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

3

1. Sketch the curve \(y = - \frac { 1 } { x ^ { 2 } }\).
2. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is translated by 2 units in the positive \(x\)-direction. State the equation of the curve after it has been translated.
3. The curve \(y = - \frac { 1 } { x ^ { 2 } }\) is stretched parallel to the \(y\)-axis with scale factor \(\frac { 1 } { 2 }\) and, as a result, the point \(\left( \frac { 1 } { 2 } , - 4 \right)\) on the curve is transformed to the point \(P\). State the coordinates of \(P\).

5 A curve has equation \(y = a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants. The curve has a stationary point at \(( - 3,2 )\).

1. State the values of \(b\) and \(c\). When the curve is translated by \(\binom { 4 } { 0 }\) the transformed curve passes through the point \(( 3 , - 18 )\).
2. Determine the value of \(a\).

5 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-04_700_727_260_242} The diagram shows a curve \(C\) for which \(y\) is inversely proportional to \(x\). The curve passes through the point \(\left( 1 , - \frac { 1 } { 2 } \right)\).

1. 1. Determine the equation of the gradient function for the curve \(C\).
   2. Sketch this gradient function on the axes in the Printed Answer Booklet.
2. The diagram indicates that the curve \(C\) has no stationary points. State what feature of your sketch in part (a)(ii) corresponds to this.
3. The curve \(C\) is translated by the vector \(\binom { - 2 } { 0 }\). Find the equation of the curve after it has been translated.

3 A Ferris wheel at a fairground rotates in a vertical plane. The height above the ground of a seat on the wheel is \(h\) metres at time \(t\) seconds after the seat is at its lowest point. The height is given by the equation \(h = 15 - 14 \cos ( k t ) ^ { \circ }\), where \(k\) is a positive constant.

1. 1. Write down the greatest height of a seat above the ground.
   2. Write down the least height of a seat above the ground.
2. Given that a seat first returns to its lowest point after 150 seconds, calculate the value of \(k\).
3. Determine for how long a seat is 20 metres or more above the ground during one complete revolution. Give your answer correct to the nearest tenth of a second.

5 \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-4_591_547_262_242} The diagram shows the graphs of \(y = 2 ^ { 3 x }\) and \(y = 2 ^ { 3 x + 2 }\). The graph of \(y = 2 ^ { 3 x }\) can be transformed to the graph of \(y = 2 ^ { 3 x + 2 }\) by means of a stretch.

1. Give details of the stretch. The point \(A\) lies on \(y = 2 ^ { 3 x }\) and the point \(B\) lies on \(y = 2 ^ { 3 x + 2 }\). The line segment \(A B\) is parallel to the \(y\)-axis and the difference between the \(y\)-coordinates of \(A\) and \(B\) is 36 .
2. Determine the \(x\)-coordinate of \(A\). Give your answer in the form \(m \log _ { 2 } n\) where \(m\) and \(n\) are constants to be determined.

9 The curve \(y = ( x - 1 ) ^ { 2 }\) maps onto the curve \(\mathrm { C } _ { 1 }\) following a stretch scale factor \(\frac { 1 } { 2 }\) in the \(x\)-direction.

1. Show that the equation of \(\mathrm { C } _ { 1 }\) can be written as \(y = 4 x ^ { 2 } - 4 x + 1\). The curve \(\mathrm { C } _ { 2 }\) is a translation of \(y = 4.25 x - x ^ { 2 }\) by \(\binom { 0 } { - 3 }\).
2. Show that the normal to the curve \(\mathrm { C } _ { 1 }\) at the point \(( 0,1 )\) is a tangent to the curve \(\mathrm { C } _ { 2 }\).

7 In this question you must show detailed reasoning.

1. Nigel is asked to determine whether \(( x + 7 )\) is a factor of \(x ^ { 3 } - 37 x + 84\). He substitutes \(x = 7\) and calculates \(7 ^ { 3 } - 37 \times 7 + 84\). This comes to 168 , so Nigel concludes that ( \(x + 7\) ) is not a factor. Nigel's conclusion is wrong.

4 The quadratic function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 3 x + 2\).

1. Write \(\mathrm { f } ( x )\) in the form \(( \mathrm { x } + \mathrm { a } ) ^ { 2 } + \mathrm { b }\), where \(a\) and \(b\) are constants.
2. Write down the coordinates of the minimum point on the graph of \(y = f ( x )\).
3. Describe fully the transformation that maps the graph of \(y = f ( x )\) onto the graph of \(y = ( x + 1 ) ^ { 2 } - \frac { 1 } { 4 }\).

5 Part of the graph of \(y = f ( x )\) is shown below. The graph is the image of \(y = \tan x ^ { \circ }\) after a stretch in the \(x\)-direction. \includegraphics[max width=\textwidth, alt={}, center]{7af62e61-c67f-4d05-b6b9-c1a110345812-4_791_1022_1014_244}

1. Find the equation of the graph.
2. Write down the period of the function \(\mathrm { f } ( x )\).
3. In this question you must show detailed reasoning. Find all the roots of the equation \(\mathrm { f } ( x ) = 1\) for \(0 ^ { \circ } \leqslant x ^ { \circ } \leqslant 360 ^ { \circ }\).

9 The graph shows the function \(\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }\). \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-6_595_732_322_242}

1. Describe the transformation of the graph of \(y = e ^ { x }\) that gives the graph of \(y = e ^ { 2 x }\). A second function is defined by \(\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }\).
2. A copy of the graph of \(\mathrm { y } = \mathrm { e } ^ { 2 \mathrm { x } }\) is given in the Printed Answer Booklet. Add a sketch of the graph of \(\mathrm { y } = \mathrm { k } + \mathrm { e } ^ { \mathrm { x } }\) in a case where \(k\) is a positive constant.
3. Show that the two graphs do not intersect for values of \(k\) less than \(- \frac { 1 } { 4 }\).
4. In the case where \(k = 2\), show that the only point of intersection occurs when \(x = \ln 2\).

10

1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.

10

1. Express \(7 \cos x - 2 \sin x\) in the form \(R \cos ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
2. Give details of a sequence of two transformations which maps the curve \(y = \sec x\) onto the curve \(y = \frac { 1 } { 7 \cos x - 2 \sin x }\).

2 The equation of a curve is \(y = e ^ { x }\). The curve is subject to a translation \(\binom { 3 } { 0 }\) and a stretch scale factor 2 parallel to the \(y\)-axis. Write down the equation of the new curve.

1 The equation of a curve is \(y = 4 x ^ { 2 } + 8 x + 1\).
The curve is stretched parallel to the \(x\)-axis with scale factor 2 .
Find the equation of the new curve, giving your answer in the form \(\mathrm { y } = a \mathrm { x } ^ { 2 } + b \mathrm { x } + c\), where \(a , b\) and \(c\) are integers to be determined.

2 The curve \(y = x ^ { 3 } - 2 x\) is translated by the vector \(\binom { 1 } { - 4 }\). Write down the equation of the translated curve. [2]

1

1. Express \(x ^ { 2 } + 8 x + 2\) in the form \(( x + a ) ^ { 2 } + b\).
2. Write down the coordinates of the turning point of the curve \(y = x ^ { 2 } + 8 x + 2\).
3. State the transformation(s) which map(s) the curve \(y = x ^ { 2 }\) onto the curve \(y = x ^ { 2 } + 8 x + 2\).

4

1. 1. Express \(x ^ { 2 } + 2 x + 5\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence show that \(x ^ { 2 } + 2 x + 5\) is always positive.
2. A curve has equation \(y = x ^ { 2 } + 2 x + 5\).
   1. Write down the coordinates of the minimum point of the curve.
   2. Sketch the curve, showing the value of the intercept on the \(y\)-axis.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 2 x + 5\).

5

1. Express \(( x - 5 ) ( x - 3 ) + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   (3 marks)
2. 1. Sketch the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
   2. Write down an equation of the tangent to the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\) at its vertex.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\).

2

1. Factorise \(x ^ { 2 } - 4 x - 12\).
2. Sketch the graph with equation \(y = x ^ { 2 } - 4 x - 12\), stating the values where the curve crosses the coordinate axes.
3. 1. Express \(x ^ { 2 } - 4 x - 12\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are positive integers.
   2. Hence find the minimum value of \(x ^ { 2 } - 4 x - 12\).
4. The curve with equation \(y = x ^ { 2 } - 4 x - 12\) is translated by the vector \(\left[ \begin{array} { r } - 3 \\ 2 \end{array} \right]\). Find an equation of the new curve. You need not simplify your answer.

4

1. 1. Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x - p ) ^ { 2 } + q\).
   2. Use the result from part (a)(i) to show that the equation \(x ^ { 2 } - 6 x + 11 = 0\) has no real solutions.
2. A curve has equation \(y = x ^ { 2 } - 6 x + 11\).
   1. Find the coordinates of the vertex of the curve.
   2. Sketch the curve, indicating the value of \(y\) where the curve crosses the \(y\)-axis.
   3. Describe the geometrical transformation that maps the curve with equation \(y = x ^ { 2 } - 6 x + 11\) onto the curve with equation \(y = x ^ { 2 }\).

2

1. Express \(x ^ { 2 } - 6 x + 16\) in the form \(( x - p ) ^ { 2 } + q\).
2. A curve has equation \(y = x ^ { 2 } - 6 x + 16\). Using your answer from part (a), or otherwise:
   1. find the coordinates of the vertex (minimum point) of the curve;
   2. sketch the curve, indicating the value where the curve crosses the \(y\)-axis;
   3. state the equation of the line of symmetry of the curve.
3. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 6 x + 16\).

2

1. Express \(x ^ { 2 } + 8 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
2. Hence, or otherwise, show that the equation \(x ^ { 2 } + 8 x + 19 = 0\) has no real solutions.
3. Sketch the graph of \(y = x ^ { 2 } + 8 x + 19\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 8 x + 19\).

4

1. Express \(x ^ { 2 } + 5 x + 7\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   (3 marks)
2. A curve has equation \(y = x ^ { 2 } + 5 x + 7\).
   1. Find the coordinates of the vertex of the curve.
   2. State the equation of the line of symmetry of the curve.
   3. Sketch the curve, stating the value of the intercept on the \(y\)-axis.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 5 x + 7\).

3 A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 14 y + 25 = 0$$

1. Write the equation of \(C\) in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where \(a , b\) and \(k\) are integers.
2. Hence, for the circle \(C\), write down:
   1. the coordinates of its centre;
   2. its radius.
3. 1. Sketch the circle \(C\).
   2. Write down the coordinates of the point on \(C\) that is furthest away from the \(x\)-axis.
4. Given that \(k\) has the same value as in part (a), describe geometrically the transformation which maps the circle with equation \(( x + 1 ) ^ { 2 } + y ^ { 2 } = k\) onto the circle \(C\).

5

1. Express \(x ^ { 2 } + 3 x + 2\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
2. A curve has equation \(y = x ^ { 2 } + 3 x + 2\).
   1. Use the result from part (a) to write down the coordinates of the vertex of the curve.
   2. State the equation of the line of symmetry of the curve.
3. The curve with equation \(y = x ^ { 2 } + 3 x + 2\) is translated by the vector \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right]\). Find the equation of the resulting curve in the form \(y = x ^ { 2 } + b x + c\).