Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Consider the graph of \(g(x)=\sqrt{x}\) Use your knowledge of rigid and nonrigid transformations to write an equation for each of the following descriptions. Verify with a graphing utility. The graph of \(g\) is reflected in the \(x\) -axis, shifted two units to the left, and shifted one unit upward.

Step-by-Step Solution

Answer

The transformed function is \(g(x) = -\sqrt{x+2} + 1\).

Step 1: Reflecting over X-axis

The operation for reflecting a function over the x-axis is to make the entire function negative. This means that \(g(x)\) becomes \(-g(x)\), so the graph of \(g(x) = \sqrt{x}\) reflected in the x-axis becomes \(g(x) = -\sqrt{x}\).

Step 2: Shifting Horizontally

Shifting a function horizontally involves changing the value inside the function that is in association with \(x\). To shift a function to the left, the value is added to \(x\) within the function. Shifting two units to the left hence gives us \(g(x) = -\sqrt{x+2}\).

Step 3: Shifting Vertically

To shift a function one unit upwards, 1 is added to the entire function. Thus, making the function as \(g(x) = -\sqrt{x+2} + 1\).

Step 4: Verification

Verify the equation by graphing the function. As per the exercise's guidelines, we can use a graphing calculator or computer software for this verification. The transformed function should behave exactly as described: flipped over the x-axis, shifted 2 units to the left, and 1 unit up.

Key Concepts

Reflection on X-axis

Reflecting a function on the x-axis involves flipping its graph along this axis. To achieve this reflection mathematically, each output value of the function is negated.

• If the original function is denoted as \(f(x)\), its reflection over the x-axis will be \(-f(x)\).

In our provided example, where the function is \(g(x) = \sqrt{x}\), the reflected function becomes \(-\sqrt{x}\). This technique effectively inverts the graph vertically, ensuring that points above the x-axis move an equivalent distance below it, mirroring the shape and spread of the original function across the x-axis.

Horizontal Shift

Horizontal shifts involve moving the graph of a function left or right. This is achieved by altering the function’s input, \(x\), within the function itself.

• To shift a function to the left, add a positive constant to \(x\).
• To shift it to the right, subtract a constant from \(x\).

For the function \(g(x) = -\sqrt{x}\), shifting it 2 units to the left transforms the function into \(-\sqrt{x+2}\). This means that every point on the original graph moves two units to the left along the x-axis, modifying the starting point and overall phase of the graph. Understanding this allows us to predict and manipulate the horizontal positioning of a function precisely.

Vertical Shift

A vertical shift moves the entire graph of a function up or down without altering its shape. This transformation is done by adding or subtracting a constant to the entire function.

• Add a constant to shift the graph upward.
• Subtract a constant to move it downward.

In the case of the function \(g(x) = -\sqrt{x+2}\), adding 1 to the entire expression shifts the graph one unit upward. Thus, the new function becomes \(-\sqrt{x+2} + 1\). This means that every point on the graph moves upward by one unit, elevating the curve's position relative to the x-axis. Vertical shifts are straightforward yet crucial for adjusting a function's vertical layout.

Graphing Utility

Graphing utilities, such as graphing calculators and software, are invaluable tools for visualizing mathematical functions and their transformations. They allow for quick graph plotting which can verify theoretical transformations. Furthermore, graphing utilities can enhance understanding through interactive engagement. For the function \(g(x) = -\sqrt{x+2} + 1\), using a graphing utility would vividly showcase the reflection, horizontal shift, and vertical shift.

• The graphing tool will flip the \(\sqrt{x}\) curve over the x-axis.
• It will correctly move it 2 units to the left.
• Finally, it will adjust it 1 unit upwards.

Such visual validation clarifies transformations and ensures the correctness of manual calculations.