Question

sketch the graph of ( g(x) ) with the following features:

1. the function ( f(x) ) has a maximum at ( (0,-3) )
2. the function ( f(x) ) is increasing for ( x < 0 )
3. the function ( f(x) ) is decreasing for ( x > 0 )
4. the function ( f(x) ) is negative for all values of ( x )
5. the graph of ( g(x) ) is the graph of ( f(x) ) after a translation 2 units right and 3 units up.

you only need to sketch ( g(x) ), not ( f(x) )

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 \(x
eq2\), \(g(x)=f(x - 2)+3\). 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 \(x\)