{x-3}, x \in \mathbb{R}, x \neq 3\).

1. Prove that \(f^{-1}(x) = f(x)\) for all \(x \in \mathbb{R}, x \neq 3\). [3]
2. Hence find, in terms of \(k\), \(ff(k)\), where \(x \neq 3\). [2]

\includegraphics{figure_3} Figure 3 shows a sketch of the one-one function \(g\), defined over the domain \(-2 \leq x \leq 2\).

1. Find the value of \(fg(-2)\). [3]
2. Sketch the graph of the inverse function \(g^{-1}\) and state its domain. [3]

The function \(h\) is defined by \(h: x \mapsto 2g(x - 1)\).

1. Sketch the graph of the function \(h\) and state its range. [3]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = f(x), x \geq 0\). The curve meets the coordinate axes at the points \((0, c)\) and \((d, 0)\). In separate diagrams sketch the curve with equation

1. \(y = f^{-1}(x)\), [2]
2. \(y = 3f(2x)\). [3]

Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that \(f\) is defined by $$f : x \mapsto 3(2^{-x}) - 1, \quad x \in \mathbb{R}, x \geq 0,$$

1. state
   1. the value of \(c\),
   2. the range of \(f\). [3]
2. Find the value of \(d\), giving your answer to 3 decimal places. [3]

The function \(g\) is defined by $$g : x \to \log_2 x, \quad x \in \mathbb{R}, x \geq 1.$$

1. Find \(fg(x)\), giving your answer in its simplest form. [3]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = f(x), -1 \leq x \leq 3\). The curve touches the \(x\)-axis at the origin \(O\), crosses the \(x\)-axis at the point \(A(2, 0)\) and has a maximum at the point \(B(\frac{4}{3}, 1)\). In separate diagrams, show a sketch of the curve with equation

1. \(y = f(x + 1)\), [3]
2. \(y = |f(x)|\), [3]
3. \(y = f(|x|)\), [4]

marking on each sketch the coordinates of points at which the curve

1. has a turning point,
2. meets the \(x\)-axis.

\includegraphics{figure_1} Figure 1 shows a sketch of a curve with equation \(y = f(x)\). The curve crosses the \(y\)-axis at \((0, 3)\) and has a minimum at \(P(4, 2)\). On separate diagrams, sketch the curve with equation

1. \(y = f(x + 4)\), [2]
2. \(y = 2f(x)\). [2]

On each diagram, show clearly the coordinates of the minimum point and any point of intersection with the \(y\)-axis.

The curve \(y = \text{f}(x)\) passes through the point \(P\) with coordinates \((2, 5)\).

1. State the coordinates of the point corresponding to \(P\) on the curve \(y = \text{f}(x) + 2\). [1]
2. State the coordinates of the point corresponding to \(P\) on the curve \(y = \text{f}(2x)\). [1]
3. Describe the transformation that transforms the curve \(y = \text{f}(x)\) to the curve \(y = \text{f}(x + 4)\). [2]

A cubic polynomial is given by \(f(x) = x^3 + x^2 - 10x + 8\).

1. Show that \((x - 1)\) is a factor of \(f(x)\). Factorise \(f(x)\) fully. Sketch the graph of \(y = f(x)\). [7]
2. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\). Write down an equation for the resulting graph. You need not simplify your answer. Find also the intercept on the \(y\)-axis of the resulting graph. [5]

\includegraphics{figure_2} Fig. 2 shows graphs \(A\) and \(B\).

1. State the transformation which maps graph \(A\) onto graph \(B\). [2]
2. The equation of graph \(A\) is \(y = f(x)\). Which one of the following is the equation of graph \(B\)? \(y = f(x) + 2\) \quad \(y = f(x) - 2\) \quad \(y = f(x + 2)\) \quad \(y = f(x - 2)\) \(y = 2f(x)\) \quad \(y = f(x + 3)\) \quad \(y = f(x - 3)\) \quad \(y = 3f(x)\) [2]

Answer the whole of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\), \(x \neq 0\).

1. Use the graph to find approximate roots of the equation \(\frac{1}{x} = 2x + 3\), showing your method clearly. [3]
2. Rearrange the equation \(\frac{1}{x} = 2x + 3\) to form a quadratic equation. Solve the resulting equation, leaving your answers in the form \(\frac{p \pm \sqrt{q}}{r}\). [5]
3. Draw the graph of \(y = \frac{1}{x} + 2\), \(x \neq 0\), on the grid used for part (i). [2]
4. Write down the values of \(x\) which satisfy the equation \(\frac{1}{x} + 2 = 2x + 3\). [2]

You are given that \(f(x) = x^3 + 6x^2 - x - 30\).

1. Use the factor theorem to find a root of \(f(x) = 0\) and hence factorise \(f(x)\) completely. [6]
2. Sketch the graph of \(y = f(x)\). [3]
3. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\). Show that the equation of the translated graph may be written as $$y = x^3 + 3x^2 - 10x - 24.$$ [3]

The point P \((5, 4)\) is on the curve \(y = f(x)\). State the coordinates of the image of P when the graph of \(y = f(x)\) is transformed to the graph of

1. \(y = f(x - 5)\), [2]
2. \(y = f(x) + 7\). [2]