1.02e Complete the square: quadratic polynomials and turning points

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 \(\frac { 1 } { x ^ { 2 } + 4 x + 14 }\).

10

1. A quadratic function is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 8\).
   Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the points where it crosses the axes. Mark the lowest point on the curve, and give its coordinates.
2. Solve the inequality \(x ^ { 2 } - 6 x + 8 < 0\).
3. On the same graph, sketch \(y = \mathrm { f } ( x + 3 )\).
4. The graph of \(y = \mathrm { f } ( x + 3 ) - 2\) is obtained from the graph of \(y = \mathrm { f } ( x )\) by a transformation. Describe the transformation and sketch the curve on the same axes as in (i) and (iii) above. Label all these curves clearly.

9 The graph shows the function \(y = x ^ { 2 } + b x + c\) where \(b\) and \(c\) are constants.
The point \(\mathrm { M } ( - 3 , - 16 )\) on the graph is the minimum point of the graph. \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-2_478_948_1871_588}

1. Write down the function \(y = \mathrm { f } ( x )\) in completed square form.
2. Hence find the coordinates of the points where the curve cuts the axes.

11 Fig. 11 shows the graph of \(y = a x ^ { 2 } + b x + c\). \begin{figure}[h] \end{figure}

1. Explain why a must be negative.
2. State two factors of \(y = a x ^ { 2 } + b x + c\).
3. Hence, or otherwise, find the values of \(a , b\) and \(c\). Another function is given by \(y = x ^ { 2 } - 4 x + 10\).
4. Write this in completed square form.
5. Explain why the graphs of these two functions never meet.

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 \(y = x ^ { 2 } + 6 x + 7\).

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

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

9. (i) Find an equation for the tangent to the curve \(y = x ^ { 2 } + 2\) at the point \(( 1,3 )\) in the form \(y = m x + c\).
(ii) Express \(x ^ { 2 } - 6 x + 11\) in the form \(( x + a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers.
(iii) Describe fully the transformation that maps the graph of \(y = x ^ { 2 } + 2\) onto the graph of \(y = x ^ { 2 } - 6 x + 11\).
(iv) Use your answers to parts (i) and (iii) to deduce an equation for the tangent to the curve \(y = x ^ { 2 } - 6 x + 11\) at the point with \(x\)-coordinate 4.

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 \(b\).

4. (i) By completing the square, find in terms of the constant \(k\) the roots of the equation $$x ^ { 2 } + 2 k x + 4 = 0 .$$ (ii) Hence find the exact roots of the equation $$x ^ { 2 } + 6 x + 4 = 0 .$$

15

1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.

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

1

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. [4] \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{973ad9eb-33f2-432e-9449-e54c1728008b-1_1292_1401_887_359} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 3 }\).
4. Draw accurately, on the copy of Fig. 12, the graph of \(y = x ^ { 2 } - 4 x + 1\) for \(- 1 \leqslant x \leqslant 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac { 1 } { x - 3 }\) and \(y = x ^ { 2 } - 4 x + 1\).
5. Show algebraically that, where the curves intersect, \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\).
6. Use the fact that \(x = 4\) is a root of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\) to find a quadratic factor of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4\). Hence find the exact values of the other two roots of this equation. [5]
7. Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
8. Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
9. For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
   (A) write down the equation of the line of symmetry,
   (B) write down the minimum \(y\)-value on the curve.

5. 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 \(g f ( - 1 )\).
3. Solve the equation $$\operatorname { fg } ( x ) = 17$$

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

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.

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

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 \(x = 8\).

6. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow 1 - a x , \quad x \in \mathbb { R } \\ & \mathrm {~g} : x \rightarrow x ^ { 2 } + 2 a x + 2 , \quad x \in \mathbb { R } \end{aligned}$$ where \(a\) is a constant.
Find, in terms of \(a\),

1. an expression for \(\mathrm { f } ^ { - 1 } ( x )\),
2. the range of g . Given that \(g f ( 3 ) = 7\),
3. find the two possible values of \(a\).

5

1. Express \(t ^ { 2 } + t + 1\) in the form \(( t + a ) ^ { 2 } + b\).
2. By using the substitution \(\tan \frac { 1 } { 2 } x = t\), show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { 1 } { 2 + \sin x } \mathrm {~d} x = \frac { \sqrt { 3 } } { 9 } \pi$$

11. Given that $$f ( x ) = 2 x ^ { 2 } + 8 x + 3$$

1. find the value of the discriminant of \(\mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in the form \(p ( x + q ) ^ { 2 } + r\) where \(p , q\) and \(r\) are integers to be found. The line \(y = 4 x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = \mathrm { f } ( x )\).
3. Calculate the value of \(c\).

6

1. Express \(5 x ^ { 2 } + 20 x - 8\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the equation of the line of symmetry of the curve \(y = 5 x ^ { 2 } + 20 x - 8\).
3. Calculate the discriminant of \(5 x ^ { 2 } + 20 x - 8\).
4. State the number of real roots of the equation \(5 x ^ { 2 } + 20 x - 8 = 0\).

1 Express \(x ^ { 2 } - 12 x + 1\) in the form \(( x - p ) ^ { 2 } + q\).

7

1. Express \(4 x ^ { 2 } + 12 x - 3\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. Solve the equation \(4 x ^ { 2 } + 12 x - 3 = 0\), giving your answers in simplified surd form.
3. The quadratic equation \(4 x ^ { 2 } + 12 x - k = 0\) has equal roots. Find the value of \(k\).

3 Given that $$5 x ^ { 2 } + p x - 8 = q ( x - 1 ) ^ { 2 } + r$$ for all values of \(x\), find the values of the constants \(p , q\) and \(r\).

6

1. Sketch the curve \(y = - \sqrt { x }\).
2. Describe fully a transformation that transforms the curve \(y = - \sqrt { x }\) to the curve \(y = 5 - \sqrt { x }\).
3. The curve \(y = - \sqrt { x }\) is stretched by a scale factor of 2 parallel to the \(x\)-axis. State the equation of the curve after it has been stretched.

8

1. Express \(2 x ^ { 2 } + 5 x\) in the form \(2 ( x + p ) ^ { 2 } + q\).
2. State the coordinates of the minimum point of the curve \(y = 2 x ^ { 2 } + 5 x\).
3. State the equation of the normal to the curve at its minimum point.
4. Solve the inequality \(2 x ^ { 2 } + 5 x > 0\).

1 Express \(3 x ^ { 2 } - 18 x + 4\) in the form \(p ( x + q ) ^ { 2 } + r\).