Problem 71 - College Algebra Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=-\sqrt{x+2}$$

Step-by-Step Solution

Answer

The function \(h(x)=-\sqrt{x+2}\) is a reflection of the square root function \(f(x)=\sqrt{x}\) over the x-axis, and it is also shifted 2 units to the left. The transformation includes a reflection across the x-axis and a horizontal translation to the left by 2 units.

Step 1: Understanding the base function \(f(x)=\sqrt{x}\)

The square root function \(f(x)=\sqrt{x}\) is defined for all non-negative x, and its graph starts at the origin (0,0) and increases slowly, bending upwards towards \(+\infty\). The shape of its graph is half of a parabola turned 90 degrees clockwise.

Step 2: Identifying the transformations

The function \(h(x)\) can be obtained by transforming \(f(x)\) in two ways: 1) Reflection across the x-axis: The negative sign in front of the square root is indicative of a reflection or an upside down flip over the x-axis. 2) Horizontal translation to the left by 2 units: The term inside the square root function, \(x+2\), indicates a shift or translation of the graph to the left by 2 units. The '+' sign inside the function actually indicates a 'left shift' in the graph.

Step 3: Applying the transformation on \(f(x)=\sqrt{x}\)

Applying the transformations to the graph of \(f(x)=\sqrt{x}\) involves two parts: 1) Reflect the graph of \(f(x)\) across the x-axis. This means that the graph will now descend from the y-axis. 2) Shift the reflected graph to the left by two units. This will move your entire graph 2 units leftwards.

Step 4: Graphing \(h(x)=-\sqrt{x+2}\)

Now it's time to plot \(h(x)\) using the transformed graph. Begin at the point (-2,0). This is the 'new' starting point after transformation. The graph will decline downwards from (-2,0) because it has been reflected and should look like an inverted version of the basic square root function, displaced to the left by two units.