Problem 77 - College Algebra

Question

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

Step-by-Step Solution

Answer

The graph of the function \(h(x) = \sqrt{x+2} - 2\) is a graph of the function \(f(x) = \sqrt{x}\) shifted 2 units to the left and 2 units down. It begins at the point \((-2,-2)\) and increases slowly, curving upwards.

Step 1: Graph the square root function

The square root function, \(f(x) = \sqrt{x}\), is graphed. It starts at the origin \((0,0)\) and increases slowly, curving upwards.

Step 2: Identify the transformations

Excercise provides the function \(h(x) = \sqrt{x+2} - 2\). This is the function \(f(x) = \sqrt{x}\) that has been horizontally shifted 2 units to the left (hence the \(+2\) inside the square root) and vertically shifted 2 units down (hence the \(-2\) outside the square root).

Step 3: Apply the horizontal shift

Shift the graph of \(f(x)\) to the left by 2 units. This means that the start of the graph is now at \((-2,0)\) rather than at \((0,0)\).

Step 4: Apply the vertical shift

Shift the graph from the previous step down by 2 units. This means that the point that was at \((-2,0)\) is now at \((-2,-2)\).

Step 5: Combine the shifts

With both shifts applied, you should now have the graph of the function \(h(x) = \sqrt{x+2} - 2\). It looks like the graph of \(f(x) = \sqrt{x}\), but moved 2 units to the left and 2 units down.

Key Concepts

Understanding the Square Root Function

The square root function, expressed as \(f(x) = \sqrt{x}\), is a fundamental function in algebra. It is characterized by its distinct curve that begins at the origin, \((0,0)\), and increases slowly as \(x\) gets larger. This function only accepts non-negative inputs, since taking the square root of a negative number results in an imaginary number. This leads to the domain of the square root function being \(x \geq 0\).

The shape of the square root function graph is not a straight line; rather, it's a gentle curve that starts at \(0\) and rises as \(x\) increases. This curve reflects the property that the square root function grows slower compared to linear functions. The output, which represents \(y\), increases at a decreasing rate.

Here's a quick recap of key points:

• The starting point or vertex of the square root function is at \((0, 0)\).
• It only exists in the first quadrant of the Cartesian plane.
• As \(x\) increases, the graph rises slowly to the right.

Graphing Techniques for Function Transformations

Graphing techniques are essential when working with transformations, as they allow us to visualize how a function changes. Begin by plotting the basic form of the function. For instance, with \(f(x) = \sqrt{x}\), you'll draw the familiar square root curve.

One of the simplest techniques is to perform translations or shifts, which means changing the position of a graph without altering its shape. Transformations typically involve shifting, stretching, compressing, or reflecting graphs.

To effectively manage this:

• Identify any constants added or subtracted inside or outside the square root. These indicate horizontal or vertical shifts.
• Practice tracing the graph of the parent function, like \(f(x) = \sqrt{x}\), then apply the shifts accordingly.
• Visualize each step by sketching the new position of the graph after applying transformations.

Horizontal and Vertical Shifts in Graphing

Horizontal and vertical shifts are common transformations used to adjust the position of a graph. They do not alter the shape of the graph, just where it sits on the coordinate plane.

A **horizontal shift** is a transformation that moves the graph left or right. If you see \(x + a\) inside a function, this indicates a shift to the left by \(a\) units, due to the counterintuitive effect of horizontal transformations. Similarly, \(x - a\) represents a shift to the right.

A **vertical shift** occurs when a constant is added or subtracted outside of the function. For instance, subtracting \(2\) from \(\sqrt{x}\) results in \(\sqrt{x} - 2\), which shifts the graph downward by 2 units.

• To shift left by 2 units in \(\sqrt{x+2}\), think of the starting point moving from \((0,0)\) to \((-2,0)\).
• To shift down by 2 units in \(\sqrt{x} - 2\), the graph's entire range is lowered by 2.

Applying both transforms will position the vertex appropriately at \((-2, -2)\), reflecting the complete transformation of \(h(x) = \sqrt{x+2} - 2\). Each shift is applied separately, but they can combine to give the final new graph location.