A function \( f(x) \) is a linear function and the function \( g(x)=f(x)+k \) .If the value of k is 4 , how does the graph of \( g(x) \) compare with the graph \( f(x) \) ?

When dealing with linear functions, any modification involving the addition of a constant affects the graph in a specific way. Given: - **\( f(x) \)** is a linear function. Its graph is a straight line. - **\( g(x) = f(x) + k \)**, where \( k = 4 \). **Impact of Adding \( k \):** - Adding a constant \( k \) to \( f(x) \) results in a vertical shift of the graph. - Specifically, \( g(x) = f(x) + 4 \) shifts the entire graph of \( f(x) \) **upward** by 4 units. **Characteristics of the Transformation:** - **Slope**: The slope of \( f(x) \) and \( g(x) \) remains the same since the addition of a constant does not affect it. - **Y-Intercept**: The y-intercept of \( g(x) \) is 4 units higher than that of \( f(x) \). **Conclusion:** The graph of \( g(x) \) will be a straight line that is **parallel** to the graph of \( f(x) \), positioned **4 units above** it. **Visual Representation:** ``` f(x): g(x): | | | | | | --+------------+-- | | | | ``` *(Imagine the right line being exactly 4 units above the left one)* --- **Summary:** Adding 4 to the linear function \( f(x) \) shifts its graph upward by 4 units, resulting in a parallel line for \( g(x) \). **Answer:** Its graph is the same straight line shifted upward by four units, running parallel to f (x ).