1.02n Sketch curves: simple equations including polynomials

6 The equation of a curve is \(y = 4 x ^ { 2 } + 20 x + 6\).

1. Express the equation in the form \(y = a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants.
2. Hence solve the equation \(4 x ^ { 2 } + 20 x + 6 = 45\).
3. Sketch the graph of \(y = 4 x ^ { 2 } + 20 x + 6\) showing the coordinates of the stationary point. You are not required to indicate where the curve crosses the \(x\) - and \(y\)-axes.

12 \includegraphics[max width=\textwidth, alt={}, center]{9803d51b-215e-4d03-884f-a67fb7ed6442-20_524_972_274_548} The diagram shows the curve with equation \(y = x ( x - 2 ) ^ { 2 }\). The minimum point on the curve has coordinates \(( a , 0 )\) and the \(x\)-coordinate of the maximum point is \(b\), where \(a\) and \(b\) are constants.

1. State the value of \(a\).
2. Calculate the value of \(b\).
3. Find the area of the shaded region.
4. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of the curve has a minimum value \(m\). Calculate the value of \(m\).

3

1. Sketch the curve \(y = ( x - 2 ) ^ { 2 }\).
2. The region enclosed by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume obtained, giving your answer in terms of \(\pi\).

7 A curve has equation \(y = x ^ { 2 } - 4 x + 4\) and a line has equation \(y = m x\), where \(m\) is a constant.

1. For the case where \(m = 1\), the curve and the line intersect at the points \(A\) and \(B\). Find the coordinates of the mid-point of \(A B\).
2. Find the non-zero value of \(m\) for which the line is a tangent to the curve, and find the coordinates of the point where the tangent touches the curve.

7 The function f is defined by \(\mathrm { f } : x \mapsto 7 - 2 x ^ { 2 } - 12 x\) for \(x \in \mathbb { R }\).

1. Express \(7 - 2 x ^ { 2 } - 12 x\) in the form \(a - 2 ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
2. State the coordinates of the stationary point on the curve \(y = \mathrm { f } ( x )\).
   The function g is defined by \(\mathrm { g } : x \mapsto 7 - 2 x ^ { 2 } - 12 x\) for \(x \geqslant k\).
3. State the smallest value of \(k\) for which g has an inverse.
4. For this value of \(k\), find \(\mathrm { g } ^ { - 1 } ( x )\).

5 The function f is such that \(\mathrm { f } ( x ) = a - b \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), where \(a\) and \(b\) are positive constants. The maximum value of \(\mathrm { f } ( x )\) is 10 and the minimum value is - 2 .

1. Find the values of \(a\) and \(b\).
2. Solve the equation \(\mathrm { f } ( x ) = 0\).
3. Sketch the graph of \(y = \mathrm { f } ( x )\).

10 \includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-4_799_1390_255_376} The diagram shows the curve \(y = \sqrt { } ( 1 + 2 x )\) meeting the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The \(y\)-coordinate of the point \(C\) on the curve is 3 .

1. Find the coordinates of \(B\) and \(C\).
2. Find the equation of the normal to the curve at \(C\).
3. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.

8 A function f is defined by \(\mathrm { f } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant 2 \pi\).

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Find the range of f .
3. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant k\).
4. State the maximum value of \(k\) for which g has an inverse.
5. Obtain an expression for \(\mathrm { g } ^ { - 1 } ( x )\).

5

1. By sketching the graphs of $$y = | 5 - 2 x | \quad \text { and } \quad y = 3 \ln x$$ on the same diagram, show that the equation \(| 5 - 2 x | = 3 \ln x\) has exactly two roots.
2. Show that the value of the larger root satisfies the equation \(x = 2.5 + 1.5 \ln x\).
3. Show by calculation that the value of the larger root lies between 4.5 and 5.0.
4. Use an iterative formula, based on the equation in part (b), to find the value of the larger root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

3

1. By sketching a suitable pair of graphs, show that the equation $$x ^ { 3 } = 11 - 2 x$$ has exactly one real root.
2. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 11 - 2 x _ { n } \right)$$ to find the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

4

1. By sketching suitable graphs, show that the equation $$4 x ^ { 2 } - 1 = \cot x$$ has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
2. Verify by calculation that this root lies between 0.6 and 1 .
3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \sqrt { } \left( 1 + \cot x _ { n } \right)$$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

9

1. Sketch the curve \(y = \ln ( x + 1 )\) and hence, by sketching a second curve, show that the equation $$x ^ { 3 } + \ln ( x + 1 ) = 40$$ has exactly one real root. State the equation of the second curve.
2. Verify by calculation that the root lies between 3 and 4 .
3. Use the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 40 - \ln \left( x _ { n } + 1 \right) \right)$$ with a suitable starting value, to find the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
4. Deduce the root of the equation $$\left( \mathrm { e } ^ { y } - 1 \right) ^ { 3 } + y = 40$$ giving the answer correct to 2 decimal places.

5 \includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-2_346_437_1155_854} The diagram shows a variable point \(P\) with coordinates \(( x , y )\) and the point \(N\) which is the foot of the perpendicular from \(P\) to the \(x\)-axis. \(P\) moves on a curve such that, for all \(x \geqslant 0\), the gradient of the curve is equal in value to the area of the triangle \(O P N\), where \(O\) is the origin.

1. State a differential equation satisfied by \(x\) and \(y\). The point with coordinates \(( 0,2 )\) lies on the curve.
2. Solve the differential equation to obtain the equation of the curve, expressing \(y\) in terms of \(x\).
3. Sketch the curve.

1 Let \(a\) be a positive constant.

1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { ax } } { \mathrm { x } + 7 }\).
2. Sketch the curve with equation \(y = \left| \frac { a x } { x + 7 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { a x } { x + 7 } \right| > \frac { a } { 2 }\).

3 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { 2 \mathrm { x } + 1 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points on \(C\).
3. Sketch \(C\).

6 The curve \(C\) has equation \(\mathrm { y } = \frac { 10 + \mathrm { x } - 2 \mathrm { x } ^ { 2 } } { 2 \mathrm { x } - 3 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no turning points.
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right| < 4\).

7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 } \right|\) and find the set of values of \(x\) for which \(2 \left| x ^ { 2 } + x + 9 \right| > 13 | x + 1 |\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right| < 3\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 }\).

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote of \(C\).
2. Find the coordinates of the stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right|\) and state the set of values of \(k\) for which \(\left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right| = k\) has 4 distinct real solutions.

1

1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { x } + 1 } { \mathrm { x } - 1 }\).
2. Sketch the curve with equation \(\mathrm { y } = \frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 }\) and find the set of values of x for which \(\frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 } < - 2\).

3 A curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { a } \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 }\), where \(a\) is a positive constant.

1. Find the equations of the asymptotes of \(C\).
2. Show that there is no point on \(C\) for which \(1 < \mathrm { y } < 1 + 4 \mathrm { a }\).
3. Sketch C. You do not need to find the coordinates of the intersections with the axes.

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|\).
5. Find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15\).

7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } + 1 } { \mathrm { x } - 3 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the turning points on \(C\).
3. Sketch \(C\).
4. Sketch the curves with equations \(y = \left| \frac { x ^ { 2 } + 2 x + 1 } { x - 3 } \right|\) and \(y ^ { 2 } = \frac { x ^ { 2 } + 2 x + 1 } { x - 3 }\) on a single diagram, clearly identifying each curve. If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 }\), where \(a > \frac { 5 } { 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the point of intersection with the \(y\)-axis and labelling the asymptotes.
4. 1. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right|\).
   2. On your sketch in part (i), draw the line \(\mathrm { y } = \mathrm { a }\).
   3. It is given that \(\left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right| < \mathrm { a }\) for \(- 5 - \sqrt { 14 } < x < - 3\) and \(- 5 + \sqrt { 14 } < x < 3\). Find the value of \(a\).

6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote. [2]
2. Find the coordinates of any stationary points on \(C\). \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-12_2715_35_144_2012} \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-13_2718_33_141_23}
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch \(y ^ { 2 } = \frac { x + 1 } { x ^ { 2 } + 3 }\), stating the coordinates of the stationary points and the intersections with the axes.