To determine which statement correctly describes the equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex], let's analyze it step-by-step:
### Step 1: Understand the nature of the equation
The given equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex] is a quadratic equation because the highest power of [tex]\( x \)[/tex] is 2. Quadratic equations are polynomial equations of degree 2.
### Step 2: Define relation and function
- A relation is a relationship between sets of values. In other words, it is a set of ordered pairs [tex]\((x, y)\)[/tex].
- A function is a specific type of relation where each input (or [tex]\( x \)[/tex]-value) has exactly one output (or [tex]\( y \)[/tex]-value).
### Step 3: Determine if the equation represents a function
To verify if [tex]\( y = -3x^2 + 4x - 11 \)[/tex] is a function, we need to ensure that for every [tex]\( x \)[/tex]-value, there is only one [tex]\( y \)[/tex]-value.
- This property is known as the vertical line test. If a vertical line intersects the graph of the equation at exactly one point for any [tex]\( x \)[/tex]-value, then the equation represents a function.
In this case, [tex]\( y \)[/tex] is expressed explicitly in terms of [tex]\( x \)[/tex] ([tex]\( y = -3x^2 + 4x - 11 \)[/tex]). For any value of [tex]\( x \)[/tex], because all quadratic equations of this form open either upwards or downwards and are continuous, they ensure that each [tex]\( x \)[/tex]-value maps to exactly one [tex]\( y \)[/tex]-value. Therefore, this quadratic equation passes the vertical line test and is a function.
### Step 4: Determine if the equation is a relation
Since all functions are inherently relations (as they describe a set of ordered pairs [tex]\((x, y)\)[/tex]), [tex]\( y = -3x^2 + 4x - 11 \)[/tex] is also a relation.
### Conclusion
Based on the steps above, we conclude that the given equation [tex]\( y = -3x^2 + 4x - 11 \)[/tex] represents both a relation and a function.
Therefore, the correct answer is:
A. It represents both a relation and a function.
