Complete square then find vertex/turning point

2

1. Express \(x ^ { 2 } - 8 x + 3\) in the form \(( x + a ) ^ { 2 } + b\).
2. Hence write down the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 8 x + 3\).

2

1. Find the constants \(a\) and \(b\) such that, for all values of \(x\), $$x ^ { 2 } + 4 x + 14 = ( x + a ) ^ { 2 } + b$$
2. Write down the greatest value of \(\frac { 1 } { x ^ { 2 } + 4 x + 14 }\).

4. (i) Express \(x ^ { 2 } + 6 x + 7\) in the form \(( x + a ) ^ { 2 } + b\).
(ii) State the coordinates of the vertex of the curve \(y = x ^ { 2 } + 6 x + 7\).

1 Express \(5 x ^ { 2 } + 15 x + 12\) in the form \(a ( x + b ) ^ { 2 } + c\).
Hence state the minimum value of \(y\) on the curve \(y = 5 x ^ { 2 } + 15 x + 12\).

6

1. Express \(4 + 12 x - 2 x ^ { 2 }\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. State the coordinates of the maximum point of the curve \(y = 4 + 12 x - 2 x ^ { 2 }\).

8 Express \(5 x ^ { 2 } + 15 x + 12\) in the form \(a ( x + b ) ^ { 2 } + c\).
Hence state the minimum value of \(y\) on the curve \(y = 5 x ^ { 2 } + 15 x + 12\).

4

1. Write \(2 x ^ { 2 } + 6 x + 7\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are constants.
2. State the coordinates of the minimum point on the graph of \(y = 2 x ^ { 2 } + 6 x + 7\).
3. Hence deduce

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = - 3 x ^ { 2 } + 12 x + 8$$

1. Write \(\mathrm { f } ( x )\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The curve \(C\) has a maximum turning point at \(M\).
2. Find the coordinates of \(M\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{235cd1dc-a3ab-473a-bf77-3e41b274dfd8-34_735_841_913_612} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of the curve \(C\).
   The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 3, is bounded by \(C , l\) and the \(y\)-axis.
3. Using algebraic integration, find the area of \(R\).

1. Given that

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

1. express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found. The curve with equation \(y = \mathrm { f } ( x )\)
   • meets the \(y\)-axis at the point \(P\)
   • has a minimum turning point at the point \(Q\)
   • Write down
     1. the coordinates of \(P\)
     2. the coordinates of \(Q\)

2

1. Express \(5 x ^ { 2 } - 20 x + 3\) in the form \(p ( x + q ) ^ { 2 } + r\), where \(p , q\) and \(r\) are integers.
2. State the coordinates of the minimum point of the curve \(y = 5 x ^ { 2 } - 20 x + 3\).
3. State the equation of the normal to the curve \(y = 5 x ^ { 2 } - 20 x + 3\) at its minimum point.

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 \(y = x ^ { 2 } - 6 x + 1\).

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. \(y = \mathrm { f } ( 2 x )\).

5

1. Factorise \(9 - 8 x - x ^ { 2 }\).
2. Show that \(25 - ( x + 4 ) ^ { 2 }\) can be written as \(9 - 8 x - x ^ { 2 }\).
3. A curve has equation \(y = 9 - 8 x - x ^ { 2 }\).
   1. Write down the equation of its line of symmetry.
   2. Find the coordinates of its vertex.
   3. Sketch the curve, indicating the values of the intercepts on the \(x\)-axis and the \(y\)-axis.

1

1. Express \(x ^ { 2 } - 8 x + 10\) in the form \(( x - a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found.
2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 10\) and the corresponding value of \(x\).

The diagram below shows a sketch of the graph of \(y = f ( x )\), where $$f ( x ) = 2 x ^ { 2 } + 12 x + 10 .$$ The graph intersects the \(x\)-axis at the points \(( p , 0 ) , ( q , 0 )\) and the \(y\)-axis at the point \(( 0,10 )\). \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-5_1004_1171_648_440}
a) Write down the value of \(f f ( p )\).
b) Determine the values of \(p\) and \(q\).
c) Express \(f ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b , c\) are constants whose values are to be found. Write down the coordinates of the minimum point.
d) Explain why \(f ^ { - 1 } ( x )\) does not exist.
e) The function \(g ( x )\) is defined as $$g ( x ) = f ( x ) \quad \text { for } \quad - 3 \leqslant x < \infty .$$ i) Find an expression for \(g ^ { - 1 } ( x )\).
ii) Sketch the graph of \(y = g ^ { - 1 } ( x )\), indicating the coordinates of the points where the graph intersects the \(x\)-axis and the \(y\)-axis.

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]

Express \(3x^2 - 12x + 5\) in the form \(a(x - b)^2 - c\). Hence state the minimum value of \(y\) on the curve \(y = 3x^2 - 12x + 5\). [5]

Express \(3x^2 - 12x + 5\) in the form \(a(x - b)^2 - c\). Hence state the minimum value of \(y\) on the curve \(y = 3x^2 - 12x + 5\). [5]

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]

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]