1.02e Complete the square: quadratic polynomials and turning points

9 The functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 3 \text { for } x > c , \text { where } c \text { is a constant, } \\ & \mathrm { g } ( x ) = \frac { 1 } { x + 1 } \quad \text { for } x > - 1 \end{aligned}$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\).
   It is given that f is a one-one function.
2. State the smallest possible value of \(c\).
   It is now given that \(c = 5\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
4. Find an expression for \(\mathrm { gf } ( x )\) and state the range of gf .

1

1. Express \(16 x ^ { 2 } - 24 x + 10\) in the form \(( 4 x + a ) ^ { 2 } + b\).
2. It is given that the equation \(16 x ^ { 2 } - 24 x + 10 = k\), where \(k\) is a constant, has exactly one root. Find the value of this root.

6 Functions f and g are both defined for \(x \in \mathbb { R }\) and are given by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 2 x + 5 \\ & \mathrm {~g} ( x ) = x ^ { 2 } + 4 x + 13 \end{aligned}$$

1. By first expressing each of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) in completed square form, express \(\mathrm { g } ( x )\) in the form \(\mathrm { f } ( x + p ) + q\), where \(p\) and \(q\) are constants.
2. Describe fully the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).

1

1. Express \(x ^ { 2 } - 8 x + 11\) in the form \(( x + p ) ^ { 2 } + q\) where \(p\) and \(q\) are constants.
2. Hence find the exact solutions of the equation \(x ^ { 2 } - 8 x + 11 = 1\).

6 The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 16 x + 23\) for \(x < 3\).

1. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\).
2. Find the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x + 4\) for \(x < - 1\).
4. Find and simplify an expression for \(\mathrm { fg } ( x )\).

11 The function f is defined by \(\mathrm { f } ( x ) = 10 + 6 x - x ^ { 2 }\) for \(x \in \mathbb { R }\).

1. By completing the square, find the range of f . \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-16_2715_37_143_2010} The function g is defined by \(\mathrm { g } ( x ) = 4 x + k\) for \(x \in \mathbb { R }\) where \(k\) is a constant.
2. It is given that the graph of \(y = \mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) meets the graph of \(y = \mathrm { g } ( x )\) at a single point \(P\). Determine the coordinates of \(P\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-18_2715_35_143_2012}

9

1. Express \(2 x ^ { 2 } + 12 x + 11\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } + 12 x + 11\) for \(x \leqslant - 4\).
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).
   The function g is defined by \(\mathrm { g } ( x ) = 2 x - 3\) for \(x \leqslant k\).
3. For the case where \(k = - 1\), solve the equation \(\operatorname { fg } ( x ) = 193\).
4. State the largest value of \(k\) possible for the composition fg to be defined.

7 Functions f and g are defined as follows: $$\begin{aligned} & \mathrm { f } : x \mapsto x ^ { 2 } + 2 x + 3 \text { for } x \leqslant - 1 , \\ & \mathrm {~g} : x \mapsto 2 x + 1 \text { for } x \geqslant - 1 . \end{aligned}$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) and state the range of f .
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
3. Solve the equation \(\operatorname { gf } ( x ) = 13\).

5

1. Express \(2 x ^ { 2 } - 8 x + 14\) in the form \(2 \left[ ( x - a ) ^ { 2 } + b \right]\).
   The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } \quad \text { for } x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 2 x ^ { 2 } - 8 x + 14 \quad \text { for } x \in \mathbb { R } \end{aligned}$$
2. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), making clear the order in which the transformations are applied. \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-08_679_1043_260_552} The circle with equation \(( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 85\) and the straight line with equation \(y = 3 x - 20\) are shown in the diagram. The line intersects the circle at \(A\) and \(B\), and the centre of the circle is at \(C\).

1

1. Express \(x ^ { 2 } + 6 x + 5\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. The curve with equation \(y = x ^ { 2 }\) is transformed to the curve with equation \(y = x ^ { 2 } + 6 x + 5\). Describe fully the transformation(s) involved.

8

1. Express \(- 3 x ^ { 2 } + 12 x + 2\) in the form \(- 3 ( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The one-one function f is defined by \(\mathrm { f } : x \mapsto - 3 x ^ { 2 } + 12 x + 2\) for \(x \leqslant k\).
2. State the largest possible value of the constant \(k\).
   It is now given that \(k = - 1\).
3. State the range of f.
4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The result of translating the graph of \(y = \mathrm { f } ( x )\) by \(\binom { - 3 } { 1 }\) is the graph of \(y = \mathrm { g } ( x )\).
5. Express \(\mathrm { g } ( x )\) in the form \(p x ^ { 2 } + q x + r\), where \(p , q\) and \(r\) are constants.

3

1. Express \(5 y ^ { 2 } - 30 y + 50\) in the form \(5 ( y + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. The function f is defined by \(\mathrm { f } ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 50 x\) for \(x \in \mathbb { R }\). Determine whether \(f\) is an increasing function, a decreasing function or neither.

9 Functions f and g are both defined for \(x \in \mathbb { R }\) and are given by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9 \\ & \mathrm {~g} ( x ) = 2 x ^ { 2 } + 4 x + 12 \end{aligned}$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\).
2. Express \(\mathrm { g } ( x )\) in the form \(2 \left[ ( x + c ) ^ { 2 } + d \right]\).
3. Express \(\mathrm { g } ( x )\) in the form \(k \mathrm { f } ( x + h )\), where \(k\) and \(h\) are integers.
4. Describe fully the two transformations that have been combined to transform the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).

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.

11

1. Find the coordinates of the minimum point of the curve \(y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18\). \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-18_675_901_1270_612} The diagram shows the curves with equations \(y = \frac { 9 } { 4 } x ^ { 2 } - 12 x + 18\) and \(y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }\). The curves intersect at the points \(( 0,18 )\) and \(( 4,6 )\).
2. Find the area of the shaded region.
3. A point \(P\) is moving along the curve \(y = 18 - \frac { 3 } { 8 } x ^ { \frac { 5 } { 2 } }\) in such a way that the \(x\)-coordinate of \(P\) is increasing at a constant rate of 2 units per second. Find the rate at which the \(y\)-coordinate of \(P\) is changing when \(x = 4\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The function f is defined by \(\mathrm { f } ( x ) = - 2 x ^ { 2 } - 8 x - 13\) for \(x < - 3\).

1. Express \(\mathrm { f } ( x )\) in the form \(- 2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers.
2. Find the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

9

1. Express \(4 x ^ { 2 } - 12 x + 13\) in the form \(( 2 x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The function f is defined by \(\mathrm { f } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(p < x < q\), where \(p\) and \(q\) are constants. The function g is defined by \(\mathrm { g } ( x ) = 3 x + 1\) for \(x < 8\).
2. Given that it is possible to form the composite function gf , find the least possible value of \(p\) and the greatest possible value of \(q\).
3. Find an expression for \(\operatorname { gf } ( x )\).
   The function h is defined by \(\mathrm { h } ( x ) = 4 x ^ { 2 } - 12 x + 13\) for \(x < 0\).
4. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).

11 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. It is given that \(a = 6\) and \(b = - 8\). Find the range of f .
2. It is given instead that \(a = 5\) and that 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 \(a ^ { 2 } < 4 ( b - a )\).

11 The equation of a curve is \(y = 8 x - x ^ { 2 }\).

1. Express \(8 x - x ^ { 2 }\) in the form \(a - ( x + b ) ^ { 2 }\), stating the numerical values of \(a\) and \(b\).
2. Hence, or otherwise, find the coordinates of the stationary point of the curve.
3. Find the set of values of \(x\) for which \(y \geqslant - 20\). The function g is defined by \(\mathrm { g } : x \mapsto 8 x - x ^ { 2 }\), for \(x \geqslant 4\).
4. State the domain and range of \(\mathrm { g } ^ { - 1 }\).
5. Find an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).

10 The function f is defined by \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 12 x + 13\) for \(0 \leqslant x \leqslant A\), where \(A\) is a constant.

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. State the value of \(A\) for which the graph of \(y = \mathrm { f } ( x )\) has a line of symmetry.
3. When \(A\) has this value, find the range of f . The function g is defined by \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 12 x + 13\) for \(x \geqslant 4\).
4. Explain why \(g\) has an inverse.
5. Obtain an expression, in terms of \(x\), for \(\mathrm { g } ^ { - 1 } ( x )\).

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

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x - b ) ^ { 2 } - c\).
2. State the range of f .
3. Find the set of values of \(x\) for which \(\mathrm { f } ( x ) < 21\). The function g is defined by \(\mathrm { g } : x \mapsto 2 x + k\) for \(x \in \mathbb { R }\).
4. Find the value of the constant \(k\) for which the equation \(\operatorname { gf } ( x ) = 0\) has two equal roots.

10 The function \(\mathrm { f } : x \mapsto 2 x ^ { 2 } - 8 x + 14\) is defined for \(x \in \mathbb { R }\).

1. Find the values of the constant \(k\) for which the line \(y + k x = 12\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
3. Find the range of f . The function \(\mathrm { g } : x \mapsto 2 x ^ { 2 } - 8 x + 14\) is defined for \(x \geqslant A\).
4. Find the smallest value of \(A\) for which g has an inverse.
5. For this value of \(A\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).

10

1. Express \(2 x ^ { 2 } - 4 x + 1\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point, \(A\), on the curve \(y = 2 x ^ { 2 } - 4 x + 1\). The line \(x - y + 4 = 0\) intersects the curve \(y = 2 x ^ { 2 } - 4 x + 1\) at points \(P\) and \(Q\). It is given that the coordinates of \(P\) are \(( 3,7 )\).
2. Find the coordinates of \(Q\).
3. Find the equation of the line joining \(Q\) to the mid-point of \(A P\).

8 The function \(\mathrm { f } : x \mapsto x ^ { 2 } - 4 x + k\) is defined for the domain \(x \geqslant p\), where \(k\) and \(p\) are constants.

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b + k\), where \(a\) and \(b\) are constants.
2. State the range of f in terms of \(k\).
3. State the smallest value of \(p\) for which f is one-one.
4. For the value of \(p\) found in part (iii), find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain \(\mathrm { f } ^ { - 1 }\), giving your answers in terms of \(k\).

8

1. Express \(2 x ^ { 2 } - 12 x + 13\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. The function f is defined by \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 12 x + 13\) for \(x \geqslant k\), where \(k\) is a constant. It is given that f is a one-one function. State the smallest possible value of \(k\). The value of \(k\) is now given to be 7 .
3. Find the range of f .
4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\).