1.02n Sketch curves: simple equations including polynomials

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.

12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).

1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
2. Find the other factors of \(\mathrm { f } ( x )\).
3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
   Solve this equation to find the \(x\)-coordinates of A and B .

8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$

1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
   1. \(\quad y = \mathrm { f } ( x + 3 )\),
   2. \(y = \mathrm { f } ( - x )\).

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

4.

1. Sketch on the same diagram the curves \(y = x ^ { 2 } - 4 x\) and \(y = - \frac { 1 } { x }\).
2. State, with a reason, the number of real solutions to the equation $$x ^ { 2 } - 4 x + \frac { 1 } { x } = 0 .$$

10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = ( x + 2 ) ^ { 3 }$$

1. Sketch the curve \(C\), showing the coordinates of any points of intersection with the coordinate axes.
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The straight line \(l\) is the tangent to \(C\) at the point \(P ( - 1,1 )\).
3. Find an equation for \(l\). The straight line \(m\) is parallel to \(l\) and is also a tangent to \(C\).
4. Show that \(m\) has the equation \(y = 3 x + 8\).

8.

1. Sketch the graphs of \(y = 2 x ^ { 4 }\) and \(y = 2 \sqrt { x } , x \geq 0\) on the same diagram and write down the coordinates of the point where they intersect.
2. Describe fully the transformation that maps the graph of \(y = 2 \sqrt { x }\) onto the graph of \(y = 2 \sqrt { x - 3 }\).
3. Find and simplify the equation of the graph obtained when the graph of \(y = 2 x ^ { 4 }\) is stretched by a factor of 2 in the \(x\)-direction, about the \(y\)-axis.

9. \(f ( x ) = 2 x ^ { 2 } + 3 x - 2\).

1. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
3. Find the coordinates of the points where the curve with equation \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\) crosses the coordinate axes. When the graph of \(y = \mathrm { f } ( x )\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.
4. Find the values of \(a , b\) and \(c\).

10. The curve with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).

1. Sketch the curve, showing the coordinates of \(A\) and \(B\).
2. Show that the tangent to the curve at \(A\) has the equation $$x + y = 2$$ Given that the curve is stationary at the points \(B\) and \(C\),
3. find the exact coordinates of \(C\).

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.

3 The curve with equation \(y = \frac { 1 } { 5 } x ( 10 - x )\) is used to model the arch of a bridge over a road, where \(x\) and \(y\) are distances in metres, with the origin as shown in Fig. 12.1. The \(x\)-axis represents the road surface. \begin{figure}[h] \end{figure}

1. State the value of \(x\) at A , where the arch meets the road.
2. Using symmetry, or otherwise, state the value of \(x\) at the maximum point B of the graph. Hence find the height of the arch.
3. Fig. 12.2 shows a lorry which is 4 m high and 3 m wide, with its cross-section modelled as a rectangle. Find the value of \(d\) when the lorry is in the centre of the road. Hence show that the lorry can pass through this arch. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fed65420-9ef9-41d6-a58f-3b0f801d6225-3_528_870_1558_717} \captionsetup{labelformat=empty} \caption{Fig. 12.2}
   \end{figure}
4. Another lorry, also modelled as having a rectangular cross-section, has height 4.5 m and just touches the arch when it is in the centre of the road. Find the width of this lorry, giving your answer in surd form.

1 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.

2 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).

1. Use the intersections with both axes to express the equation of the curve in a factorised form.
2. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
3. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection.

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

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. State the roots of \(\mathrm { f } ( x - 2 ) = 0\).
3. You are also given that \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + 15\).
   (A) Show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 2 x - 9\).
   (B) Show that \(\mathrm { g } ( 1 ) = 0\) and hence factorise \(\mathrm { g } ( x )\) completely.

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

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. State the values of \(x\) which satisfy \(\mathrm { f } ( x + 3 ) = 0\).

5 A cubic curve has equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis where \(x = - , \frac { 1 } { 2 }\) and 5 .

1. Write down three linear factors of \(\mathrm { f } ( x )\). Hence find the equation of the curve in the form \(y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 0 } { - 8 }\). State the coordinates of the point where the translated curve intersects the \(y\)-axis.
4. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression in factorised form for \(\mathrm { g } ( x )\) and state the coordinates of the point where the curve \(y = \mathrm { g } ( x )\) intersects the \(y\)-axis.

2 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).

1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
2. Sketch the curve \(y = \mathrm { f } ( x )\).
3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).

3 \begin{figure}[h] \end{figure} Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).

1. On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
   [0pt] [3]
2. Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
   [0pt] [6]
3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).
4. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
   (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
5. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
   (A) Show that \(\mathrm { g } ( 5 ) = 0\).
   (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
   (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
6. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).

1 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).

1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$

2 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).

1. Sketch the graph of \(y = \mathrm { f } ( x )\).
2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).

3

1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
   (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
   (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). State an equation for the resulting graph. You need not simplify your answer.
   Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis.
2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.

4 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).

1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
3. Sketch the graph of \(y = \mathrm { f } ( x )\).
4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).