1.02e Complete the square: quadratic polynomials and turning points

8. Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A \mathrm {~cm} ^ { 2 }\), after \(t\) seconds is given by $$A = ( p + q t ) ^ { 2 } ,$$ where \(p\) and \(q\) are positive constants.
Given that when \(t = 0 , A = 4\) and that when \(t = 5 , A = 9\),

1. find the value of \(p\) and show that \(q = \frac { 1 } { 5 }\),
2. find \(\frac { \mathrm { d } A } { \mathrm {~d} t }\) in terms of \(t\),
3. find the rate at which the area of the stain is increasing when \(t = 15\).

9. The curve \(C\) has the equation \(y = x ^ { 2 } + 2 x + 4\).

1. Express \(x ^ { 2 } + 2 x + 4\) in the form \(a ( x + b ) ^ { 2 } + c\) and hence state the coordinates of the minimum point of \(C\). The straight line \(l\) has the equation \(x + y = 8\).
2. Sketch \(l\) and \(C\) on the same set of axes.
3. Find the coordinates of the points where \(I\) and \(C\) intersect.

5. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1 .$$

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.

7. \(\quad f ( x ) = x ^ { 2 } - 2 x + 5 , x \in \mathbb { R } , x \geq 1\).

1. Express \(\mathrm { f } ( x )\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. State the range of f.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
4. Describe fully two transformations that would map the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) onto the graph of \(y = \sqrt { x } , x \geq 0\).
5. Find an equation for the normal to the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point where \(x = 8\).

2. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv x ^ { 2 } - 3 x + 7 , \quad x \in \mathbb { R } \\ & \mathrm {~g} ( x ) \equiv 2 x - 1 , \quad x \in \mathbb { R } \end{aligned}$$

1. Find the range of f .
2. Evaluate \(\operatorname { gf } ( - 1 )\).
3. Solve the equation $$\mathrm { fg } ( x ) = 17$$
   1. \(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 } , x \in \mathbb { R }\).
   2. Find the values of the constants \(A\), \(B\), \(C\) and \(D\) such that
   $$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$ The point \(P\) on the curve \(y = \mathrm { f } ( x )\) has \(x\)-coordinate 1.
4. Show that the normal to the curve \(y = \mathrm { f } ( x )\) at \(P\) has the equation $$x + 5 y + 9 = 0$$
   1. (a) Given that
   $$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$
5. Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).

1. (a) Write \(x ^ { 2 } + 4 x - 5\) in the form \(( x + p ) ^ { 2 } + q\) where \(p\) and \(q\) are integers.
   (b) Hence use a standard integral from the formula book to find

$$\int \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \mathrm {~d} x$$ (c) Determine the mean value of the function $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \quad 3 \leqslant x \leqslant 13$$ giving your answer in the form \(A \ln B\) where \(A\) and \(B\) are constants in simplest form.

1

1. Show that \(4 x ^ { 2 } - 12 x + 3 = 4 \left( x - \frac { 3 } { 2 } \right) ^ { 2 } - 6\).
2. State the coordinates of the minimum point of the curve \(y = 4 x ^ { 2 } - 12 x + 3\).

4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 y + 20 = 0\).

1. By completing the square, express this equation in the form $$x ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle, leaving your answer in surd form.
3. A line has equation \(y = 2 x\).
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation \(x ^ { 2 } - 4 x + 4 = 0\).
   2. Hence show that the line is a tangent to the circle and find the coordinates of the point of contact, \(P\).
4. Prove that the point \(Q ( - 1,4 )\) lies inside the circle.

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.

3

1. 1. Express \(x ^ { 2 } + 10 x + 19\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Write down the coordinates of the vertex (minimum point) of the curve with equation \(y = x ^ { 2 } + 10 x + 19\).
   3. Write down the equation of the line of symmetry of the curve \(y = x ^ { 2 } + 10 x + 19\).
   4. Describe geometrically the transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 10 x + 19\).
2. Determine the coordinates of the points of intersection of the line \(y = x + 11\) and the curve \(y = x ^ { 2 } + 10 x + 19\).

4

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

6

1. 1. Express \(x ^ { 2 } - 8 x + 17\) in the form \(( x - p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 17\).
   3. State the value of \(x\) for which the minimum value of \(x ^ { 2 } - 8 x + 17\) occurs.
      (1 mark)
2. The point \(A\) has coordinates (5,4) and the point \(B\) has coordinates ( \(x , 7 - x\) ).
   1. Expand \(( x - 5 ) ^ { 2 }\).
   2. Show that \(A B ^ { 2 } = 2 \left( x ^ { 2 } - 8 x + 17 \right)\).
   3. Use your results from part (a) to find the minimum value of the distance \(A B\) as \(x\) varies.

7

1. Describe a geometrical transformation by which the hyperbola $$x ^ { 2 } - 4 y ^ { 2 } = 1$$ can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).
2. The diagram shows the hyperbola \(H\) with equation $$x ^ { 2 } - y ^ { 2 } - 4 x + 3 = 0$$
   \includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-4_951_1216_824_402}
   By completing the square, describe a geometrical transformation by which the hyperbola \(H\) can be obtained from the hyperbola \(x ^ { 2 } - y ^ { 2 } = 1\).

2

1. Find the transformation which maps the curve \(y = x ^ { 2 }\) to the curve \(y = x ^ { 2 } + 8 x - 7\).
2. Write down the coordinates of the turning point of \(y = x ^ { 2 } + 8 x - 7\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) Given that \(C _ { 1 }\)

• has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a quadratic function
• cuts the \(x\)-axis at the origin and at \(x = 4\)
• has a minimum turning point at ( \(2 , - 4.8\) )
  1. find \(\mathrm { f } ( x )\)

Given that \(C _ { 2 }\)

The curves \(C _ { 1 }\) and \(C _ { 2 }\) meet in the first quadrant at the point \(P\), shown in Figure 1.

Use algebra to find the coordinates of \(P\).

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.

10

1. Given that \(10 + 4 x - x ^ { 2 } \equiv p - ( x - q ) ^ { 2 }\), show that \(q = 2\) and find the value of \(p\).
2. Hence find the coordinates of all the points of intersection of the curve \(y = 10 + 4 x - x ^ { 2 }\) and the circle \(( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 25\).

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 \(x\).

4

1. Show that \(2 x ^ { 2 } - 10 x - 3\) may be expressed in the form \(a ( x + b ) ^ { 2 } + c\) where \(a , b\) and \(c\) are real numbers to be found. Hence write down the co-ordinates of the minimum point on the curve.
2. Solve the equation \(4 x ^ { 4 } - 13 x ^ { 2 } + 9 = 0\).

2

1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } + 6 x + 5\).
3. Find the value of the constant \(k\) for which the line \(y = k - 2 x\) is a tangent to the curve \(y = 2 x ^ { 2 } + 6 x + 5\).

4

1. Show that \(2 x ^ { 2 } - 10 x - 3\) may be expressed in the form \(a ( x + b ) ^ { 2 } + c\) where \(a , b\) and \(c\) are real numbers to be found. Hence write down the coordinates of the minimum point on the curve.
2. Solve the equation \(4 x ^ { 4 } - 13 x ^ { 2 } + 9 = 0\).

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 equation of a curve is \(y = x^2 - 8x + 5\).

1. Find the coordinates of the minimum point of the curve. [2]

The curve is stretched by a factor of 2 parallel to the \(y\)-axis and then translated by \(\begin{pmatrix} 4 \\ 1 \end{pmatrix}\).

1. Find the coordinates of the minimum point of the transformed curve. [2]
2. Find the equation of the transformed curve. Give the answer in the form \(y = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]