Question

use the graph of ( y = f(x) ) to graph the function ( g(x)=-f(x)+2 ). choose the correct graph of ( g ) below.

Explanation:

Step1: Reflection about the x - axis

When we have \(y = -f(x)\), it is a reflection of the graph \(y = f(x)\) about the \(x\) - axis. If the original function \(y = f(x)\) has a point \((x,y)\), then for \(y=-f(x)\), the point becomes \((x, - y)\). The original graph \(y = f(x)\) has a horizontal line segment at \(y = 3\). After reflection about the \(x\) - axis, the \(y\) - values of the points on the line segment change sign, so the line segment will be at \(y=-3\).

Step2: Vertical translation

When we have \(y=-f(x)+2\), it is a vertical translation of the graph \(y = - f(x)\) by \(2\) units up. Using the transformation rule \(y = g(x)+k\) (where \(k = 2\) and \(g(x)=-f(x)\)), if a point on \(y=-f(x)\) is \((x,z)\), then a point on \(y=-f(x)+2\) is \((x,z + 2)\). Since the line segment of \(y=-f(x)\) is at \(y=-3\), after a vertical translation of \(2\) units up (\(y=-3 + 2\)), the line segment will be at \(y=-1\).

Answer:

C.