A linear function is given. Complete parts (a)-(d). \( g(x)=\frac{1}{3} x-3 \) (a) Determine the slope and \( y \)-intercept of the function. The slope is \( \frac{1}{3} \). (Type an integer or a simplified fraction.) The \( y \)-intercept is \( \square \). (Type an integer or a simplified fraction.)

The slope of the function \( g(x) = \frac{1}{3} x - 3 \) is \( \frac{1}{3} \), which indicates that for every 3 units you move horizontally to the right, the function rises by 1 unit. The \( y \)-intercept, where the graph crosses the \( y \)-axis, occurs when \( x = 0 \). By plugging in \( 0 \) for \( x \), we find that \( g(0) = -3 \), so the \( y \)-intercept is \( -3 \). For anyone grappling with linear functions, keep this in mind: the slope determines the steepness of the line while the \( y \)-intercept tells you where it hits the \( y \)-axis. A common mistake is overlooking the negative sign in the \( y \)-intercept—always check your calculations!