On the same axes sketch the graphs of the curves with equations (i) $y = x^2(x - 2)$, (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 2 - 2006 - Paper 1

To sketch the graph of the equation $y = x(6 - x)$:

1. Identify the roots: Setting $y = 0$ gives $x(6 - x) = 0$, resulting in $x = 0$ and $x = 6$.
2. The graph crosses the x-axis at both $x = 0$ and $x = 6$.
3. The leading coefficient is positive, and the degree is 2, indicating it opens upwards and has a single maximum at $x = 3$ where $y = 3(6 - 3) = 9$.
4. Thus, the vertex is at $(3, 9)$.

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

$x^2(x - 2) = x(6 - x)$

1. Expand both sides:

   • Left side: $x^3 - 2x^2$
   • Right side: $6x - x^2$

2. Rearranging gives: $x^3 - 2x^2 - 6x + x^2 = 0$ $x^3 - x^2 - 6x = 0$

   • Factor out $x$: $x(x^2 - x - 6) = 0$

3. Further factorization:

   • The quadratic factors as $(x - 3)(x + 2)$.

4. Solve for $x$: $x = 0$, $x = 3$, $x = -2$. Only $x = 0$ and $x = 6$ are valid intersections.

5. Substitute back to find $y$: At $x = 0$, $y = 0$ and at $x = 6$, $y = 0$.

   • Hence, the points of intersection are $(0, 0)$ and $(6, 0)$.