1.02e Complete the square: quadratic polynomials and turning points

4 The quadratic function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 3 x + 2\).

1. Write \(\mathrm { f } ( x )\) in the form \(( \mathrm { x } + \mathrm { a } ) ^ { 2 } + \mathrm { b }\), where \(a\) and \(b\) are constants.
2. Write down the coordinates of the minimum point on the graph of \(y = f ( x )\).
3. Describe fully the transformation that maps the graph of \(y = f ( x )\) onto the graph of \(y = ( x + 1 ) ^ { 2 } - \frac { 1 } { 4 }\).

4

1. Express \(x ^ { 2 } + 4 x + 7\) in the form \(( x + b ) ^ { 2 } + c\).
2. Explain why the minimum point on the curve \(y = ( x + b ) ^ { 2 } + c\) occurs when \(x = - b\).

2

1. Express \(x ^ { 2 } - 6 x + 1\) in the form \(( \mathrm { x } - \mathrm { a } ) ^ { 2 } - \mathrm { b }\), where \(a\) and \(b\) are integers to be determined.
2. Hence state the coordinates of the turning point on the graph of \(y = x ^ { 2 } - 6 x + 1\).

16 In the first year of a course, an A-level student, Aaishah, has a mathematics test each week. The night before each test she revises for \(t\) hours. Over the course of the year she realises that her percentage mark for a test, \(p\), may be modelled by the following formula, where \(A , B\) and \(C\) are constants. $$p = A - B ( t - C ) ^ { 2 }$$

• Aaishah finds that, however much she revises, her maximum mark is achieved when she does 2 hours revision. This maximum mark is 62 .
• Aaishah had a mark of 22 when she didn't spend any time revising.
  1. Find the values of \(A , B\) and \(C\).
  2. According to the model, if Aaishah revises for 45 minutes on the night before the test, what mark will she achieve?
  3. What is the maximum amount of time that Aaishah could have spent revising for the model to work?

In an attempt to improve her marks Aaishah now works through problems for a total of \(t\) hours over the three nights before the test. After taking a number of tests, she proposes the following new formula for \(p\). $$p = 22 + 68 \left( 1 - \mathrm { e } ^ { - 0.8 t } \right)$$ For the next three tests she recorded the data in Fig. 16. \begin{table}[h] \end{table}

Verify that the data is consistent with the new formula.

Aaishah's tutor advises her to spend a minimum of twelve hours working through problems in future. Determine whether or not this is good advice.

1

1. Express \(x ^ { 2 } + 8 x + 2\) in the form \(( x + a ) ^ { 2 } + b\).
2. Write down the coordinates of the turning point of the curve \(y = x ^ { 2 } + 8 x + 2\).
3. State the transformation(s) which map(s) the curve \(y = x ^ { 2 }\) onto the curve \(y = x ^ { 2 } + 8 x + 2\).

3

1. 1. Express \(x ^ { 2 } - 4 x + 9\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence, or otherwise, state the coordinates of the minimum point of the curve with equation \(y = x ^ { 2 } - 4 x + 9\).
2. The line \(L\) has equation \(y + 2 x = 12\) and the curve \(C\) has equation \(y = x ^ { 2 } - 4 x + 9\).
   1. Show that the \(x\)-coordinates of the points of intersection of \(L\) and \(C\) satisfy the equation $$x ^ { 2 } - 2 x - 3 = 0$$
   2. Hence find the coordinates of the points of intersection of \(L\) and \(C\).

4

1. 1. Express \(x ^ { 2 } + 2 x + 5\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence show that \(x ^ { 2 } + 2 x + 5\) is always positive.
2. A curve has equation \(y = x ^ { 2 } + 2 x + 5\).
   1. Write down the coordinates of the minimum point of the curve.
   2. Sketch the curve, showing the value of the intercept on the \(y\)-axis.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 2 x + 5\).

5

1. Express \(( x - 5 ) ( x - 3 ) + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   (3 marks)
2. 1. Sketch the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
   2. Write down an equation of the tangent to the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\) at its vertex.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = ( x - 5 ) ( x - 3 ) + 2\).

7

1. 1. Express \(4 - 10 x - x ^ { 2 }\) in the form \(p - ( x + q ) ^ { 2 }\).
   2. Hence write down the equation of the line of symmetry of the curve with equation \(y = 4 - 10 x - x ^ { 2 }\).
2. The curve \(C\) has equation \(y = 4 - 10 x - x ^ { 2 }\) and the line \(L\) has equation \(y = k ( 4 x - 13 )\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinates of any points of intersection of the curve \(C\) with the line \(L\) satisfy the equation $$x ^ { 2 } + 2 ( 2 k + 5 ) x - ( 13 k + 4 ) = 0$$
   2. Given that the curve \(C\) and the line \(L\) intersect in two distinct points, show that $$4 k ^ { 2 } + 33 k + 29 > 0$$
   3. Solve the inequality \(4 k ^ { 2 } + 33 k + 29 > 0\).

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 \(x ^ { 2 } - 6 x + 11\) in the form \(( x - p ) ^ { 2 } + q\).
   2. Use the result from part (a)(i) to show that the equation \(x ^ { 2 } - 6 x + 11 = 0\) has no real solutions.
2. A curve has equation \(y = x ^ { 2 } - 6 x + 11\).
   1. Find the coordinates of the vertex of the curve.
   2. Sketch the curve, indicating the value of \(y\) where the curve crosses the \(y\)-axis.
   3. Describe the geometrical transformation that maps the curve with equation \(y = x ^ { 2 } - 6 x + 11\) onto the curve with equation \(y = x ^ { 2 }\).

2

1. Express \(x ^ { 2 } - 6 x + 16\) in the form \(( x - p ) ^ { 2 } + q\).
2. A curve has equation \(y = x ^ { 2 } - 6 x + 16\). Using your answer from part (a), or otherwise:
   1. find the coordinates of the vertex (minimum point) of the curve;
   2. sketch the curve, indicating the value where the curve crosses the \(y\)-axis;
   3. state the equation of the line of symmetry of the curve.
3. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } - 6 x + 16\).

2

1. Express \(x ^ { 2 } + 8 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
2. Hence, or otherwise, show that the equation \(x ^ { 2 } + 8 x + 19 = 0\) has no real solutions.
3. Sketch the graph of \(y = x ^ { 2 } + 8 x + 19\), stating the coordinates of the minimum point and the point where the graph crosses the \(y\)-axis.
4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 8 x + 19\).

7

1. 1. Express \(2 x ^ { 2 } - 20 x + 53\) in the form \(2 ( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Use your result from part (a)(i) to explain why the equation \(2 x ^ { 2 } - 20 x + 53 = 0\) has no real roots.
2. The quadratic equation \(( 2 k - 1 ) x ^ { 2 } + ( k + 1 ) x + k = 0\) has real roots.
   1. Show that \(7 k ^ { 2 } - 6 k - 1 \leqslant 0\).
   2. Hence find the possible values of \(k\).

4

1. Express \(x ^ { 2 } + 5 x + 7\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   (3 marks)
2. A curve has equation \(y = x ^ { 2 } + 5 x + 7\).
   1. Find the coordinates of the vertex of the curve.
   2. State the equation of the line of symmetry of the curve.
   3. Sketch the curve, stating the value of the intercept on the \(y\)-axis.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 5 x + 7\).

5

1. 1. Express \(x ^ { 2 } - 3 x + 5\) in the form \(( x - p ) ^ { 2 } + q\).
   2. Hence write down the equation of the line of symmetry of the curve with equation \(y = x ^ { 2 } - 3 x + 5\).
2. The curve \(C\) with equation \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = x + 5\) intersect at the point \(A ( 0,5 )\) and at the point \(B\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-4_471_707_653_676}
   1. Find the coordinates of the point \(B\).
   2. Find \(\int \left( x ^ { 2 } - 3 x + 5 \right) \mathrm { d } x\).
   3. Find the area of the shaded region \(R\) bounded by the curve \(C\) and the line segment \(A B\).

5

1. 1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(2 ( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   2. Hence write down the minimum value of \(2 x ^ { 2 } + 6 x + 5\).
2. The point \(A\) has coordinates \(( - 3,5 )\) and the point \(B\) has coordinates \(( x , 3 x + 9 )\).
   1. Show that \(A B ^ { 2 } = 5 \left( 2 x ^ { 2 } + 6 x + 5 \right)\).
   2. Use your result from part (a)(ii) to find the minimum value of the length \(A B\) as \(x\) varies, giving your answer in the form \(\frac { 1 } { 2 } \sqrt { n }\), where \(n\) is an integer.

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]

5

1. Express \(x ^ { 2 } + 3 x + 2\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
2. A curve has equation \(y = x ^ { 2 } + 3 x + 2\).
   1. Use the result from part (a) to write down the coordinates of the vertex of the curve.
   2. State the equation of the line of symmetry of the curve.
3. The curve with equation \(y = x ^ { 2 } + 3 x + 2\) is translated by the vector \(\left[ \begin{array} { l } 2 \\ 4 \end{array} \right]\). Find the equation of the resulting curve in the form \(y = x ^ { 2 } + b x + c\).

3

1. 1. Express \(x ^ { 2 } - 7 x + 2\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are rational numbers.
   2. Hence write down the minimum value of \(x ^ { 2 } - 7 x + 2\).
2. Describe the geometrical transformation which maps the graph of \(y = x ^ { 2 } - 7 x + 2\) onto the graph of \(y = ( x - 4 ) ^ { 2 }\).
   [0pt] [3 marks]

6. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$

1. Write down the maximum value of \(\mathrm { f } ( x )\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
5. Find the value of \(k\).

8. $$f ( x ) = 9 - ( x - 2 ) ^ { 2 }$$

1. Write down the maximum value of \(\mathrm { f } ( x )\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points at which the graph meets the coordinate axes. The points \(A\) and \(B\) on the graph of \(y = \mathrm { f } ( x )\) have coordinates \(( - 2 , - 7 )\) and \(( 3,8 )\) respectively.
3. Find, in the form \(y = m x + c\), an equation of the straight line through \(A\) and \(B\).
4. Find the coordinates of the point at which the line \(A B\) crosses the \(x\)-axis. The mid-point of \(A B\) lies on the line with equation \(y = k x\), where \(k\) is a constant.
5. Find the value of \(k\).

2. The curve \(C\) has the equation $$y = x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants. Given that the minimum point of \(C\) has coordinates \(( - 2,5 )\), find the values of \(a\) and \(b\).

6. $$f ( x ) = 9 + 6 x - x ^ { 2 } .$$

1. Find the values of \(A\) and \(B\) such that $$\mathrm { f } ( x ) = A - ( x + B ) ^ { 2 }$$
2. State the maximum value of \(\mathrm { f } ( x )\).
3. Solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are integers.
4. Sketch the curve \(y = \mathrm { f } ( x )\).

6. \(f ( x ) = x ^ { 2 } - 10 x + 17\).

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. State the coordinates of the minimum point of the curve \(y = \mathrm { f } ( x )\).
3. Deduce the coordinates of the minimum point of each of the following curves:
   1. \(\quad y = \mathrm { f } ( x ) + 4\),
   2. \(y = \mathrm { f } ( 2 x )\).