To identify which of the given equations represents a linear function, let's evaluate each one:
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\( x = 8 \): This represents a vertical line where \( x \) is always 8. It is not a function of \( y \), thus it does not represent a linear function.
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\( x - 3 = 5 \): This simplifies to \( x = 8 \), which is again a vertical line. It does not represent a linear function.
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\( y = -4x^2 \): This equation represents a quadratic function because of the \( x^2 \) term. Therefore, it is not a linear function.
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\( y = \frac{1}{4}x + 5 \): This is in the slope-intercept form \( y = mx + b \), where \( m = \frac{1}{4} \) (the slope) and \( b = 5 \) (the y-intercept). This is a linear function.
The equation that represents a linear function is:
\( y = \frac{1}{4}x + 5 \).