To determine which function, $f(x) = x(x - 6)$ or $g(x) = (x + 3)^2 - 8$, has the lower minimum, let us analyze each function.

1. Analyze $f(x) = x(x - 6)$:

Simplify: $f(x) = x^2 - 6x$

This is a quadratic function in standard form: $f(x) = ax^2 + bx + c$ Here, $a = 1$, $b = -6$, and $c = 0$. Since $a > 0$, the parabola opens upwards, and the vertex represents the minimum value.

The $x$-coordinate of the vertex is given by: $x = -\frac{b}{2a} = -\frac{-6}{2(1)} = 3$

Substitute $x = 3$ back into $f(x)$ to find the minimum value: $f(3) = 3(3 - 6) = 3(-3) = -9$

Thus, the minimum value of $f(x)$ is: $\text{Minimum of } f(x) = -9$

2. Analyze $g(x) = (x + 3)^2 - 8$:

This is already in vertex form: $g(x) = (x + 3)^2 - 8$

The vertex occurs at $x = -3$, and the value of $g(x)$ at the vertex is: $g(-3) = ((-3) + 3)^2 - 8 = 0 - 8 = -8$

Thus, the minimum value of $g(x)$ is: $\text{Minimum of } g(x) = -8$

Comparison of Minimum Values:

• Minimum of $f(x)$: $-9$
• Minimum of $g(x)$: $-8$

Since $-9 < -8$, $f(x)$ has the lower minimum.

Let me know if you want a deeper explanation or any additional details!

Here are 5 related questions you might consider:

1. How do we determine the vertex of a quadratic function?
2. Can the minimum of a quadratic function change if the equation is shifted horizontally?
3. What is the significance of the parabola's direction (upward vs downward)?
4. How can we graphically compare the two functions $f(x)$ and $g(x)$?
5. What are the general steps to convert a quadratic function to vertex form?

Tip: Always identify the direction of a parabola (upward or downward) before calculating its minimum or maximum!