To sketch the curve for part (a), which is y = x2 − 6x. To sketch a curve, we identify its key features such as its shape, intercepts, and vertex.

The equation $y = x^2 - 6x$ is a quadratic function of the form $y = ax^2 + bx + c$.
This means the curve is a parabola. Since the coefficient of $x^2$ is $a=1$ (which is positive), the parabola opens upwards.

To find the x-intercepts, set $y=0$:

Factor out $x$:

This gives two x-intercepts:

The x-intercepts are $(0, 0)$ and $(6, 0)$.

To find the y-intercept, set $x=0$:

The y-intercept is $(0, 0)$.

The x-coordinate of the vertex of a parabola $y = ax^2 + bx + c$ is given by the formula $x = -\frac{b}{2a}$.
For $y = x^2 - 6x$, we have $a=1$ and $b=-6$.

Now, substitute $x=3$ back into the original equation to find the y-coordinate of the vertex:

The vertex of the parabola is $(3, -9)$.

The key features for sketching the curve $y = x^2 - 6x$ are:

• Type of curve: Parabola opening upwards.
• x-intercepts: $(0, 0)$ and $(6, 0)$.
• y-intercept: $(0, 0)$.
• Vertex: $(3, -9)$.