Question

begin by graphing the square root function, ( f(x)=sqrt{x} ). then, use transformations of this graph to graph the given function ( h(x)=-sqrt{x + 8} ). what transformations are needed to graph the function ( h(x)=-sqrt{x + 8} )? choose the correct answer below. a. the graph of ( sqrt{x} ) should be horizontally shifted to the right by 8 units and reflected about the ( y )-axis. b. the graph of ( sqrt{x} ) should be horizontally shifted to the right by 8 units and reflected about the ( x )-axis. c. the graph of ( sqrt{x} ) should be horizontally shifted to the left by 8 units and reflected about the ( y )-axis. d. the graph of ( sqrt{x} ) should be horizontally shifted to the left by 8 units and reflected about the ( x )-axis.

Explanation:

For the function \(y = f(x + c)\) (\(c>0\)), the graph of \(y = f(x)\) is shifted to the left by \(c\) units. For the function \(y=-f(x)\), the graph of \(y = f(x)\) is reflected about the \(x -\)axis.

For \(h(x)=-\sqrt{x + 8}\), compared with \(f(x)=\sqrt{x}\), we have \(y = f(x+8)\) (horizontal shift) and then \(y=-f(x + 8)\) (reflection).

When we change \(f(x)=\sqrt{x}\) to \(y=\sqrt{x + 8}\), using the rule \(y = f(x + c)\) (\(c = 8>0\)), the graph of \(y=\sqrt{x}\) is shifted to the left by 8 units.

When we change \(y=\sqrt{x+8}\) to \(h(x)=-\sqrt{x + 8}\), using the rule \(y=-f(x)\), the graph of \(y=\sqrt{x+8}\) is reflected about the \(x -\)axis.

Answer:

D. The graph of \(\sqrt{x}\) should be horizontally shifted to the left by 8 units and reflected about the \(x -\)axis.