1.02n Sketch curves: simple equations including polynomials

\includegraphics{figure_5} \includegraphics{figure_6} A suspension bridge cable \(PQR\) hangs between the tops of two vertical towers, \(AP\) and \(BR\), as shown in Figure 5. A walkway \(AOB\) runs between the bases of the towers, directly under the cable. The towers are 100 m apart and each tower is 24 m high. At the point \(O\), midway between the towers, the cable is 4 m above the walkway. The points \(P\), \(Q\), \(R\), \(A\), \(O\) and \(B\) are assumed to lie in the same vertical plane and \(AOB\) is assumed to be horizontal. Figure 6 shows a symmetric quadratic curve \(PQR\) used to model this cable. Given that \(O\) is the origin,

1. find an equation for the curve \(PQR\). [3] Lee can safely inspect the cable up to a height of 12 m above the walkway. A defect is reported on the cable at a location 19 m horizontally from one of the towers.
2. Determine whether, according to the model, Lee can safely inspect this defect. [2]

A cubic curve \(C\) has equation $$y = (3-x)(4+x)^2.$$

1. Sketch the graph of \(C\). [3] The sketch must include any points where the graph meets the coordinate axes.
2. Sketch in separate diagrams the graph of \(\ldots\)
   1. \(\ldots y = (3-2x)(4+2x)^2\). [2]
   2. \(\ldots y = (3+x)(4-x)^2\). [2]
   3. \(\ldots y = (2-x)(5+x)^2\). [2]
   Each of the sketches must include any points where the graph meets the coordinate axes.

\includegraphics{figure_1} Figure 1 shows a sketch of a curve C with equation \(y = \text{f}(x)\), where f(x) is a quartic expression in \(x\). The curve • has maximum turning points at \((-1, 0)\) and \((5, 0)\) • crosses the \(y\)-axis at \((0, -75)\) • has a minimum turning point at \(x = 2\)

1. Find the set of values of \(x\) for which $$\text{f}'(x) \geq 0$$ writing your answer in set notation. [2]
2. Find the equation of C. You may leave your answer in factorised form. [3]

The curve \(C_1\) has equation \(y = \text{f}(x) + k\), where \(k\) is a constant. Given that the graph of \(C_1\) intersects the \(x\)-axis at exactly four places,

1. find the range of possible values for \(k\). [2]

Sketch the graph of $$y = (x - k)^2(x + 2k)$$ where \(k\) is a positive constant. Label the coordinates of the points where the graph meets the axes. \includegraphics{figure_6} [3]

\(f(x) = 2x^2 + 4x + 9 \quad x \in \mathbb{R}\)

1. Write \(f(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are integers to be found. [3]
2. Sketch the curve with equation \(y = f(x)\) showing any points of intersection with the coordinate axes and the coordinates of any turning point. [3]
3. 1. Describe fully the transformation that maps the curve with equation \(y = f(x)\) onto the curve with equation \(y = g(x)\) where $$g(x) = 2(x - 2)^2 + 4x - 3 \quad x \in \mathbb{R}$$
   2. Find the range of the function $$h(x) = \frac{21}{2x^2 + 4x + 9} \quad x \in \mathbb{R}$$ [4]

\(\text{f}(x)\) is a cubic polynomial in which the coefficient of \(x^3\) is 1. The equation \(\text{f}(x) = 0\) has exactly two roots.

1. Sketch a possible graph of \(y = \text{f}(x)\). [2]

It is now given that the two roots are \(x = 2\) and \(x = 3\).

1. Find, in expanded form, the two possible polynomials \(\text{f}(x)\). [3]

A publisher has to choose the price at which to sell a certain new book. The total profit, \(£t\), that the publisher will make depends on the price, \(£p\). 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. \includegraphics{figure_3}

1. Explain how model A is inconsistent with one of the assumptions given above. [1]
2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k(12p - p^2)\), and find the value of the constant \(k\). [2]
3. The publisher needs to make a total profit of at least £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]
4. Comment briefly on how realistic model B may be in the following cases. • \(p = 0\) • \(p = 12.1\) [2]

Sketch the curve with equation \(y = \frac{x^2 + 4x}{2x - 1}\), justifying all significant features. [11]

A curve has equation \(y = \frac{2x^2 + 5x - 25}{x - 3}\).

1. Determine the equations of the asymptotes. [3]
2. Find the coordinates of the turning points. [5]
3. Sketch the curve. [3]

The curve \(C\) has equation $$y = \frac{x^2 - 2x - 3}{x + 2}.$$

1. Find the equations of the asymptotes of \(C\). [4]
2. Draw a sketch of \(C\), which should include the asymptotes, and state the coordinates of the points of intersection of \(C\) with the \(x\)-axis. [5]