To determine the turning point of the function $y = x^2 - 4x$, you can either complete the square or utilize differentiation.

Completing the Square

Start with the equation:

$y = x^2 - 4x.$

We want to rewrite it in the form:

$y = (x - a)^2 + b.$

To do this, take the coefficient of $x$, which is $-4$. Halve this value to get $-2$, and then square it to obtain $4$. We will then add and subtract this square within the equation:

$y = x^2 - 4x + 4 - 4.$

This can be simplified to:

$y = (x - 2)^2 - 4.$

Now, the equation is in the desired form $y = (x - 2)^2 - 4$. From this representation, we can identify that the turning point occurs at $(2, -4)$. The term $(x - 2)^2$ is always non-negative and reaches its minimum value of $0$ when $x = 2$. At this point, the corresponding $y$-value is $-4$.

Using Differentiation

Alternatively, you can find the turning point by differentiating the function. Differentiate $y = x^2 - 4x$ with respect to $x$:

$\frac{dy}{dx} = 2x - 4.$

To find the critical points, set the derivative equal to zero:

$2x - 4 = 0.$

Solving for $x$:

$2x = 4$ $x = 2.$

Next, substitute $x = 2$ back into the original equation to determine the corresponding $y$-value:

$y = (2)^2 - 4(2)$ $y = 4 - 8$ $y = -4.$

Thus, we conclude that the turning point is at $(2, -4)$. This point represents a minimum because the coefficient of $x^2$ in the original equation is positive, indicating that the parabola opens upwards.