sketch the graph of ( g(x) ) with the following features:\n1) the function ( f(x) ) has a maximum at (…

Explanation:

Step1: Analyze the transformation of the maximum point

The maximum point of (f(x)) is ((0,-3)). For a translation (2) units right and (3) units up, we use the transformation rule ((x,y)\to(x + 2,y+3)). So the new maximum point for (g(x)) is ((0 + 2,-3+3)=(2,0)).

Step2: Analyze the increasing - decreasing nature

Since (f(x)) is increasing for (x\lt0) and decreasing for (x\gt0), for (g(x)=f(x - 2)+3) (using the horizontal translation (x\to x - 2) for a shift of (2) units to the right), (g(x)) is increasing for (x-2\lt0) (i.e., (x\lt2)) and decreasing for (x - 2\gt0) (i.e., (x\gt2))

Step3: Analyze the sign of the function

Since (f(x)\lt0) for all (x), and (g(x)=f(x - 2)+3). The maximum value of (g(x)) is (0) (from the transformed maximum point ((2,0))). For (x\neq2), (g(x)=f(x - 2)+3). Since (f(x-2)\lt0), then (g(x)=f(x - 2)+3\lt3) and (g(x)) is non - positive for all (x)

To sketch (g(x)):

• Plot the maximum point at ((2,0))
• Draw a curve that is increasing for (x\lt2) and decreasing for (x\gt2)
• Ensure that the entire graph is on or below the (x) - axis.

Answer:

Sketch a curve with maximum at ((2,0)), increasing for (x\lt2), decreasing for (x\gt2) and non - positive for all (x)