Question

use the graph of ( y = f(x) ) to graph the function ( g(x)=-f(x)-1 ). choose the correct graph of ( g ) below.

Explanation:

Step1: Reflect the graph about the x - axis

When we have \(y = -f(x)\), for each point \((x,y)\) on the graph of \(y = f(x)\), the corresponding point on the graph of \(y=-f(x)\) is \((x, - y)\). If the original function \(y = f(x)\) has a horizontal line segment at \(y = 3\) (assuming the \(y\) - value of the horizontal line in \(y = f(x)\) is \(y = 3\)), then after reflection about the \(x\) - axis, the \(y\) - value of the corresponding line segment in \(y=-f(x)\) is \(y=-3\).

Step2: Translate the graph vertically

When we have \(y=-f(x)-1\), we take the graph of \(y = - f(x)\) and translate it down by 1 unit. Using the result from Step 1 (\(y=-f(x)\) with \(y\) - value \(- 3\)), after the vertical translation \(y=-f(x)-1\), the \(y\) - value of the line segment is \(y=-3 - 1=-4\)

Since the original function \(y = f(x)\) has a horizontal line segment. After reflection about the \(x\) - axis (\(y=-f(x)\)) and then translation down by 1 unit (\(y=-f(x)-1\)), we look for a graph with a horizontal line segment at a lower \(y\) - value.

Answer:

B.