Question

11 mark for review
$f(x) = (x - 44)(x - 46)$
the function $f$ is defined by the given equation. for what value of $x$ does $f(x)$ reach its minimum?
a 46
b 45
c 44
d -1

Explanation:

Step1: Identify the roots of the quadratic function

The function \( f(x) = (x - 44)(x - 46) \) is a quadratic function in factored form. The roots (where \( f(x) = 0 \)) are at \( x = 44 \) and \( x = 46 \) because when \( x = 44 \) or \( x = 46 \), one of the factors becomes zero, making the product zero.

Step2: Find the vertex (minimum point) of the parabola

For a quadratic function in the form \( f(x) = a(x - r_1)(x - r_2) \) (where \( a > 0 \), so the parabola opens upwards and has a minimum), the vertex (which is the minimum point) lies on the axis of symmetry. The axis of symmetry of a parabola with roots \( r_1 \) and \( r_2 \) is the midpoint of the roots. The formula for the midpoint of two numbers \( r_1 \) and \( r_2 \) is \( \frac{r_1 + r_2}{2} \).

Substituting \( r_1 = 44 \) and \( r_2 = 46 \) into the midpoint formula:

$$ \frac{44 + 46}{2} = \frac{90}{2} = 45 $$

So the function \( f(x) \) reaches its minimum at \( x = 45 \).

Answer:

B. 45