On separate axes sketch the graphs of (i) $y = -3x + c$, where $c$ is a positive constant, (ii) $y = \frac{1}{x} + 5$ On each sketch show the coordinates of any point at which the graph crosses the $y$-axis and the equation of any horizontal asymptote - Edexcel - A-Level Maths Pure - Question 2 - 2017 - Paper 1

1. The graph has two branches with a horizontal asymptote at $y = 5$. This is because as $x$ approaches infinity, $\frac{1}{x}$ approaches $0$, making $y$ approach $5$.
2. The graph crosses the $y$-axis at $(0, 5)$ since plugging $x = 0$ into $y = \frac{1}{x} + 5$ is undefined, indicating an asymptote at $x = 0$.
3. The branches will approach the horizontal asymptote near $y = 5$.

Starting from the equation:

$\frac{1}{x} + 5 = -3x + c$

1. Rearranging gives: $5 - c = -3x - \frac{1}{x}$
2. Squaring both sides: $(5 - c)^2 = ( -3x - \frac{1}{x})^2$
3. Solving gives: $b^2 - 4ac \Rightarrow b^2 - 4ac = (5 - c)^2 - 4(1)(12) > 0$

1. From the inequality $(5 - c^2) > 12$, we can rearrange it to yield:
   $5 - c^2 > 12$ $-c^2 > 7$ $c^2 < -7$ $c < -\sqrt{7}$ or $c > \sqrt{7}$.
2. Therefore, the possible values for $c$ are: $c < -\sqrt{7}$ or $c > \sqrt{7}$, excluding values where $c$ is negative, hence the valid range is $c > \sqrt{7}$.