1.02d Quadratic functions: graphs and discriminant conditions

5 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 17 x + 6\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\).
2. Given that \(\mathrm { f } ( 2 ) = 0\), express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
3. Determine the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.

1 Determine the set of values of \(k\) such that the equation \(x ^ { 2 } + 4 x + ( k + 3 ) = 0\) has two distinct real roots.

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 \(k\).

3 A publisher has to choose the price at which to sell a certain new book. The total profit, \(\pounds t\), that the publisher will make depends on the price, \(\pounds p\). He decides to use a model that includes the following assumptions.

• If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small.
• If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small.

The graphs below show two possible models. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Explain how model A is inconsistent with one of the assumptions given above.
2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k \left( 12 p - p ^ { 2 } \right)\), and find the value of the constant \(k\).
3. The publisher needs to make a total profit of at least \(\pounds 6400\). Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie.
4. Comment briefly on how realistic model B may be in the following cases.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = g ( x )\).
The curve has a single turning point, a minimum, at the point \(M ( 4 , - 1.5 )\).
The curve crosses the \(x\)-axis at two points, \(P ( 2,0 )\) and \(Q ( 7,0 )\).
The curve crosses the \(y\)-axis at a single point \(R ( 0,5 )\).

1. State the coordinates of the turning point on the curve with equation \(y = 2 \mathrm {~g} ( x )\).
2. State the largest root of the equation $$g ( x + 1 ) = 0$$
3. State the range of values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) \leqslant 0\) Given that the equation \(\mathrm { g } ( x ) + k = 0\), where \(k\) is a constant, has no real roots,
4. state the range of possible values for \(k\).

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.

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 \(R\).

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.

6. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 2 shows the entrance to a road tunnel. The maximum height of the tunnel is measured as 5 metres and the width of the base of the tunnel is measured as 6 metres. Figure 3 shows a quadratic curve \(B C A\) used to model this entrance.
The points \(A , O , B\) and \(C\) are assumed to lie in the same vertical plane and the ground \(A O B\) is assumed to be horizontal.

1. Find an equation for curve \(B C A\). A coach has height 4.1 m and width 2.4 m .
2. Determine whether or not it is possible for the coach to enter the tunnel.
3. Suggest a reason why this model may not be suitable to determine whether or not the coach can pass through the tunnel.

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.

6. Complete the table below. The first one has been done for you. For each statement you must state if it is always true, sometimes true or never true, giving a reason in each case.

1. The line \(l\) has equation

$$3 x - 2 y = k$$ where \(k\) is a real constant.
Given that the line \(l\) intersects the curve with equation $$y = 2 x ^ { 2 } - 5$$ at two distinct points, find the range of possible values for \(k\).

4

1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\).
2. The equation \(x ^ { 3 } - 6 x ^ { 2 } + 9 x + k = 0\) has exactly one real root. Using your answers from part (a) or otherwise, find the range of possible values of \(k\).

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 \(p > 0\).

1 The quadratic equation \(k x ^ { 2 } + 3 x + k = 0\) has no real roots. Determine the set of possible values of \(k\).

6 The polynomial \(x ^ { 3 } - 4 x ^ { 2 } + 10 x - 21\) is denoted by \(\mathrm { f } ( x )\).

1. Use the factor theorem to show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. The polynomial \(\mathrm { f } ( x )\) can be written as \(( \mathrm { x } - 3 ) \left( \mathrm { x } ^ { 2 } + \mathrm { bx } + \mathrm { c } \right)\) where \(b\) and \(c\) are constants. Find the values of \(b\) and \(c\).
3. Show that \(x = 3\) is the only real root of the equation \(\mathrm { f } ( x ) = 0\).

11 In this question you must show detailed reasoning.
Determine for what values of \(k\) the graphs \(y = 2 x ^ { 2 } - k x\) and \(y = x ^ { 2 } - k\) intersect.

2

1. Find the discriminant of the equation \(3 x ^ { 2 } - 2 x + 5 = 0\).
2. Use your answer to part (a) to find the number of real roots of the equation \(3 x ^ { 2 } - 2 x + 5 = 0\).

13 A projectile is fired from ground level at \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal.

1. State a modelling assumption that is used in the standard projectile model.
2. Find the cartesian equation of the trajectory of the projectile. The projectile travels above horizontal ground towards a wall that is 110 m away from the point of projection and 5 m high. The projectile reaches a maximum height of 22.5 m .
3. Determine whether the projectile hits the wall.

7

1. Simplify \(( k + 5 ) ^ { 2 } - 12 k ( k + 2 )\).
2. The quadratic equation \(3 ( k + 2 ) x ^ { 2 } + ( k + 5 ) x + k = 0\) has real roots.
   1. Show that \(( k - 1 ) ( 11 k + 25 ) \leqslant 0\).
   2. Hence find the possible values of \(k\).

4 The quadratic equation \(x ^ { 2 } + ( m + 4 ) x + ( 4 m + 1 ) = 0\), where \(m\) is a constant, has equal roots.

1. Show that \(m ^ { 2 } - 8 m + 12 = 0\).
2. Hence find the possible values of \(m\).

4

1. 1. Express \(x ^ { 2 } + 2 x + 5\) in the form \(( x + p ) ^ { 2 } + q\), where \(p\) and \(q\) are integers.
   2. Hence show that \(x ^ { 2 } + 2 x + 5\) is always positive.
2. A curve has equation \(y = x ^ { 2 } + 2 x + 5\).
   1. Write down the coordinates of the minimum point of the curve.
   2. Sketch the curve, showing the value of the intercept on the \(y\)-axis.
3. Describe the geometrical transformation that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = x ^ { 2 } + 2 x + 5\).