To determine which equation represents a linear function, we need to understand the characteristics of linear functions. A linear function has the form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable raised to the power of 1 only. This means there should be no higher powers of [tex]\( x \)[/tex] such as [tex]\( x^2 \)[/tex], [tex]\( x^3 \)[/tex], etc.
Let's analyze each given equation:
1. [tex]\( y = 2x^2 \)[/tex]
In this equation, the variable [tex]\( x \)[/tex] is raised to the power of 2. Therefore, this is not a linear equation, but a quadratic equation.
2. [tex]\( y = -4x + 5 \)[/tex]
Here, the variable [tex]\( x \)[/tex] is raised to the power of 1. This fits the form of a linear equation [tex]\( y = mx + b \)[/tex] with [tex]\( m = -4 \)[/tex] and [tex]\( b = 5 \)[/tex]. Hence, this is a linear equation.
3. [tex]\( y = x^3 \)[/tex]
In this equation, the variable [tex]\( x \)[/tex] is raised to the power of 3. Therefore, this is not a linear equation, but a cubic equation.
4. [tex]\( y = -3x^2 + 2 \)[/tex]
Here, the variable [tex]\( x \)[/tex] is raised to the power of 2. Therefore, this is not a linear equation, but a quadratic equation.
Based on these analyses, the equation that represents a linear function is:
[tex]\[ y = -4x + 5 \][/tex]
Thus, the correct choice number is:
[tex]\[ 2 \][/tex]
