Sketch quadratic curve

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 \(C\).

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}

8. $$4 x - 5 - x ^ { 2 } = q - ( x + p ) ^ { 2 }$$ where \(p\) and \(q\) are integers.

1. Find the value of \(p\) and the value of \(q\).
2. Calculate the discriminant of \(4 x - 5 - x ^ { 2 }\)
3. On the axes on page 17, sketch the curve with equation \(y = 4 x - 5 - x ^ { 2 }\) showing clearly the coordinates of any points where the curve crosses the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{089c3b5b-22ab-4fa2-8383-4f30cefa792a-11_1143_1143_260_388}

12

1. Write \(4 x ^ { 2 } - 24 x + 27\) in the form \(a ( x - b ) ^ { 2 } + c\).
2. State the coordinates of the minimum point on the curve \(y = 4 x ^ { 2 } - 24 x + 27\).
3. Solve the equation \(4 x ^ { 2 } - 24 x + 27 = 0\).
4. Sketch the graph of the curve \(y = 4 x ^ { 2 } - 24 x + 27\).

9

1. Sketch the curve \(y = 12 - x - x ^ { 2 }\), giving the coordinates of all intercepts with the axes.
2. Solve the inequality \(12 - x - x ^ { 2 } > 0\).
3. Find the coordinates of the points of intersection of the curve \(y = 12 - x - x ^ { 2 }\) and the line \(3 x + y = 4\).

10

1. Solve the equation \(9 x ^ { 2 } + 18 x - 7 = 0\).
2. Find the coordinates of the stationary point on the curve \(y = 9 x ^ { 2 } + 18 x - 7\).
3. Sketch the curve \(y = 9 x ^ { 2 } + 18 x - 7\), giving the coordinates of all intercepts with the axes.
4. For what values of \(x\) does \(9 x ^ { 2 } + 18 x - 7\) increase as \(x\) increases?

7

1. Solve the equation \(( x - 2 ) ^ { 2 } = 9\).
2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } - 9\), showing the coordinates of its intersections with the axes and its turning point.

2

1. Factorise \(x ^ { 2 } - 4 x - 12\).
2. Sketch the graph with equation \(y = x ^ { 2 } - 4 x - 12\), stating the values where the curve crosses the coordinate axes.
3. 1. Express \(x ^ { 2 } - 4 x - 12\) in the form \(( x - p ) ^ { 2 } - q\), where \(p\) and \(q\) are positive integers.
   2. Hence find the minimum value of \(x ^ { 2 } - 4 x - 12\).
4. The curve with equation \(y = x ^ { 2 } - 4 x - 12\) is translated by the vector \(\left[ \begin{array} { r } - 3 \\ 2 \end{array} \right]\). Find an equation of the new curve. You need not simplify your answer.

4

1. 1. Express \(16 - 6 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\) where \(p\) and \(q\) are integers.
   2. Hence write down the maximum value of \(16 - 6 x - x ^ { 2 }\).
2. 1. Factorise \(16 - 6 x - x ^ { 2 }\).
   2. Sketch the curve with equation \(y = 16 - 6 x - x ^ { 2 }\), stating the values of \(x\) where the curve crosses the \(x\)-axis and the value of the \(y\)-intercept.
      [0pt] [3 marks]

6. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),

1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).

3

1. Express \(x ^ { 2 } + 2 x - 3\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
2. Sketch the graph of \(y = x ^ { 2 } + 2 x - 3\) giving the coordinates of the vertex and of any intersections with the coordinate axes.

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]

Express \(x^2 - 6x\) in the form \((x - a)^2 - b\). Sketch the graph of \(y = x^2 - 6x\), giving the coordinates of its minimum point and the intersections with the axes. [5]