Sure, let's break down the problem step-by-step to determine which statement is true about the given equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex].
1. Identify the Nature of the Equation:
- The given equation is [tex]\( y = -3x^2 + 4x - 11 \)[/tex].
- This is a quadratic equation, which is a polynomial equation of degree 2.
2. Understand the Definition of a Function:
- A function is a specific type of relation where each input (or x-value) has exactly one output (or y-value).
3. Analyze if the Given Equation is a Function:
- Consider the form of our equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex].
- For each value of [tex]\( x \)[/tex], this equation will produce exactly one value for [tex]\( y \)[/tex]. There are no [tex]\( x \)[/tex] values that will produce more than one [tex]\( y \)[/tex] value.
- Therefore, this equation defines a function.
4. Understand the Definition of a Relation:
- A relation is simply any set of ordered pairs. Functions are special kinds of relations, but not all relations are functions.
5. Analyze if the Given Equation is a Relation:
- Since we already established that the equation is a function, and functions are a subset of relations, it must also be a relation.
6. Conclusion:
- Given that the equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex] represents exactly one [tex]\( y \)[/tex] for each [tex]\( x \)[/tex], it is both a function and a relation.
Thus, the correct statement is:
A. It represents both a relation and a function.
