To determine which of the given equations represents a linear function, let's analyze the form of a linear function and then examine each equation.
A linear function has the general form:
[tex]\[ y = mx + b \][/tex]
where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable. The key characteristic of a linear function is that the highest power of [tex]\( x \)[/tex] is 1.
Now, let's examine each of the given options:
1. Option A: [tex]\( y = 4x^3 + 1 \)[/tex]
- This equation includes the term [tex]\( 4x^3 \)[/tex], where the highest power of [tex]\( x \)[/tex] is 3.
- Since the highest power of [tex]\( x \)[/tex] is 3, this is not a linear function.
2. Option B: [tex]\( y = 2x^2 - 5 \)[/tex]
- This equation includes the term [tex]\( 2x^2 \)[/tex], where the highest power of [tex]\( x \)[/tex] is 2.
- Since the highest power of [tex]\( x \)[/tex] is 2, this is not a linear function.
3. Option C: [tex]\( y = 2x + 4 \)[/tex]
- This equation is in the form [tex]\( y = mx + b \)[/tex] with [tex]\( m = 2 \)[/tex] and [tex]\( b = 4 \)[/tex].
- The highest power of [tex]\( x \)[/tex] is 1, which matches the form of a linear function.
- Therefore, this equation represents a linear function.
4. Option D: [tex]\( y = x^2 \)[/tex]
- This equation includes the term [tex]\( x^2 \)[/tex], where the highest power of [tex]\( x \)[/tex] is 2.
- Since the highest power of [tex]\( x \)[/tex] is 2, this is not a linear function.
Based on this analysis, the equation that represents a linear function is:
[tex]\[ y = 2x + 4 \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{3} \][/tex]
