1.02e Complete the square: quadratic polynomials and turning points

\includegraphics{figure_6} Fig. 11 shows a sketch of the curve with equation \(y = (x - 4)^2 - 3\).

1. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point. [2]
2. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary. [4]
3. The curve is translated by \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\). Show that the equation of the translated curve may be written as \(y = x^2 - 12x + 33\). [2]
4. Show that the line \(y = 8 - 2x\) meets the curve \(y = x^2 - 12x + 33\) at just one point, and find the coordinates of this point. [5]

The function f is even and has domain \(\mathbb{R}\). For \(x \geq 0\), f(x) = \(x^2 - 4ax\), where \(a\) is a positive constant.

1. In the space below, sketch the curve with equation \(y = \text{f}(x)\), showing the coordinates of all the points at which the curve meets the axes. [3]
2. Find, in terms of \(a\), the value of f(2a) and the value of f(-2a). [2]

Given that \(a = 3\),

1. use algebra to find the values of \(x\) for which f(x) = 45. [4]

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

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

f(x) = \(x^2 - 2x - 3\), \(x \in \mathbb{R}\), \(x \geq 1\).

1. Find the range of f. [1]
2. Write down the domain and range of \(f^{-1}\). [2]
3. Sketch the graph of \(f^{-1}\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. [4]

Given that g(x) = \(|x - 4|\), \(x \in \mathbb{R}\),

1. find an expression for gf(x). [2]
2. Solve gf(x) = 8. [5]

\includegraphics{figure_3} Fig. 7 shows the graph of \(y = \frac{1}{100}(100 + 15x - x^2)\). For \(0 \leq x < 20\), this graph shows the trajectory of a small stone projected from the point Q where \(y\) m is the height of the stone above horizontal ground and \(x\) m is the horizontal displacement of the stone from O. The stone hits the ground at the point R.

1. Write down the height of Q above the ground. [1]
2. Find the horizontal distance from O of the highest point of the trajectory and show that this point is \(1.5625\) m above the ground. [5]
3. Show that the time taken for the stone to fall from its highest point to the ground is \(0.565\) seconds, correct to 3 significant figures. [3]
4. Show that the horizontal component of the velocity of the stone is \(22.1\text{ms}^{-1}\), correct to 3 significant figures. Deduce the time of flight from Q to R. [5]
5. Calculate the speed at which the stone hits the ground. [4]

The functions f and g are defined for all real values of x by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).

1. Find the range of f. [3]
2. Give a reason why f has no inverse. [1]
3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]

A fire crew is tackling a grass fire on horizontal ground. The crew directs a single jet of water which flows continuously from point \(A\). \includegraphics{figure_11} The path of the jet can be modelled by the equation $$y = -0.0125x^2 + 0.5x - 2.55$$ where \(x\) metres is the horizontal distance of the jet from the fire truck at \(O\) and \(y\) metres is the height of the jet above the ground. The coordinates of point \(A\) are \((a, 0)\)

1. 1. Find the value of \(a\). [3 marks]
   2. Find the horizontal distance from \(A\) to the point where the jet hits the ground. [1 mark]
2. Calculate the maximum vertical height reached by the jet. [4 marks]
3. A vertical wall is located 11 metres horizontally from \(A\) in the direction of the jet. The height of the wall is 2.3 metres. Using the model, determine whether the jet passes over the wall, stating any necessary modelling assumption. [3 marks]

Find the coordinates, in terms of \(a\), of the minimum point on the curve \(y = x^2 - 5x + a\), where \(a\) is a constant. Fully justify your answer. [3 marks]

A curve \(C\) has equation \(y = x^2 - 4x + k\), where \(k\) is a constant. It crosses the \(x\)-axis at the points \((2 + \sqrt{5}, 0)\) and \((2 - \sqrt{5}, 0)\)

1. Find the value of \(k\). [2 marks]
2. Sketch the curve \(C\), labelling the exact values of all intersections with the axes. [3 marks]

\includegraphics{figure_1} A stone is thrown over level ground from the top of a tower, \(X\). The height, \(h\), in meters, of the stone above the ground level after \(t\) seconds is modelled by the function. $$h(t) = 7 + 21t - 4.9t^2, \quad t \geq 0$$ A sketch of \(h\) against \(t\) is shown in Figure 1. Using the model,

1. give a physical interpretation of the meaning of the constant term 7 in the model. [1]
2. find the time taken after the stone is thrown for it to reach ground level. [3]
3. Rearrange \(h(t)\) into the form \(A - B(t - C)^2\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
4. Using your answer to part c or otherwise, find the maximum height of the stone above the ground, and the time after which this maximum height is reached. [2]

Given that $$f(x) = x^2 - 4x + 5 \quad x \in \mathbb{R}$$

1. express \(f(x)\) in the form \((x + a)^2 + b\) where \(a\) and \(b\) are integers to be found. [2]

The curve with equation \(y = f(x)\) • meets the \(y\)-axis at the point \(P\) • has a minimum turning point at the point \(Q\)

1. Write down
   1. the coordinates of \(P\)
   2. the coordinates of \(Q\)
   [2]

A curve \(C\) has equation \(y = f(x)\) where $$f(x) = -3x^2 + 12x + 8$$

1. Write \(f(x)\) in the form $$a(x + b)^2 + c$$ where \(a\), \(b\) and \(c\) are constants to be found. [3]

The curve \(C\) has a maximum turning point at \(M\).

1. Find the coordinates of \(M\). [2]