1.02e Complete the square: quadratic polynomials and turning points

8 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

1. In the case where \(a = 6\) and \(b = - 8\), find the range of f .
2. In the case where \(a = 5\), the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots, then \(a ^ { 2 } < 4 ( b - a )\).

3

1. Express \(3 x ^ { 2 } - 6 x + 2\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. The function f , where \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 7 x - 8\), is defined for \(x \in \mathbb { R }\). Find \(\mathrm { f } ^ { \prime } ( x )\) and state, with a reason, whether f is an increasing function, a decreasing function or neither.

1

1. Express \(x ^ { 2 } + 6 x + 2\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. Hence, or otherwise, find the set of values of \(x\) for which \(x ^ { 2 } + 6 x + 2 > 9\).

11

1. Express \(2 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 11\) for \(x \leqslant k\).
2. State the largest value of the constant \(k\) for which f is a one-one function.
3. For this value of \(k\) find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   The function g is defined by \(\mathrm { g } ( x ) = x + 3\) for \(x \leqslant p\).
4. With \(k\) now taking the value 1 , find the largest value of the constant \(p\) which allows the composite function fg to be formed, and find an expression for \(\mathrm { fg } ( x )\) whenever this composite function exists.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-18_650_611_260_762} The diagram shows a shaded region bounded by the \(y\)-axis, the line \(y = - 1\) and the part of the curve \(y = x ^ { 2 } + 4 x + 3\) for which \(x \geqslant - 2\).

1. Express \(y = x ^ { 2 } + 4 x + 3\) in the form \(y = ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants. Hence, for \(x \geqslant - 2\), express \(x\) in terms of \(y\).
2. Hence, showing all necessary working, find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 2 x ^ { 2 } + 8 x + 1 \quad \text { for } x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 2 x - k \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(k\) is a constant.

1. Find the value of \(k\) for which the line \(y = \mathrm { g } ( x )\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
2. In the case where \(k = - 9\), find the set of values of \(x\) for which \(\mathrm { f } ( x ) < \mathrm { g } ( x )\).
3. In the case where \(k = - 1\), find \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) and solve the equation \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = 0\).
4. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants, and hence state the least value of \(\mathrm { f } ( x )\).

2 The function g is defined by \(\mathrm { g } ( x ) = x ^ { 2 } - 6 x + 7\) for \(x > 4\). By first completing the square, find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\).

9

1. Express \(- x ^ { 2 } + 6 x - 5\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
   The function \(\mathrm { f } : x \mapsto - x ^ { 2 } + 6 x - 5\) is defined for \(x \geqslant m\), where \(m\) is a constant.
2. State the smallest value of \(m\) for which f is one-one.
3. For the case where \(m = 5\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{097c5d00-9f92-4c3e-8056-7de09347fbb6-16_771_636_260_756} The diagram shows a cuboid \(O A B C P Q R S\) with a horizontal base \(O A B C\) in which \(A B = 6 \mathrm {~cm}\) and \(O A = a \mathrm {~cm}\), where \(a\) is a constant. The height \(O P\) of the cuboid is 10 cm . The point \(T\) on \(B R\) is such that \(B T = 8 \mathrm {~cm}\), and \(M\) is the mid-point of \(A T\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O P\) respectively.

4. \begin{figure}[h] \end{figure} The points \(P\) and \(Q\), as shown in Figure 2, have coordinates ( \(- 2,13\) ) and ( \(4 , - 5\) ) respectively. The straight line \(l\) passes through \(P\) and \(Q\).

1. Find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are integers to be found. The quadratic curve \(C\) passes through \(P\) and has a minimum point at \(Q\).
2. Find an equation for \(C\). The region \(R\), shown shaded in Figure 2, lies in the second quadrant and is bounded by \(C\) and \(l\) only.
3. Use inequalities to define region \(R\). \includegraphics[max width=\textwidth, alt={}, center]{6a5d0ffc-a725-404b-842a-f3b6000e6fed-11_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

2. $$f ( x ) = 11 - 4 x - 2 x ^ { 2 }$$

1. Express \(\mathrm { f } ( x )\) in the form $$a + b ( x + c ) ^ { 2 }$$ where \(a , b\) and \(c\) are integers to be found.
2. Sketch the graph of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), showing clearly the coordinates of the point where the curve crosses the \(y\)-axis.
3. Write down the equation of the line of symmetry of \(C\).

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the straight line \(l\) and the curve \(C\).
Given that \(l\) cuts the \(y\)-axis at - 12 and cuts the \(x\)-axis at 4 , as shown in Figure 2,

1. find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. Given that \(C\)
   The region \(R\) is shown shaded in Figure 2.
2. Use inequalities to define \(R\).

1. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• \(C _ { 1 }\) has a maximum turning point at \(( 2,20 )\)
• \(C _ { 1 }\) passes through the origin
  1. sketch a graph of \(C _ { 1 }\) showing the coordinates of any points where \(C _ { 1 }\) cuts the coordinate axes,
  2. find an expression for \(\mathrm { f } ( x )\).

The curve \(C _ { 2 }\) has equation \(y = x \left( x ^ { 2 } - 4 \right)\) Curve \(C _ { 1 }\) and \(C _ { 2 }\) meet at the origin, and at the points \(P\) and \(Q\) Given that the \(x\) coordinate of the point \(P\) is negative,

using algebra and showing all stages of your working, find the coordinates of \(P\)

5. \begin{figure}[h] \end{figure} The share value of two companies, company \(A\) and company \(B\), has been monitored over a 15-year period. The share value \(P _ { A }\) of company \(\boldsymbol { A }\), in millions of pounds, is modelled by the equation $$P _ { A } = 53 - 0.4 ( t - 8 ) ^ { 2 } \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. The share value \(P _ { B }\) of company \(B\), in millions of pounds, is modelled by the equation $$P _ { B } = - 1.6 t + 44.2 \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. Figure 2 shows a graph of both models. Use the equations of one or both models to answer parts (a) to (d).

1. Find the difference between the share value of company \(\boldsymbol { A }\) and the share value of company \(\boldsymbol { B }\) at the point monitoring began.
2. State the maximum share value of company \(\boldsymbol { A }\) during the 15-year period.
3. Find, using algebra and showing your working, the times during this 15-year period when the share value of company \(\boldsymbol { A }\) was greater than the share value of company \(\boldsymbol { B }\).
4. Explain why the model for company \(\boldsymbol { A }\) should not be used to predict its share value when \(t = 20\) \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-17_2644_1838_121_116}

8. The curve \(C _ { 1 }\) has equation $$y = 3 x ^ { 2 } + 6 x + 9$$

1. Write \(3 x ^ { 2 } + 6 x + 9\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The point \(P\) is the minimum point of \(C _ { 1 }\)
2. Deduce the coordinates of \(P\). A different curve \(C _ { 2 }\) has equation $$y = A x ^ { 3 } + B x ^ { 2 } + C x + D$$ where \(A\), \(B\), \(C\) and \(D\) are constants. Given that \(C _ { 2 }\)
   \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-29_2646_1838_121_116}

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\)

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• the maximum turning point on \(C\) has coordinates \(( - 2,12 )\)
• \(C\) cuts the negative \(x\)-axis at - 5
  1. find \(\mathrm { f } ( x )\)

The line \(l _ { 1 }\) has equation \(y = \frac { 4 } { 5 } x\) Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(( - 5,0 )\)

find an equation for \(l _ { 2 }\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.
(3) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-10_983_712_1126_616} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve \(C\) and the lines \(l _ { 1 }\) and \(l _ { 2 }\)

Define the region \(R\), shown shaded in Figure 2, using inequalities.

1. (a) Express \(3 x ^ { 2 } + 12 x + 13\) in the form

$$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are integers to be found.
(b) Hence sketch the curve with equation \(y = 3 x ^ { 2 } + 12 x + 13\) On your sketch show clearly

• the coordinates of the \(y\) intercept
• the coordinates of the turning point of the curve

3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation \(y = x ^ { 2 } - 5 x + 13\) The point \(M\) is the minimum point of \(C\). The straight line \(l\) passes through the origin \(O\) and intersects \(C\) at the points \(M\) and \(N\) as shown. Find, showing your working,

1. the coordinates of \(M\),
2. the coordinates of \(N\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-06_531_561_1793_680} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows the curve \(C\) and the line \(l\). The finite region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(y\)-axis.
3. Use inequalities to define the region \(R\).

2. $$f ( x ) = 3 + 12 x - 2 x ^ { 2 }$$

1. Express \(\mathrm { f } ( x )\) in the form
   2. \(\mathrm { f } ( x ) = 3 + 12 x - 2 x ^ { 2 }\)
2. Express \(\mathrm { f } ( x )\) in the form $$\begin{aligned} & \qquad a - b ( x + c ) ^ { 2 } \\ & \text { where } a , b \text { and } c \text { are integers to be found. } \\ & \text { he curve with equation } y = \mathrm { f } ( x ) - 7 \text { crosses the } x \text {-axis at the points } P \text { and } Q \text { and crosses } \\ & \text { te } y \text {-axis at the point } R \text {. } \\ & \text { F) Find the area of the triangle } P Q R \text {, giving your answer in the form } m \sqrt { n } \text { where } m \text { and } \\ & n \text { are integers to be found. } \end{aligned}$$ \(\_\_\_\_\) "

7. \begin{figure}[h] \end{figure} Figure 3 shows the design for a sign at a bird sanctuary.
The design consists of a kite \(O A B C\) joined to a sector \(O C X A\) of a circle centre \(O\).
In the design

• \(O A = O C = 0.6 \mathrm {~m}\)
• \(A B = C B = 1.4 \mathrm {~m}\)
• Angle \(O A B =\) Angle \(O C B = 2\) radians
• Angle \(A O C = \theta\) radians, as shown in Figure 3

Making your method clear,

1. show that \(\theta = 1.64\) radians to 3 significant figures,
2. find the perimeter of the sign, in metres to 2 significant figures,
3. find the area of the sign, in \(\mathrm { m } ^ { 2 }\) to 2 significant figures.

8. \section*{Diagram NOT to scale} \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a design for a pond.
The design consists of a sector \(A O B X\) of a circle centre \(O\) joined to a quadrilateral \(A O B C\).

• \(B C = 8 \mathrm {~m}\)
• \(O A = O B = 3 \mathrm {~m}\)
• angle \(A O B\) is \(\frac { 2 \pi } { 3 }\) radians
• angle \(B C A\) is \(\frac { \pi } { 6 }\) radians
  1. Calculate (i) the exact area of the sector \(A O B X\),
     (ii) the exact perimeter of the sector \(A O B X\).
  2. Calculate the exact area of the triangle \(A O B\).
  3. Show that the length \(A B\) is \(3 \sqrt { 3 } \mathrm {~m}\).
  4. Find the total surface area of the pond. Give your answer in \(\mathrm { m } ^ { 2 }\) correct to 2 significant figures.

11. \begin{figure}[h] \end{figure} Figure 5 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = 2 x ^ { 2 } - 12 x + 14$$

1. Write \(2 x ^ { 2 } - 12 x + 14\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. Given that \(C\) has a minimum at the point \(P\)
2. state the coordinates of \(P\) The line \(l\) intersects \(C\) at \(( - 1,28 )\) and at \(P\) as shown in Figure 5.
3. Find the equation of \(l\) giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found. The finite region \(R\), shown shaded in Figure 5, is bounded by the \(x\)-axis, \(l\), the \(y\)-axis, and \(C\).
4. Use inequalities to define the region \(R\).

10. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geqslant 0 ,$$

1. express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , x \geqslant 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).
2. In the space provided on page 19, sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). The line \(y = 41\) meets \(C\) at the point \(R\).
3. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q \sqrt { } 2\), where \(p\) and \(q\) are integers.

10. $$x ^ { 2 } + 2 x + 3 \equiv ( x + a ) ^ { 2 } + b .$$

1. Find the values of the constants \(a\) and \(b\).
2. In the space provided below, sketch the graph of \(y = x ^ { 2 } + 2 x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes.
3. Find the value of the discriminant of \(x ^ { 2 } + 2 x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). The equation \(x ^ { 2 } + k x + 3 = 0\), where \(k\) is a constant, has no real roots.
4. Find the set of possible values of \(k\), giving your answer in surd form.

10. $$\mathrm { f } ( x ) = x ^ { 2 } + 4 k x + ( 3 + 11 k ) , \quad \text { where } k \text { is a constant. }$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found in terms of \(k\). Given that the equation \(\mathrm { f } ( x ) = 0\) has no real roots,
2. find the set of possible values of \(k\). Given that \(k = 1\),
3. sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of any point at which the graph crosses a coordinate axis.

10. $$4 x ^ { 2 } + 8 x + 3 \equiv a ( x + b ) ^ { 2 } + c$$

1. Find the values of the constants \(a , b\) and \(c\).
2. On the axes on page 27, sketch the curve with equation \(y = 4 x ^ { 2 } + 8 x + 3\), showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{099016ad-e742-4679-9669-47dcd1d9cc5f-15_1283_1284_319_322}