Question

consider the following function.

m(x)=-\sqrt{x - 1}-2

step 1 of 2: graph the original function by indicating how the more basic function has been shifted, reflected, stretched, or compressed.

Explanation:

Step1: Identify the basic function

The basic function is \(y = \sqrt{x}\).

Step2: Analyze horizontal shift

For the function \(y=\sqrt{x - h}\), a shift of \(h\) units to the right. In \(m(x)=-\sqrt{x - 1}-2\), \(h = 1\), so shift \(y = \sqrt{x}\) 1 unit to the right to get \(y=\sqrt{x - 1}\).

Step3: Analyze reflection

The negative sign in front of the square - root, \(y=-\sqrt{x - 1}\), reflects the graph of \(y=\sqrt{x - 1}\) about the \(x\) - axis.

Step4: Analyze vertical shift

For the function \(y=f(x)+k\), a shift of \(k\) units. In \(m(x)=-\sqrt{x - 1}-2\), \(k=-2\), so shift the graph of \(y =-\sqrt{x - 1}\) 2 units down.

Answer:

Start with the graph of \(y = \sqrt{x}\). Shift it 1 unit to the right (because of \(x-1\) inside the square root), then reflect it about the \(x\) - axis (because of the negative sign in front of the square root), and finally shift it 2 units down (because of \(- 2\) outside the square root) to get the graph of \(m(x)=-\sqrt{x - 1}-2\).