1.02e Complete the square: quadratic polynomials and turning points

2

1. Express \(4 x ^ { 2 } - 12 x\) in the form \(( 2 x + a ) ^ { 2 } + b\).
2. Hence, or otherwise, find the set of values of \(x\) satisfying \(4 x ^ { 2 } - 12 x > 7\).

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x - 3 , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \mapsto x ^ { 2 } + 4 x , \quad x \in \mathbb { R } . \end{aligned}$$

1. Solve the equation \(\mathrm { ff } ( x ) = 11\).
2. Find the range of g .
3. Find the set of values of \(x\) for which \(\mathrm { g } ( x ) > 12\).
4. Find the value of the constant \(p\) for which the equation \(\mathrm { gf } ( x ) = p\) has two equal roots. Function h is defined by \(\mathrm { h } : x \mapsto x ^ { 2 } + 4 x\) for \(x \geqslant k\), and it is given that h has an inverse.
5. State the smallest possible value of \(k\).
6. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

8

1. Express \(2 x ^ { 2 } - 10 x + 8\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and use your answer to state the minimum value of \(2 x ^ { 2 } - 10 x + 8\).
2. Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } - 10 x + 8 = k x\) has no real roots.

1 Express \(2 x ^ { 2 } - 12 x + 7\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.

5 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_364_625_1873_762} A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown in the diagram. Each sheep pen measures \(x \mathrm {~m}\) by \(y \mathrm {~m}\) and is fully enclosed by metal fencing. The farmer uses 480 m of fencing.

1. Show that the total area of land used for the sheep pens, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 384 x - 9.6 x ^ { 2 }$$
2. Given that \(x\) and \(y\) can vary, find the dimensions of each sheep pen for which the value of \(A\) is a maximum. (There is no need to verify that the value of \(A\) is a maximum.)

11 The function f is defined by \(\mathrm { f } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \in \mathbb { R }\).

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 3\).
2. Given that the line \(y = m x + c\) is a tangent to the curve \(y = \mathrm { f } ( x )\), show that \(4 c = m ^ { 2 } - 12 m + 16\). The function g is defined by \(\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5\) for \(x \geqslant k\), where \(k\) is a constant.
3. Express \(6 x - x ^ { 2 } - 5\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
4. State the smallest value of \(k\) for which g has an inverse.
5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). {www.cie.org.uk} after the live examination series. }

7 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 - x ^ { 2 } - 6 x\) passes through the point \(( 3 , - 10 )\).

1. Find the equation of the curve.
2. Express \(7 - x ^ { 2 } - 6 x\) in the form \(a - ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
3. Find the set of values of \(x\) for which the gradient of the curve is positive.

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 \(\mathrm { g } ^ { - 1 } ( x )\).

1 Express \(3 x ^ { 2 } - 12 x + 7\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.

5 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } + 12 x - 3\) for \(x \in \mathbb { R }\).

1. Express \(- 2 x ^ { 2 } + 12 x - 3\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. State the greatest value of \(\mathrm { f } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x + 5\) for \(x \in \mathbb { R }\).
3. Find the values of \(x\) for which \(\operatorname { gf } ( x ) + 1 = 0\).

9 The curve \(C _ { 1 }\) has equation \(y = x ^ { 2 } - 4 x + 7\). The curve \(C _ { 2 }\) has equation \(y ^ { 2 } = 4 x + k\), where \(k\) is a constant. The tangent to \(C _ { 1 }\) at the point where \(x = 3\) is also the tangent to \(C _ { 2 }\) at the point \(P\). Find the value of \(k\) and the coordinates of \(P\).

8 The function f is such that \(\mathrm { f } ( x ) = a ^ { 2 } x ^ { 2 } - a x + 3 b\) for \(x \leqslant \frac { 1 } { 2 a }\), where \(a\) and \(b\) are constants.

1. For the case where \(\mathrm { f } ( - 2 ) = 4 a ^ { 2 } - b + 8\) and \(\mathrm { f } ( - 3 ) = 7 a ^ { 2 } - b + 14\), find the possible values of \(a\) and \(b\).
2. For the case where \(a = 1\) and \(b = - 1\), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and give the domain of \(\mathrm { f } ^ { - 1 }\).

11

1. Express \(2 x ^ { 2 } + 8 x - 10\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. For the curve \(y = 2 x ^ { 2 } + 8 x - 10\), state the least value of \(y\) and the corresponding value of \(x\).
3. Find the set of values of \(x\) for which \(y \geqslant 14\). Given that \(\mathrm { f } : x \mapsto 2 x ^ { 2 } + 8 x - 10\) for the domain \(x \geqslant k\),
4. find the least value of \(k\) for which f is one-one,
5. express \(\mathrm { f } ^ { - 1 } ( x )\) in terms of \(x\) in this case.

9 The function f : \(x \mapsto 2 x - a\), where \(a\) is a constant, is defined for all real \(x\).

1. In the case where \(a = 3\), solve the equation \(\mathrm { ff } ( x ) = 11\). The function \(\mathrm { g } : x \mapsto x ^ { 2 } - 6 x\) is defined for all real \(x\).
2. Find the value of \(a\) for which the equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real solution. The function \(\mathrm { h } : x \mapsto x ^ { 2 } - 6 x\) is defined for the domain \(x \geqslant 3\).
3. Express \(x ^ { 2 } - 6 x\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are constants.
4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { h } ^ { - 1 }\).

10 The function f is defined by \(\mathrm { f } : x \mapsto x ^ { 2 } - 3 x\) for \(x \in \mathbb { R }\).

1. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) > 4\).
2. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } - b\), stating the values of \(a\) and \(b\).
3. Write down the range of f .
4. State, with a reason, whether f has an inverse. The function g is defined by \(\mathrm { g } : x \mapsto x - 3 \sqrt { } x\) for \(x \geqslant 0\).
5. Solve the equation \(\mathrm { g } ( x ) = 10\).

11 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 11\) for \(x \in \mathbb { R }\).

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are constants.
2. State the range of f .
3. Explain why f does not have an inverse. The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 8 x + 11\) for \(x \leqslant A\), where \(A\) is a constant.
4. State the largest value of \(A\) for which g has an inverse.
5. When \(A\) has this value, obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\) and state the range of \(\mathrm { g } ^ { - 1 }\).

10 The function f is defined by $$\mathrm { f } : x \mapsto 3 x - 2 \text { for } x \in \mathbb { R } .$$

1. Sketch, in a single diagram, the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the two graphs. The function g is defined by $$\mathrm { g } : x \mapsto 6 x - x ^ { 2 } \text { for } x \in \mathbb { R }$$
2. Express \(\operatorname { gf } ( x )\) in terms of \(x\), and hence show that the maximum value of \(\operatorname { gf } ( x )\) is 9 . The function h is defined by $$\mathrm { h } : x \mapsto 6 x - x ^ { 2 } \text { for } x \geqslant 3$$
3. Express \(6 x - x ^ { 2 }\) in the form \(a - ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants.
4. Express \(\mathrm { h } ^ { - 1 } ( x )\) in terms of \(x\).

7 The function f is defined by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 7 \text { for } x > 2$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\) and hence state the range of f .
2. Obtain an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\). The function g is defined by $$\mathrm { g } ( x ) = x - 2 \text { for } x > 2$$ The function h is such that \(\mathrm { f } = \mathrm { hg }\) and the domain of h is \(x > 0\).
3. Obtain an expression for \(\mathrm { h } ( x )\).

7 \includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-3_534_895_255_625} The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48 m .

1. Find an expression for \(y\) in terms of \(x\).
2. Given that the area of the garden is \(A \mathrm {~m} ^ { 2 }\), show that \(A = 48 x - 8 x ^ { 2 }\).
3. Given that \(x\) can vary, find the maximum area of the garden, showing that this is a maximum value rather than a minimum value.

11 Functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 10 & \text { for } 0 \leqslant x \leqslant 2 \\ \mathrm {~g} : x \mapsto x & \text { for } 0 \leqslant x \leqslant 10 \end{array}$$

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. State the range of f .
3. State the domain of \(\mathrm { f } ^ { - 1 }\).
4. Sketch on the same diagram the graphs of \(y = \mathrm { f } ( x ) , y = \mathrm { g } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.
5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

10 The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 24 x + 11\), for \(x \in \mathbb { R }\).

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x - b ) ^ { 2 } + c\) and hence state the coordinates of the vertex of the graph of \(y = \mathrm { f } ( x )\). The function g is defined by \(\mathrm { g } ( x ) = 4 x ^ { 2 } - 24 x + 11\), for \(x \leqslant 1\).
2. State the range of g .
3. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { g } ^ { - 1 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-2_485_755_751_696} The diagram shows a plan for a rectangular park \(A B C D\), in which \(A B = 40 \mathrm {~m}\) and \(A D = 60 \mathrm {~m}\). Points \(X\) and \(Y\) lie on \(B C\) and \(C D\) respectively and \(A X , X Y\) and \(Y A\) are paths that surround a triangular playground. The length of \(D Y\) is \(x \mathrm {~m}\) and the length of \(X C\) is \(2 x \mathrm {~m}\).

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the playground is given by $$A = x ^ { 2 } - 30 x + 1200$$
2. Given that \(x\) can vary, find the minimum area of the playground.

10 A curve has equation \(y = 2 x ^ { 2 } - 3 x\).

1. Find the set of values of \(x\) for which \(y > 9\).
2. Express \(2 x ^ { 2 } - 3 x\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants, and state the coordinates of the vertex of the curve. The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 2 x ^ { 2 } - 3 x \quad \text { and } \quad \mathrm { g } ( x ) = 3 x + k$$ where \(k\) is a constant.
3. Find the value of \(k\) for which the equation \(\mathrm { gf } ( x ) = 0\) has equal roots.

10

1. Express \(x ^ { 2 } - 2 x - 15\) in the form \(( x + a ) ^ { 2 } + b\). The function f is defined for \(p \leqslant x \leqslant q\), where \(p\) and \(q\) are positive constants, by $$f : x \mapsto x ^ { 2 } - 2 x - 15$$ The range of f is given by \(c \leqslant \mathrm { f } ( x ) \leqslant d\), where \(c\) and \(d\) are constants.
2. State the smallest possible value of \(c\). For the case where \(c = 9\) and \(d = 65\),
3. find \(p\) and \(q\),
4. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

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 \(\mathrm { f } ^ { - 1 }\).
   [0pt] [Questions 10 and 11 are printed on the next page.] {www.cie.org.uk} after the live examination series. } \includegraphics[max width=\textwidth, alt={}, center]{a9e04003-1e43-40c4-991a-36aa3a93654b-4_773_641_260_753} 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.