1.02e Complete the square: quadratic polynomials and turning points

4

1. Express \(2 x ^ { 2 } - 20 x + 49\) in the form \(p ( x - q ) ^ { 2 } + r\).
2. State the coordinates of the vertex of the curve \(y = 2 x ^ { 2 } - 20 x + 49\).

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 \(y = 5 x ^ { 2 } + 15 x + 12\).

11

1. Express \(x ^ { 2 } - 5 x + 6\) in the form \(( x - a ) ^ { 2 } - b\). Hence state the coordinates of the turning point of the curve \(y = x ^ { 2 } - 5 x + 6\).
2. Find the coordinates of the intersections of the curve \(y = x ^ { 2 } - 5 x + 6\) with the axes and sketch this curve.
3. Solve the simultaneous equations \(y = x ^ { 2 } - 5 x + 6\) and \(x + y = 2\). Hence show that the line \(x + y = 2\) is a tangent to the curve \(y = x ^ { 2 } - 5 x + 6\) at one of the points where the curve intersects the axes.

10 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B . \begin{figure}[h] \end{figure}

1. Write down the coordinates of B .
2. Find the radius of the circle and hence write down the equation of the circle.
3. AD is a diameter of the circle. Find the coordinates of D .
4. Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-4_643_853_269_589} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).
5. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
6. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
7. The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
8. Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-5_775_1461_317_296} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).
9. Use the intersections with both axes to express the equation of the curve in a factorised form.
10. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
11. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
12. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0 .$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. \section*{END OF QUESTION PAPER}

1 Find the values of \(A , B\) and \(C\) in the identity \(4 x ^ { 2 } - 16 x + C \equiv A ( x + B ) ^ { 2 } + 2\).

6 By first completing the square, find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x ^ { 2 } + 4 x + 8 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.

3 By first completing the square, find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \frac { 1 } { \sqrt { 2 x - x ^ { 2 } } } \mathrm {~d} x\).

9 The function f is defined for all real values of \(x\) as \(\mathrm { f } ( x ) = c + 8 x - x ^ { 2 }\), where \(c\) is a constant.

1. Given that the range of f is \(\mathrm { f } ( x ) \leqslant 19\), find the value of \(c\).
2. Given instead that \(\mathrm { ff } ( 2 ) = 8\), find the possible values of \(c\).

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

1. Express \(2 x ^ { 2 } - 12 x + 23\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Use your result to show that the equation \(2 x ^ { 2 } - 12 x + 23 = 0\) has no real roots.
3. Given that the equation \(2 x ^ { 2 } - 12 x + k = 0\) has repeated roots, find the value of the constant \(k\).

1. Given that the point \(A\) has position vector \(4 \mathbf { i } - 5 \mathbf { j }\) and the point \(B\) has position vector \(- 5 \mathbf { i } - 2 \mathbf { j }\), (a) find the vector \(\overrightarrow { A B }\),
   (b) find \(| \overrightarrow { A B } |\).

Give your answer as a simplified surd.

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 \(R\).

5. The function f is defined by $$\mathrm { f } : x \rightarrow \frac { 2 x - 3 } { x - 1 } \quad x \in R , x \neq 1$$ a. Find \(f ^ { - 1 } ( 3 )\).
b. Show that $$\mathrm { ff } ( x ) = \frac { x + p } { x - 2 } \quad x \in R , \quad x \neq 2$$ where \(p\) is an integer to be found. The function g is defined by $$g : x \rightarrow x ^ { 2 } - 5 x \quad x \in R , 0 \leq x \leq 6$$ c. Find the range of g .
d. Explain why the function g does not have an inverse.

8. \begin{figure}[h] \end{figure} In a competition, competitors are going to kick a ball over the barrier walls. The height of the barrier walls are each 9 metres high and 50 cm wide and stand on horizontal ground. The figure 2 is a graph showing the motion of a ball. The ball reaches a maximum height of 12 metres and hits the ground at a point 80 metres from where its kicked.
a. Find a quadratic equation linking \(Y\) with \(x\) that models this situation. The ball pass over the barrier walls.
b. Use your equation to deduce that the ball should be placed about 20 m from the first barrier wall.

5. $$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$

1. Write \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are integers to be found.
2. Sketch the curve with equation \(y = \mathrm { f } ( x )\) showing any points of intersection with the coordinate axes and the coordinates of any turning point.
3. 1. Describe fully the transformation that maps the curve with equation \(y = \mathrm { f } ( x )\) onto the curve with equation \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
   2. Find the range of the function $$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\).
The curve \(C\) meets \(l\) at the points \(( - 2,13 )\) and \(( 0,25 )\) as shown.
The shaded region \(R\) is bounded by \(C\) and \(l\) as shown in Figure 1.
Given that

• \(\mathrm { f } ( x )\) is a quadratic function in \(x\)
• ( \(- 2,13\) ) is the minimum turning point of \(y = \mathrm { f } ( x )\) use inequalities to define \(R\).

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 \(Q\)

11. An archer shoots an arrow. The height, \(H\) metres, of the arrow above the ground is modelled by the formula $$H = 1.8 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ where \(d\) is the horizontal distance of the arrow from the archer, measured in metres.
Given that the arrow travels in a vertical plane until it hits the ground,

1. find the horizontal distance travelled by the arrow, as given by this model.
2. With reference to the model, interpret the significance of the constant 1.8 in the formula.
3. Write \(1.8 + 0.4 d - 0.002 d ^ { 2 }\) in the form $$A - B ( d - C ) ^ { 2 }$$ where \(A , B\) and \(C\) are constants to be found. It is decided that the model should be adapted for a different archer.
   The adapted formula for this archer is $$H = 2.1 + 0.4 d - 0.002 d ^ { 2 } , \quad d \geqslant 0$$ Hence or otherwise, find, for the adapted model
4. 1. the maximum height of the arrow above the ground.
   2. the horizontal distance, from the archer, of the arrow when it is at its maximum height.

8. \begin{figure}[h] \end{figure} Figure 1 is a graph showing the trajectory of a rugby ball. The height of the ball above the ground, \(H\) metres, has been plotted against the horizontal distance, \(x\) metres, measured from the point where the ball was kicked. The ball travels in a vertical plane. The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked.

1. Find a quadratic equation linking \(H\) with \(x\) that models this situation. The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.
2. Use your equation to find the greatest horizontal distance of the bar from \(O\).
3. Give one limitation of the model.

9. \begin{figure}[h] \end{figure} The graph in Figure 3 shows the path of a small ball.
The ball travels in a vertical plane above horizontal ground.
The ball is thrown from the point represented by \(A\) and caught at the point represented by \(B\). The height, \(H\) metres, of the ball above the ground has been plotted against the horizontal distance, \(x\) metres, measured from the point where the ball was thrown. With respect to a fixed origin \(O\), the point \(A\) has coordinates \(( 0,2 )\) and the point \(B\) has coordinates (20, 0.8), as shown in Figure 3. The ball reaches its maximum height when \(x = 9\) A quadratic function, linking \(H\) with \(x\), is used to model the path of the ball.

1. Find \(H\) in terms of \(x\).
2. Give one limitation of the model. Chandra is standing directly under the path of the ball at a point 16 m horizontally from \(O\). Chandra can catch the ball if the ball is less than 2.5 m above the ground.
3. Use the model to determine if Chandra can catch the ball.

4

1. Express \(4 x ^ { 2 } - 12 x + 11\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. State the number of real roots of the equation \(4 x ^ { 2 } - 12 x + 11 = 0\).
3. Explain fully how the value of \(r\) is related to the number of real roots of the equation \(p ( x + q ) ^ { 2 } + r = 0\) where \(p , q\) and \(r\) are real constants and \(p > 0\).

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.

5 A curve has equation \(y = a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants. The curve has a stationary point at \(( - 3,2 )\).

1. State the values of \(b\) and \(c\). When the curve is translated by \(\binom { 4 } { 0 }\) the transformed curve passes through the point \(( 3 , - 18 )\).
2. Determine the value of \(a\).

9 The curve \(y = ( x - 1 ) ^ { 2 }\) maps onto the curve \(\mathrm { C } _ { 1 }\) following a stretch scale factor \(\frac { 1 } { 2 }\) in the \(x\)-direction.

1. Show that the equation of \(\mathrm { C } _ { 1 }\) can be written as \(y = 4 x ^ { 2 } - 4 x + 1\). The curve \(\mathrm { C } _ { 2 }\) is a translation of \(y = 4.25 x - x ^ { 2 }\) by \(\binom { 0 } { - 3 }\).
2. Show that the normal to the curve \(\mathrm { C } _ { 1 }\) at the point \(( 0,1 )\) is a tangent to the curve \(\mathrm { C } _ { 2 }\).