Sure, let's analyze the given equations to determine which one represents a linear function.
1. Equation: [tex]\( x = 8 \)[/tex]
- This equation states that [tex]\( x \)[/tex] is always 8, regardless of the value of [tex]\( y \)[/tex].
- It describes a vertical line in the Cartesian plane, which is parallel to the y-axis.
- Since it does not fit the form [tex]\( y = mx + b \)[/tex], it is technically not a linear function of [tex]\( y \)[/tex]; it's an equation in terms of [tex]\( x \)[/tex].
2. Equation: [tex]\( x - 3 = 5 \)[/tex]
- Simplifying this equation, we get [tex]\( x = 8 \)[/tex], identical to the first equation.
- This, again, represents a vertical line where [tex]\( x \)[/tex] is always 8.
- As before, while it describes a line, it does not represent a linear function of [tex]\( y \)[/tex].
3. Equation: [tex]\( y = -4x^2 \)[/tex]
- Here, the term [tex]\( x^2 \)[/tex] indicates that this equation is quadratic.
- The presence of the squared term means that [tex]\( y \)[/tex] changes with the square of [tex]\( x \)[/tex].
- Quadratic equations do not represent linear functions.
4. Equation: [tex]\( y = \frac{1}{4}x + 5 \)[/tex]
- This is in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope (in this case, [tex]\( \frac{1}{4} \)[/tex]) and [tex]\( b \)[/tex] is the y-intercept (here, 5).
- This is the standard form of a linear function.
From the analysis, we can see that out of all the given equations, the one that explicitly represents a linear function is:
[tex]\[ y = \frac{1}{4}x + 5 \][/tex]
Thus, the equation that represents a linear function is:
[tex]\[ y = \frac{1}{4} x + 5 \][/tex]
This corresponds to option 4.
