1.02e Complete the square: quadratic polynomials and turning points

The function f is defined by f\((x) = 3 + 6x - 2x^2\) for \(x \in \mathbb{R}\).

1. Express f\((x)\) in the form \(a - b(x - c)^2\), where \(a\), \(b\) and \(c\) are constants, and state the range of f. [3]
2. The graph of \(y = \)f\((x)\) is transformed to the graph of \(y = \)h\((x)\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \)h\((x)\) has a minimum point at the origin. Give details of the reflection and translation involved. [2] The function g is defined by g\((x) = 3 + 6x - 2x^2\) for \(x \leqslant 0\).
3. Sketch the graph of \(y = \)g\((x)\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes. [2]
4. Sketch the graph of \(y = \)g\(^{-1}(x)\) on your diagram in (c), and find an expression for g\(^{-1}(x)\). You should label the two graphs in your diagram appropriately and show any relevant mirror line. [4]

1. Express \(3x^2 - 12x + 14\) in the form \(3(x + a)^2 + b\), where \(a\) and \(b\) are constants to be found. [2]

The function f(x) = \(3x^2 - 12x + 14\) is defined for \(x \geqslant k\), where \(k\) is a constant.

1. Find the least value of \(k\) for which the function \(\text{f}^{-1}\) exists. [1]

For the rest of this question, you should assume that \(k\) has the value found in part (b).

1. Find an expression for \(\text{f}^{-1}(x)\). [3]
2. Hence or otherwise solve the equation \(\text{f f}(x) = 29\). [3]

The functions f and g are defined for \(x \in \mathbb{R}\) by $$f : x \mapsto 4x - 2x^2,$$ $$g : x \mapsto 5x + 3.$$

1. Find the range of f. [2]
2. Find the value of the constant \(k\) for which the equation \(gf(x) = k\) has equal roots. [3]

The function \(f\) is such that \(f(x) = 8 - (x - 2)^2\), for \(x \in \mathbb{R}\).

1. Find the coordinates and the nature of the stationary point on the curve \(y = f(x)\). [3]

The function \(g\) is such that \(g(x) = 8 - (x - 2)^2\), for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.

1. State the smallest value of \(k\) for which \(g\) has an inverse. [1]

For this value of \(k\),

1. find an expression for \(g^{-1}(x)\), [3]
2. sketch, on the same diagram, the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\). [3]

A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r\) cm.

1. Show that the area of the sector, \(A\) cm\(^2\), is given by \(A = 12r - r^2\). [3]
2. Express \(A\) in the form \(a - (r - b)^2\), where \(a\) and \(b\) are constants. [2]
3. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2]

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

1. Find the set of values of \(p\) for which the equation \(\mathrm{f}(x) = p\) has no real roots. [3]

The function g is defined by \(\mathrm{g} : x \mapsto 2x^2 - 6x + 5\) for \(0 \leqslant x \leqslant 4\).

1. Express \(\mathrm{g}(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
2. Find the range of g. [2]

The function h is defined by \(\mathrm{h} : x \mapsto 2x^2 - 6x + 5\) for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.

1. State the smallest value of \(k\) for which h has an inverse. [1]
2. For this value of \(k\), find an expression for \(\mathrm{h}^{-1}(x)\). [3]

1. Express \(9x^2 - 6x + 6\) in the form \((ax + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]

The function f is defined by \(\text{f}(x) = 9x^2 - 6x + 6\) for \(x \geqslant p\), where \(p\) is a constant.

1. State the smallest value of \(p\) for which f is a one-one function. [1]
2. For this value of \(p\), obtain an expression for \(\text{f}^{-1}(x)\), and state the domain of \(\text{f}^{-1}\). [4]
3. State the set of values of \(q\) for which the equation \(\text{f}(x) = q\) has no solution. [1]

The function f is defined by \(\text{f}(x) = x^2 - 4x + 8\) for \(x \in \mathbb{R}\).

1. Express \(x^2 - 4x + 8\) in the form \((x - a)^2 + b\). [2]
2. Hence find the set of values of \(x\) for which \(\text{f}(x) < 9\), giving your answer in exact form. [3]

1. Express \(x^2 - 4x + 7\) in the form \((x + a)^2 + b\). [2]

The function \(f\) is defined by \(f(x) = x^2 - 4x + 7\) for \(x < k\), where \(k\) is a constant.

1. State the largest value of \(k\) for which \(f\) is a decreasing function. [1]

The value of \(k\) is now given to be \(1\).

1. Find an expression for \(f^{-1}(x)\) and state the domain of \(f^{-1}\). [3]
2. The function \(g\) is defined by \(g(x) = \frac{2}{x-1}\) for \(x > 1\). Find an expression for \(gf(x)\) and state the range of \(gf\). [4]

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

1. Express \(2x^2 - 12x + 7\) in the form \(2(x + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
2. State the range of f. [1]

The function g is defined by \(\text{g} : x \mapsto 2x^2 - 12x + 7\) for \(x \leqslant k\).

1. State the largest value of \(k\) for which g has an inverse. [1]
2. Given that g has an inverse, find an expression for \(\text{g}^{-1}(x)\). [3]

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

1. Express \(7 - 2x^2 - 12x\) in the form \(a - 2(x + b)^2\), where \(a\) and \(b\) are constants. [2]
2. State the coordinates of the stationary point on the curve \(y = \mathrm{f}(x)\). [1]

The function g is defined by \(\mathrm{g} : x \mapsto 7 - 2x^2 - 12x\) for \(x \geqslant k\).

1. State the smallest value of \(k\) for which g has an inverse. [1]
2. For this value of \(k\), find \(\mathrm{g}^{-1}(x)\). [3]

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

Given that $$f(x) = x^2 - 6x + 18, \quad x \geq 0,$$

1. express \(f(x)\) in the form \((x - a)^2 + b\), where \(a\) and \(b\) are integers. [3]

The curve \(C\) with equation \(y = f(x)\), \(x \geq 0\), meets the \(y\)-axis at \(P\) and has a minimum point at \(Q\).

1. Sketch the graph of \(C\), showing the coordinates of \(P\) and \(Q\). [4]

The line \(y = 41\) meets \(C\) at the point \(R\).

1. Find the \(x\)-coordinate of \(R\), giving your answer in the form \(p + q\sqrt{2}\), where \(p\) and \(q\) are integers. [5]

\(x^2 - 8x - 29 = (x + a)^2 + b\), where \(a\) and \(b\) are constants.

1. Find the value of \(a\) and the value of \(b\). [3]
2. Hence, or otherwise, show that the roots of $$x^2 - 8x - 29 = 0$$ are \(c \pm d\sqrt{5}\), where \(c\) and \(d\) are integers to be found. [3]

\(x^2 + 2x + 3 \equiv (x + a)^2 + b\).

1. Find the values of the constants \(a\) and \(b\). [2]
2. Sketch the graph of \(y = x^2 + 2x + 3\), indicating clearly the coordinates of any intersections with the coordinate axes. [3]
3. Find the value of the discriminant of \(x^2 + 2x + 3\). Explain how the sign of the discriminant relates to your sketch in part (b). [2]

The equation \(x^2 + kx + 3 = 0\), where \(k\) is a constant, has no real roots.

1. Find the set of possible values of \(k\), giving your answer in surd form. [4]

1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{k^2 - c}\). [4]

The equation \(x^2 + 2kx + 81 = 0\) has equal roots.

1. Find the possible values of \(k\). [2]

\(f(x) = x^2 - kx + 9\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the equation \(f(x) = 0\) has no real solutions. [4]

Given that \(k = 4\),

1. express \(f(x)\) in the form \((x - p)^2 + q\), where \(p\) and \(q\) are constants to be found, [3]

Given that $$x^2 + 10x + 36 = (x + a)^2 + b$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\). [3]
2. Hence show that the equation \(x^2 + 10x + 36 = 0\) has no real roots. [2]

The equation \(x^2 + 10x + k = 0\) has equal roots.

1. Find the value of \(k\). [2]
2. For this value of \(k\), sketch the graph of \(y = x^2 + 10x + k\), showing the coordinates of any points at which the graph meets the coordinate axes. [4]

The functions \(f\) and \(g\) are defined by $$f: x \mapsto x^2 - 2x + 3, x \in \mathbb{R}, \quad 0 \leq x \leq 4,$$ $$g: x \mapsto \lambda x^2 + 1, \text{ where } \lambda \text{ is a constant, } x \in \mathbb{R}.$$

1. Find the range of \(f\). [3]
2. Given that \(gf(2) = 16\), find the value of \(\lambda\). [3]

$$4x^2 + 4x + 17 \equiv (ax + b)^2 + c, \quad a > 0.$$

1. Find the values of \(a\), \(b\) and \(c\). [3]
2. Find the exact value of $$\int_{-0.5}^{1.5} \frac{1}{4x^2 + 4x + 17} \, dx.$$ [4]