To determine which statement must be true about [tex]\( D \)[/tex], the discriminant of function [tex]\( h \)[/tex], given that function [tex]\( h \)[/tex] has an [tex]\( x \)[/tex]-intercept at [tex]\((4,0)\)[/tex], let's recall some key concepts about the discriminant and roots of quadratic functions.
1. Understanding the Discriminant:
- For a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], the discriminant, [tex]\( D \)[/tex], is given by the formula:
[tex]\[
D = b^2 - 4ac
\][/tex]
- The discriminant helps us determine the nature of the roots of the quadratic equation:
- If [tex]\( D > 0 \)[/tex], there are two distinct real roots.
- If [tex]\( D = 0 \)[/tex], there is exactly one real root (a repeated root).
- If [tex]\( D < 0 \)[/tex], there are no real roots (the roots are complex or imaginary).
2. Examining the Given Information:
- The function [tex]\( h \)[/tex] has an [tex]\( x \)[/tex]-intercept at [tex]\( (4,0) \)[/tex]. This means that the function crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 4 \)[/tex].
- An [tex]\( x \)[/tex]-intercept indicates that there is at least one real root of the function [tex]\( h(x) = 0 \)[/tex].
3. Implications for the Discriminant [tex]\( D \)[/tex]:
- Since the function [tex]\( h \)[/tex] has at least one real root, the discriminant [tex]\( D \)[/tex] cannot be negative because negative discriminants correspond to no real roots.
- Therefore, the discriminant [tex]\( D \)[/tex] must be non-negative.
From this reasoning, the appropriate statement about [tex]\( D \)[/tex] is:
[tex]\[
\boxed{B. \, D \geq 0}
\][/tex]
