The problem asks us to identify the coefficients $a$, $b$, and $c$ of several quadratic functions in the form $f(x) = ax^2 + bx + c$, and then to calculate the $x$-coordinate of the vertex (denoted $x_s$) of each parabola.

1. Problem Description

The problem asks us to identify the coefficients $a$, $b$, and $c$ of several quadratic functions in the form $f(x) = ax^2 + bx + c$, and then to calculate the $x$-coordinate of the vertex (denoted $x_s$) of each parabola.

2. Solution Steps

For a quadratic function $f(x) = ax^2 + bx + c$, the $x$-coordinate of the vertex is given by the formula:

$x_s = -\frac{b}{2a}$

1) $f(x) = x^2 - 5x + 6$

Here, $a = 1$, $b = -5$, and $c = 6$.

$x_s = -\frac{-5}{2(1)} = \frac{5}{2} = 2.5$

2) $g(x) = -2x^2 + 12x - 18$

Here, $a = -2$, $b = 12$, and $c = -18$.

$x_s = -\frac{12}{2(-2)} = -\frac{12}{-4} = 3$

3) $h(x) = 45 + 5x^2 - 30x = 5x^2 - 30x + 45$

Here, $a = 5$, $b = -30$, and $c = 45$.

$x_s = -\frac{-30}{2(5)} = -\frac{-30}{10} = 3$

4) $l(x) = -3x + 4 + 0.5x^2 = 0.5x^2 - 3x + 4$

Here, $a = 0.5$, $b = -3$, and $c = 4$.

$x_s = -\frac{-3}{2(0.5)} = -\frac{-3}{1} = 3$

3. Final Answer

1) $f(x) = x^2 - 5x + 6$

$a = 1$; $b = -5$; $c = 6$; $x_s = 2.5$

2) $g(x) = -2x^2 + 12x - 18$

$a = -2$; $b = 12$; $c = -18$; $x_s = 3$

3) $h(x) = 45 + 5x^2 - 30x$

$a = 5$; $b = -30$; $c = 45$; $x_s = 3$

4) $l(x) = -3x + 4 + 0.5x^2$

$a = 0.5$; $b = -3$; $c = 4$; $x_s = 3$