1.02n Sketch curves: simple equations including polynomials

1 Part of the graph of \(y = \mathrm { f } ( x )\) is shown below, where \(\mathrm { f } ( x )\) is a cubic polynomial. \includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-04_681_679_475_694}

1. Find \(\mathrm { f } ( - 1 )\).
2. Write down three linear factors of \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d\).
3. Show that \(a = - 2\).
4. Find \(b , c\) and \(d\).

5

1. Factorise \(9 - 8 x - x ^ { 2 }\).
2. Show that \(25 - ( x + 4 ) ^ { 2 }\) can be written as \(9 - 8 x - x ^ { 2 }\).
3. A curve has equation \(y = 9 - 8 x - x ^ { 2 }\).
   1. Write down the equation of its line of symmetry.
   2. Find the coordinates of its vertex.
   3. Sketch the curve, indicating the values of the intercepts on the \(x\)-axis and the \(y\)-axis.

1 The straight line \(L\) has equation \(y = 3 x - 1\) and the curve \(C\) has equation $$y = ( x + 3 ) ( x - 1 )$$

1. Sketch on the same axes the line \(L\) and the curve \(C\), showing the values of the intercepts on the \(x\)-axis and the \(y\)-axis.
2. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation \(x ^ { 2 } - x - 2 = 0\).
3. Hence find the coordinates of the points of intersection of \(L\) and \(C\).

6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\).

1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
2. 1. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x )\) as the product of linear factors.
3. 1. The curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) passes through the point \(( 0 , k )\). State the value of \(k\).
   2. Sketch the graph of \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\), indicating the values of \(x\) where the curve touches or crosses the \(x\)-axis.

6

1. 1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
   3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
      (1 mark)
2. The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
   1. Expand \(( x - 5 ) ^ { 2 }\).
   2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
   3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.

5 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$$ and its asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{a0a30197-ca11-40d9-9ccd-30281c5e0fb4-03_531_1013_616_516}

1. Write down the equations of the two asymptotes.
2. The points on the hyperbola for which \(x = 4\) are denoted by \(A\) and \(B\). Find, in surd form, the \(y\)-coordinates of \(A\) and \(B\).
3. The hyperbola and its asymptotes are translated by two units in the positive \(y\) direction. Write down:
   1. the \(y\)-coordinates of the image points of \(A\) and \(B\) under this translation;
   2. the equations of the hyperbola and the asymptotes after the translation.

9 A curve \(C\) has equation $$y = \frac { 2 } { x ( x - 4 ) }$$

1. Write down the equations of the three asymptotes of \(C\).
2. The curve \(C\) has one stationary point. By considering an appropriate quadratic equation, find the coordinates of this stationary point.
   (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).

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

1. 1. Write down the equations of the asymptotes of the curve \(C\).
   2. Sketch the curve \(C\).
2. The line \(y = x - 3\) intersects the curve \(C\) at a point which has \(x\)-coordinate \(\alpha\).
   1. Show that \(\alpha\) lies within the interval \(3 < x < 4\).
   2. Starting from the interval \(3 < x < 4\), use interval bisection twice to obtain an interval of width 0.25 within which \(\alpha\) must lie.

9 The diagram shows the hyperbola $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ and its asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{3c141dcb-4a5e-45ff-9c8e-e06762c03d10-6_798_939_612_555} The constants \(a\) and \(b\) are positive integers.
The point \(A\) on the hyperbola has coordinates ( 2,0 ).
The equations of the asymptotes are \(y = 2 x\) and \(y = - 2 x\).

1. Show that \(a = 2\) and \(b = 4\).
2. The point \(P\) has coordinates ( 1,0 ). A straight line passes through \(P\) and has gradient \(m\). Show that, if this line intersects the hyperbola, the \(x\)-coordinates of the points of intersection satisfy the equation $$\left( m ^ { 2 } - 4 \right) x ^ { 2 } - 2 m ^ { 2 } x + \left( m ^ { 2 } + 16 \right) = 0$$
3. Show that this equation has equal roots if \(3 m ^ { 2 } = 16\).
4. There are two tangents to the hyperbola which pass through \(P\). Find the coordinates of the points at which these tangents touch the hyperbola.
   (No credit will be given for solutions based on differentiation.)

8 The diagram shows a part of the curve $$\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 6 } = 1$$ and a chord \(P Q\) of the curve, where \(P\) lies on the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-05_751_1072_680_459}

1. Write down the coordinates of \(P\).
2. The gradient of the chord \(P Q\) is 2 . Find the coordinates of \(Q\).

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 4 x } { x ^ { 2 } + 9 }$$

1. 1. The graph of \(y = \mathrm { f } ( x )\) has an asymptote which is parallel to the \(x\)-axis. Find the equation of this asymptote.
   2. Explain why the graph of \(y = \mathrm { f } ( x )\) has no asymptotes parallel to the \(y\)-axis.
2. Show that the equation \(\mathrm { f } ( x ) = k\) has two equal roots if \(9 k ^ { 2 } - 9 k - 4 = 0\).
3. Hence find the coordinates of the two stationary points on the graph of \(y = \mathrm { f } ( x )\).

   Surname	Other Names
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   General Certificate of Education
   June 2005
   Advanced Subsidiary Examination MATHEMATICS
   MFP1
   Unit Further Pure 1 ASSESSMENT and
   QUALIFICATIONS
   ALLIANCE Wednesday 22 June 2005 Afternoon Session Insert for use in Question 7.
   Fill in the boxes at the top of this page.
   Fasten this insert securely to your answer book.

9 A curve \(C\) has equation $$y = \frac { ( x + 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. 1. Write down the coordinates of the points where \(C\) intersects the \(x\)-axis. (2 marks)
   2. Write down the equations of all the asymptotes of \(C\).
2. 1. Show that, if the line \(y = k\) intersects \(C\), then $$( k - 1 ) ( k - 4 ) \geqslant 0$$
   2. Given that there is only one stationary point on \(C\), find the coordinates of this stationary point.
      (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).

5

1. Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
2. Show that \(y\) cannot take values between - 3 and 1 .

8

1. Sketch the graph of \(y = \frac { 1 } { x ^ { 2 } }\) \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-12_1273_1083_404_482} 8
2. The graph of \(y = \frac { 1 } { x ^ { 2 } }\) can be transformed onto the graph of \(y = \frac { 9 } { x ^ { 2 } }\) using a stretch in one direction. Beth thinks the stretch should be in the \(y\)-direction.
   Paul thinks the stretch should be in the \(x\)-direction.
   State, giving reasons for your answer, whether Beth is correct, Paul is correct, both are correct or neither is correct.
   [0pt] [3 marks]

16 Curve \(C\) has equation \(y = \frac { a x } { x + b }\) where \(a\) and \(b\) are constants.
The equations of the asymptotes to \(C\) are \(x = - 2\) and \(y = 3\) \includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-20_796_819_459_609} 16

1. Write down the value of \(a\) and the value of \(b\) 16
2. The gradient of \(C\) at the origin is \(\frac { 3 } { 2 }\) With reference to the graph, explain why there is exactly one root of the equation $$\frac { a x } { x + b } = \frac { 3 x } { 2 }$$ 16
3. Using the values found in part (a), solve the inequality $$\frac { a x } { x + b } \leq 1 - x$$ [4 marks]

13

1. Write down the equations of the asymptotes of curve \(C _ { 1 }\) 13 A curve \(C _ { 1 }\) has equation 13
2. On the axes below, sketch the graph of curve \(C _ { 1 }\) Indicate the values of the intercepts of the curve with the axes. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-20_885_898_1192_571} 13
3. Hence, or otherwise, solve the inequality $$\frac { 2 x + 7 } { 3 x + 5 } \geq 0$$ 13
4. Curve \(C _ { 2 }\) is a reflection of curve \(C _ { 1 }\) in the line \(y = - x\) Find an equation for curve \(C _ { 2 }\) in the form \(y = \mathrm { f } ( x )\)

14 The function f is defined by $$\mathrm { f } ( x ) = \frac { x ^ { 2 } - 3 } { x ^ { 2 } + p x + 7 } \quad x \in \mathbb { R }$$ where \(p\) is a constant.
The graph of \(y = \mathrm { f } ( x )\) has only one asymptote.
14

1. Write down the equation of the asymptote.
   14
2. Find the set of possible values of \(p\) □
   14
3. Find the coordinates of the points at which the graph of \(y = \mathrm { f } ( x )\) intersects the axes. \section*{Question 14 continues on the next page} 14
4. \(\quad A\) curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 } { x ^ { 2 } - 3 x + 7 }$$ The curve \(C\) has a local minimum at the point \(M\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-24_371_835_587_605} The line \(y = k\) intersects curve \(C\) 14 (d) (i) Show that $$19 k ^ { 2 } - 16 k - 12 \leq 0$$ 14 (d) (ii) Hence, find the \(y\)-coordinate of point \(M\)

10

1. Write down the equations of the asymptotes of \(C\) 10
2. The line \(L\) has equation $$y = - \frac { 2 } { 5 } x + 2$$ 10 (b) (i) Draw the line \(L\) on Figure 1 10 (b) (ii) Hence, or otherwise, solve the inequality $$\frac { 2 x - 10 } { 3 x - 5 } \leq - \frac { 2 } { 5 } x + 2$$

9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9

1. Find the interval ( \(a , b\) ) in which \(\mathrm { f } ( x )\) does not take any values.
   Fully justify your answer.
   9
2. Find the coordinates of the two stationary points of the graph of \(y = \mathrm { f } ( x )\) 9
3. Show that the graph of \(y = \mathrm { f } ( x )\) has an oblique asymptote and find its equation.
   \section*{Question 9 continues on the next page} 9
4. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
   [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-16_1100_1100_406_470} \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-17_2493_1732_214_139}
   1. Fird \(\begin{aligned} & \text { Do not write } \\ & \text { outside the } \end{aligned}\)

6 The ellipse \(E _ { 1 }\) has equation $$x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$$ \(E _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]\) to give the ellipse \(E _ { 2 }\) 6

1. Write down the equation of \(E _ { 2 }\) 6
2. The ellipse \(E _ { 3 }\) has equation $$\frac { x ^ { 2 } } { 4 } + ( y - 3 ) ^ { 2 } = 1$$ Describe the transformation that maps \(E _ { 2 }\) to \(E _ { 3 }\) 6
3. Each of the lines \(L _ { A }\) and \(L _ { B }\) is a tangent to both \(E _ { 2 }\) and \(E _ { 3 }\) \(L _ { A }\) is closer to the origin than \(L _ { B }\) \(E _ { 2 }\) and \(E _ { 3 }\) both lie between \(L _ { A }\) and \(L _ { B }\) Sketch and label \(E _ { 2 } , E _ { 3 } , L _ { A }\) and \(L _ { B }\) on the axes below.
   You do not need to show the values of the axis intercepts for \(L _ { A }\) and \(L _ { B }\) \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-09_1095_1095_726_475} 6
4. Explain, without doing any calculations, why \(L _ { A }\) has an equation of the form $$x + y = c$$ where \(c\) is a constant.

10 The curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1$$ The curve \(C _ { 2 }\) has equation $$x ^ { 2 } - 25 y ^ { 2 } - 6 x - 200 y - 416 = 0$$ 10

1. Find a sequence of transformations that maps the graph of \(C _ { 1 }\) onto the graph of \(C _ { 2 }\) [4 marks]
   10
2. Find the equations of the asymptotes to \(C _ { 2 }\) Give your answers in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

8 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + k x } { x + 1 }\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes.

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where \(a\) is constant and \(a > 1\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) intersects the \(x\)-axis twice.
3. Justifying your answer, find the number of stationary points on \(C\).
4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis.