On the same axes sketch the graphs of the curves with equations (i) $y = x^{3}(x-2),$ (ii) $y = x(6-x),$ and indicate on your sketches the coordinates of all the points where the curves cross the $x$-axis - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 2

To sketch the graph of this equation, we follow these steps:

1. Set the equation equal to zero to find $x$-intercepts:

   \Rightarrow x = 0 \quad \text{or} \quad 6 - x = 0\\ \Rightarrow x = 0 \quad \text{or} \quad x = 6$$

2. So, it crosses the $x$-axis at $(0, 0)$ and $(6, 0)$.

3. This is a downward-opening parabola with the vertex at $(3, 9)$, which is above the $x$-axis, showcasing a maximum point.

To find where the two graphs intersect, we set the equations equal to each other:

$x^{3}(x-2) = x(6-x)$

1. Rearranging gives us:

$x^{3}(x-2) - x(6-x) = 0$ 2. Simplifying this, we factor out $x$:

$x(x^{2}(x-2) - (6-x)) = 0$ 3. Expanding the second part:

$x^{2}(x-2) - 6 + x = 0 \rightarrow x^{3} - 2x^{2} + x - 6 = 0$ 4. To solve for $x$ values, we can use numerical methods or trial and error: Possible roots include $x = 2$ and one other (found through further calculations, possibly using the Rational Root Theorem). 5. Evaluating the function yields:

For $x = 2$:

$y = 2(6-2) = 8$,
Intersection point is $(2, 8)$. Similarly, one needs to verify values to find additional intersection points.