Let's analyze each of the given equations to determine which one represents a linear function. A linear function is typically written in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants and [tex]\( x \)[/tex] is the variable.
1. Equation: [tex]\( x = -2 \)[/tex]
- This equation represents a vertical line where [tex]\( x \)[/tex] is always [tex]\(-2\)[/tex], irrespective of [tex]\( y \)[/tex]. It is not in the form [tex]\( y = mx + b \)[/tex] since it does not define [tex]\( y \)[/tex]. Therefore, it does not represent a linear function in the context of [tex]\( y = f(x) \)[/tex].
2. Equation: [tex]\( y = 5x - 6 \)[/tex]
- This equation is in the form [tex]\( y = mx + b \)[/tex] with the slope [tex]\( m = 5 \)[/tex] and the y-intercept [tex]\( b = -6 \)[/tex]. It clearly represents a linear function.
3. Equation: [tex]\( y = -\frac{1}{5} x^2 \)[/tex]
- This equation includes an [tex]\( x^2 \)[/tex] term, meaning it is a quadratic function, not a linear function. Linear functions do not have exponents other than 1 on the variable [tex]\( x \)[/tex].
4. Equation: [tex]\( -3x + 2 = 4 \)[/tex]
- To determine if this represents a linear function, we need to rearrange it into the form [tex]\( y = mx + b \)[/tex].
- First, isolate [tex]\( x \)[/tex]:
[tex]\[
-3x + 2 = 4
\][/tex]
- Subtract 2 from both sides:
[tex]\[
-3x = 2
\][/tex]
- Divide by -3:
[tex]\[
x = -\frac{2}{3}
\][/tex]
- This is a vertical line equation and does not define [tex]\( y \)[/tex]. It is similar to the first equation and does not represent [tex]\( y = mx + b \)[/tex]. Therefore, it does not represent a linear function in the context of [tex]\( y = f(x) \)[/tex].
After evaluating all the equations, the only one that fits the form [tex]\( y = mx + b \)[/tex] and represents a linear function is the second equation:
[tex]\[ y = 5x - 6 \][/tex]
Hence, the equation [tex]\( y = 5x - 6 \)[/tex] represents a linear function. Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
