1.02d Quadratic functions: graphs and discriminant conditions

6 A curve has equation \(y = k x ^ { 2 } + 1\) and a line has equation \(y = k x\), where \(k\) is a non-zero constant.

1. Find the set of values of \(k\) for which the curve and the line have no common points.
2. State the value of \(k\) for which the line is a tangent to the curve and, for this case, find the coordinates of the point where the line touches the curve.

5 Find the set of values of \(k\) for which the line \(y = 2 x - k\) meets the curve \(y = x ^ { 2 } + k x - 2\) at two distinct points.

6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.

1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).

8 The function f is defined, for \(x \in \mathbb { R }\), by \(\mathrm { f } : x \mapsto x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants.

1. In the case where \(a = 6\) and \(b = - 8\), find the range of f .
2. In the case where \(a = 5\), the roots of the equation \(\mathrm { f } ( x ) = 0\) are \(k\) and \(- 2 k\), where \(k\) is a constant. Find the values of \(b\) and \(k\).
3. Show that if the equation \(\mathrm { f } ( x + a ) = a\) has no real roots, then \(a ^ { 2 } < 4 ( b - a )\).

1 A line has equation \(y = 2 x - 7\) and a curve has equation \(y = x ^ { 2 } - 4 x + c\), where \(c\) is a constant. Find the set of possible values of \(c\) for which the line does not intersect the curve.

3 A curve has equation \(y = 2 x ^ { 2 } - 6 x + 5\).

1. Find the set of values of \(x\) for which \(y > 13\).
2. Find the value of the constant \(k\) for which the line \(y = 2 x + k\) is a tangent to the curve.

7 Points \(A\) and \(B\) lie on the curve \(y = x ^ { 2 } - 4 x + 7\). Point \(A\) has coordinates \(( 4,7 )\) and \(B\) is the stationary point of the curve. The equation of a line \(L\) is \(y = m x - 2\), where \(m\) is a constant.

1. In the case where \(L\) passes through the mid-point of \(A B\), find the value of \(m\).
2. Find the set of values of \(m\) for which \(L\) does not meet the curve.

2 A line has equation \(y = x + 1\) and a curve has equation \(y = x ^ { 2 } + b x + 5\). Find the set of values of the constant \(b\) for which the line meets the curve.

9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 2 x ^ { 2 } + 8 x + 1 \quad \text { for } x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 2 x - k \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(k\) is a constant.

1. Find the value of \(k\) for which the line \(y = \mathrm { g } ( x )\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
2. In the case where \(k = - 9\), find the set of values of \(x\) for which \(\mathrm { f } ( x ) < \mathrm { g } ( x )\).
3. In the case where \(k = - 1\), find \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) and solve the equation \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = 0\).
4. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants, and hence state the least value of \(\mathrm { f } ( x )\).

6 A line has equation \(y = 3 k x - 2 k\) and a curve has equation \(y = x ^ { 2 } - k x + 2\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the line and curve meet at two distinct points.
2. For each of two particular values of \(k\), the line is a tangent to the curve. Show that these two tangents meet on the \(x\)-axis.

6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.

1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x + a$$ where \(a\) is a constant.

1. Given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
2. When \(a\) has this value, find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.

3

1. Find the quotient when $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 13$$ is divided by \(x ^ { 2 } + 6\) and show that the remainder is 1 .
2. Show that the equation $$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 12 = 0$$ has no real roots.

3 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 14$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).

1. Find the value of \(a\).
2. Show that, when \(a\) has this value, the equation \(\mathrm { f } ( x ) = 0\) has only one real root.

8. The straight line \(l\) has equation \(y = k ( 2 x - 1 )\), where \(k\) is a constant. The curve \(C\) has equation \(y = x ^ { 2 } + 2 x + 11\) Find the set of values of \(k\) for which \(l\) does not cross or touch \(C\).
(6)

2. $$f ( x ) = 11 - 4 x - 2 x ^ { 2 }$$

1. Express \(\mathrm { f } ( x )\) in the form $$a + b ( x + c ) ^ { 2 }$$ where \(a , b\) and \(c\) are integers to be found.
2. Sketch the graph of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), showing clearly the coordinates of the point where the curve crosses the \(y\)-axis.
3. Write down the equation of the line of symmetry of \(C\).

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the straight line \(l\) and the curve \(C\).
Given that \(l\) cuts the \(y\)-axis at - 12 and cuts the \(x\)-axis at 4 , as shown in Figure 2,

1. find an equation for \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. Given that \(C\)
   The region \(R\) is shown shaded in Figure 2.
2. Use inequalities to define \(R\).

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

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• \(C _ { 1 }\) has a maximum turning point at \(( 2,20 )\)
• \(C _ { 1 }\) passes through the origin
  1. sketch a graph of \(C _ { 1 }\) showing the coordinates of any points where \(C _ { 1 }\) cuts the coordinate axes,
  2. find an expression for \(\mathrm { f } ( x )\).

The curve \(C _ { 2 }\) has equation \(y = x \left( x ^ { 2 } - 4 \right)\) Curve \(C _ { 1 }\) and \(C _ { 2 }\) meet at the origin, and at the points \(P\) and \(Q\) Given that the \(x\) coordinate of the point \(P\) is negative,

using algebra and showing all stages of your working, find the coordinates of \(P\)

5. \begin{figure}[h] \end{figure} The share value of two companies, company \(A\) and company \(B\), has been monitored over a 15-year period. The share value \(P _ { A }\) of company \(\boldsymbol { A }\), in millions of pounds, is modelled by the equation $$P _ { A } = 53 - 0.4 ( t - 8 ) ^ { 2 } \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. The share value \(P _ { B }\) of company \(B\), in millions of pounds, is modelled by the equation $$P _ { B } = - 1.6 t + 44.2 \quad t \geqslant 0$$ where \(t\) is the number of years after monitoring began. Figure 2 shows a graph of both models. Use the equations of one or both models to answer parts (a) to (d).

1. Find the difference between the share value of company \(\boldsymbol { A }\) and the share value of company \(\boldsymbol { B }\) at the point monitoring began.
2. State the maximum share value of company \(\boldsymbol { A }\) during the 15-year period.
3. Find, using algebra and showing your working, the times during this 15-year period when the share value of company \(\boldsymbol { A }\) was greater than the share value of company \(\boldsymbol { B }\).
4. Explain why the model for company \(\boldsymbol { A }\) should not be used to predict its share value when \(t = 20\) \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-17_2644_1838_121_116}

8. The curve \(C _ { 1 }\) has equation $$y = 3 x ^ { 2 } + 6 x + 9$$

1. Write \(3 x ^ { 2 } + 6 x + 9\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a\), \(b\) and \(c\) are constants to be found. The point \(P\) is the minimum point of \(C _ { 1 }\)
2. Deduce the coordinates of \(P\). A different curve \(C _ { 2 }\) has equation $$y = A x ^ { 3 } + B x ^ { 2 } + C x + D$$ where \(A\), \(B\), \(C\) and \(D\) are constants. Given that \(C _ { 2 }\)
   \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-29_2646_1838_121_116}

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the quadratic curve \(C\) with equation $$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$ The line \(l _ { 1 }\) also shown in Figure 5,

• has gradient \(\frac { 1 } { 2 }\)
• intersects \(C\) on the negative \(x\)-axis and at the point \(P\)
• find, in terms of \(b\), an equation for \(l _ { 2 }\) Given also that \(l _ { 2 }\) intersects \(C\) at the point \(P\)
• show that another equation for \(l _ { 2 }\) is $$y = - 2 x + \frac { 5 b } { 2 } - 4$$
• Hence, or otherwise, find the value of \(b\)

8. Find the range of values of \(k\) for which the quadratic equation $$k x ^ { 2 } + 8 x + 2 ( k + 7 ) = 0$$ has no real roots.

13. The curve \(C\) has equation $$y = 3 x ^ { 2 } - 4 x + 2$$ The line \(l _ { 1 }\) is the normal to the curve \(C\) at the point \(P ( 1,1 )\)

1. Show that \(l _ { 1 }\) has equation $$x + 2 y - 3 = 0$$ The line \(l _ { 1 }\) meets curve \(C\) again at the point \(Q\).
2. By solving simultaneous equations, determine the coordinates of the point \(Q\). Another line \(l _ { 2 }\) has equation \(k x + 2 y - 3 = 0\), where \(k\) is a constant.
3. Show that the line \(l _ { 2 }\) meets the curve \(C\) once only when $$k ^ { 2 } - 16 k + 40 = 0$$
4. Find the two exact values of \(k\) for which \(l _ { 2 }\) is a tangent to \(C\).