Problem 57 - 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 shifted four units to the right and three units downward.

Step-by-Step Solution

Answer

The equation of the function that results after shifting the graph of \(g(x) = \sqrt{x}\) four units to the right and three units downward is \(f(x) = \sqrt{x - 4} - 3\).

Step 1: Understand the Effect of Shifts on The Function

The necessary transformations on \(g(x)\) involve shifts along the X-axis and Y-axis. Specifically, performing a shift to the right is achieved by replacing \(x\) in the original function with \(x - c\), where \(c\) is the number of units to shift. Similarly, shifting a function downward is done by subtracting a given number \(d\) from the function. In this case, \(c = 4\) and \(d = 3\). Therefore, the new function will replace \(x\) with \(x - 4\) and then subtract \(3\) from \(g(x)\).

Step 2: Perform The Shifts

Replace \(x\) by \(x - 4\) in the original function \(g(x) = \sqrt{x}\) to shift the graph to the right. This gives a new function, \(h(x) = \sqrt{x - 4}\). Then, subtract 3 from \(h(x)\) to shift the graph downward, resulting in the transformed function \(f(x) = \sqrt{x - 4} - 3\).

Step 3: Verify The Equation With a Graphing Utility

The final step is to verify the obtained equation using a graphing utility. Plot both the original function \(g(x) = \sqrt{x}\) and the transformed function \(f(x) = \sqrt{x - 4} - 3\). You should observe that the graph of \(f(x)\) is exactly the graph of \(g(x)\), but shifted 4 units to the right and 3 units downward.

Key Concepts

Rigid Transformation

Rigid transformations are transformations that change the position of a graph but not its shape or size. This includes translations (shifts), reflections, and rotations. When dealing with functions, rigid transformations can be easily identified and performed.

In the context of functions like the square root function, \(g(x) = \sqrt{x}\), rigid transformations allow us to move the graph on the coordinate plane without altering its general shape.

For example:

• Translation: Moving the graph left, right, up, or down. These are simple directional shifts.
• Reflection: Flipping the graph over a specific axis.
• Rotation: Turning the graph around a point, although this is less common with basic functions.

In the original problem, the graph undergoes a specific translation - 4 units to the right and 3 units downward. This form of translation retains the shape and orientation of the graph while adjusting its position on the plane.

Nonrigid Transformation

Nonrigid transformations affect the shape or size of the graph. They include stretching and compressing (or dilating) the graph vertically or horizontally.

Such transformations alter the proportions of the graph, making it look wider, narrower, taller, or shorter.

In contrast to rigid transformations:

• Horizontal stretching or compressing happens when you multiply the input by a constant. If the constant is greater than 1, the graph compresses; if it is between 0 and 1, it stretches.
• Vertical stretching or compressing occurs by multiplying the output by a constant. Again, a greater than 1 constant will stretch the graph vertically, while a constant between 0 and 1 will compress it.

When analyzing transformations for \(g(x) = \sqrt{x}\), this type of transformation was not requested.
Instead, the problem focused solely on shifting the graph. But noting this distinction can help in recognizing problems where multiple transformations may occur.

Graph Shifts

Graph shifts are among the most common transformations you will encounter.

They are typically straightforward since they only involve changes in the location of the graph along the \(x\)-axis and \(y\)-axis.

Graph Shifts include:

• Horizontal Shifts: Achieved by adding or subtracting a constant to the input \(x\). For shifting right, replace \(x\) with \(x - c\); for shifting left, use \(x + c\).
• Vertical Shifts: Done by adding or subtracting a constant to the entire function. For upward shifts, add \(d\) to the function, and for downward shifts, subtract \(d\).

In the problem, we performed a right shift of 4 units and a downward shift of 3 units:

• Right Shift: By replacing \(x\) with \(x - 4\) in the function \(g(x) = \sqrt{x}\), it results in \(h(x) = \sqrt{x - 4}\).
• Downward Shift: Subtracting 3 from \(h(x)\) yields the final function \(f(x) = \sqrt{x - 4} - 3\).

This showcases the simplicity and clarity of graph shifts in adjusting the graph's position while preserving its inherent characteristics.