To determine which statement is true about the equation
[tex]\[ y = -3x^2 + 4x - 11 \][/tex],
we need to analyze the nature of this equation in terms of relations and functions.
### Understanding Relations and Functions
- Relation: A relation is a set of ordered pairs [tex]\((x, y)\)[/tex]. In simpler terms, it shows how two quantities are related.
- Function: A function is a special type of relation where every input [tex]\( x \)[/tex] is related to exactly one output [tex]\( y \)[/tex].
### Analyzing the Given Equation
The given equation is a quadratic equation in the form
[tex]\[ y = ax^2 + bx + c, \][/tex]
where [tex]\( a = -3 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -11 \)[/tex].
1. Relation:
- This quadratic equation can be expressed as a set of ordered pairs [tex]\((x, y)\)[/tex]. For every value of [tex]\( x \)[/tex] that we substitute into the equation, we get a corresponding value of [tex]\( y \)[/tex]. Therefore, it clearly defines a relation.
2. Function:
- For an equation to be a function, every [tex]\( x \)[/tex] must map to exactly one [tex]\( y \)[/tex].
- A quadratic equation is a polynomial of degree 2, which forms a parabolic curve. Each value of [tex]\( x \)[/tex] corresponds to exactly one value of [tex]\( y \)[/tex]. There are no instances where a single [tex]\( x \)[/tex] value produces multiple [tex]\( y \)[/tex] values in a quadratic equation.
### Conclusion
Given that the equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex] fits both the criteria of a relation and the stricter criteria of a function:
The correct statement is:
A. It represents both a relation and a function.
