1.02d Quadratic functions: graphs and discriminant conditions

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.

7. (i) Calculate the discriminant of \(x ^ { 2 } - 6 x + 12\).
(ii) State the number of real roots of the equation \(x ^ { 2 } - 6 x + 12 = 0\) and hence, explain why \(x ^ { 2 } - 6 x + 12\) is always positive.
(iii) Show that the line \(y = 8 - 2 x\) is a tangent to the curve \(y = x ^ { 2 } - 6 x + 12\).

1. Find the set of values of the constant \(k\) such that the equation

$$x ^ { 2 } - 6 x + k = 0$$ has real and distinct roots.

6. $$f ( x ) = 4 x ^ { 2 } + 12 x + 9 .$$

1. Determine the number of real roots that exist for the equation \(\mathrm { f } ( x ) = 0\).
2. Solve the equation \(\mathrm { f } ( x ) = 8\), giving your answers in the form \(a + b \sqrt { 2 }\) where \(a\) and \(b\) are rational.

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 \(b\).

5. Given that the equation $$x ^ { 2 } + 4 k x - k = 0$$ has no real roots,

1. show that $$4 k ^ { 2 } + k < 0 ,$$
2. find the set of possible values of \(k\).

3 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).

5 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).

1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).

4

1. Solve, by factorising, the equation \(2 x ^ { 2 } - x - 3 = 0\).
2. Sketch the graph of \(y = 2 x ^ { 2 } - x - 3\).
3. Show that the equation \(x ^ { 2 } - 5 x + 10 = 0\) has no real roots.
4. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = 2 x ^ { 2 } - x - 3\) and \(y = x ^ { 2 } - 5 x + 10\). Give your answer in the form \(a \pm \sqrt { b }\).

3 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 1 } { x - 2 }\).

1. Draw accurately the graph of \(y = 2 x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\).
2. Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\) satisfy the equation \(2 x ^ { 2 } - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection.
3. Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = - x + k\). Hence find the exact values of \(k\) for which \(y = - x + k\) is a tangent to \(y = \frac { 1 } { x - 2 }\). [4]

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 \(5 x ^ { 2 } + 20 x - 8 = 0\).

7 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Each diagram shows a quadratic curve. State which diagram corresponds to the curve
   1. \(y = ( 3 - x ) ^ { 2 }\),
   2. \(y = x ^ { 2 } + 9\),
   3. \(y = ( 3 - x ) ( x + 3 )\).
   4. Give the equation of the curve which does not correspond to any of the equations in part (i).

10 The quadratic equation \(k x ^ { 2 } - 30 x + 25 k = 0\) has equal roots. Find the possible values of \(k\).

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 \(k\).

7 A curve has equation \(y = ( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\).

1. Find the coordinates of the minimum point, justifying that it is a minimum.
2. Calculate the discriminant of \(x ^ { 2 } - 3 x + 5\).
3. Explain why \(( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\) is always positive for \(x > - 2\).

8

1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\), giving the coordinates of all points of intersection with the axes.
2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - x - 3 > 0\).
3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of the constant \(k\).

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 \(y = 4 + 12 x - 2 x ^ { 2 }\).

9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.

8 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 2 = 0\) has no real roots.

9 Prove that the line \(y = 3 x - 10\) does not intersect the curve \(y = x ^ { 2 } - 5 x + 7\).

12 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).

1. Write down the radius of the circle and the coordinates of its centre.
2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0 .$$
4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

12 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } - x ^ { 3 } + x ^ { 2 } + 9 x - 10\).

1. Show that \(x = 1\) is a root of \(\mathrm { f } ( x ) = 0\) and hence express \(\mathrm { f } ( x )\) as a product of a linear factor and a cubic factor.
2. Hence or otherwise find another root of \(\mathrm { f } ( x ) = 0\).
3. Factorise \(\mathrm { f } ( x )\), showing that it has only two linear factors. Show also that \(\mathrm { f } ( x ) = 0\) has only two real roots. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

11

1. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) with the axes.
2. Find the coordinates of the points of intersection of the curve \(y = 2 x ^ { 2 } - 5 x - 3\) and the line \(y = x + 3\).
3. Find the set of values of \(k\) for which the line \(y = x + k\) does not intersect the curve \(y = 2 x ^ { 2 } - 5 x - 3\). \section*{END OF QUESTION PAPER}