Question

graph the given functions, f and g, in the same rectangular coordinate system. describe how the graph of g is related to the graph of f.\\(f(x) = -x^2\\)\\(g(x) = -x^2 + 12\\)\\(\dots\\)\\(\\)use the graphing tool to graph the functions.\\(\\)how is the graph of g related to the graph of f?\\(\\)\\(\bigcirc\\) a. the graph of g is the graph of f shifted 12 units vertically down.\\(\\)\\(\bigcirc\\) b. the graph of g is the graph of f shifted 12 units horizontally right\\(\\)\\(\bigcirc\\) c. the graph of g is the graph of f shifted 12 units horizontally left.\\(\\)\\(\bigcirc\\) d. the graph of g is the graph of f shifted 12 units vertically up

Explanation:

To determine the relationship between the graphs of \( f(x) = -x^2 \) and \( g(x) = -x^2 + 12 \), we use the concept of vertical shifts in function graphs. For a function \( y = f(x) + k \), if \( k > 0 \), the graph of \( f(x) \) is shifted \( k \) units vertically up. Here, \( g(x) = f(x) + 12 \) (since \( f(x) = -x^2 \) and \( g(x) = -x^2 + 12 \)), so \( k = 12 > 0 \). This means the graph of \( g \) is the graph of \( f \) shifted 12 units vertically up.

Answer:

D. The graph of g is the graph of f shifted 12 units vertically up