Problem 59 - 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 vertically shrunk by a factor of \(\frac{1}{2}\) and shifted three units to the right.

Step-by-Step Solution

Answer

The graph of \(g\) when it's vertically shrunk by a factor of \(\frac{1}{2}\) and shifted ‎three units to the right ‎is represented by the equation \(h(x) = \frac{1}{2}\sqrt{x-3}\).

Step 1: Apply the vertical shrink

A vertical shrink by a factor of \(\frac{1}{2}\) involves multiplying the function \(\sqrt{x}\) by \(\frac{1}{2}\). This gives us the function \(h(x) = \frac{1}{2}\sqrt{x}\)

Step 2: Apply the shift to the right

A shift three units to the right involves replacing \(x\) in the function \(h(x)\) with \(x-3\). Therefore, the transformed function is \(h(x) = \frac{1}{2}\sqrt{x-3}\).

Step 3: Verification with a graphing utility

Using a graphing calculator or software, plot the original function \(g(x) = \sqrt{x}\) and the transformed function \(h(x) = \frac{1}{2}\sqrt{x-3}\). Ensure that the graph of \(h(x)\) shows a vertical shrink and a shift three units to the right relative to the graph of \(g(x)\).

Key Concepts

Vertical Shrink

Understanding vertical shrink is essential when dealing with transformations of functions. A vertical shrink scales the graph of a function towards the x-axis, making it narrower than the original. Imagine the y-values of a function getting squished! For a square root function like \(g(x) = \sqrt{x}\), a vertical shrink by a factor means we multiply the output of the function by that factor. For example, if we have a shrink factor of \(\frac{1}{2}\), every y-value of \(g(x)\) is halved, resulting in the new function \(h(x) = \frac{1}{2}\sqrt{x}\). This transformation makes the graph flatter since the y-values are reduced, but it doesn't alter the x-values or the shape of the graph.

Using a graphing utility can help visually confirm the shrink, and it's a handy way to check our work. To truly grasp this concept, keep practicing with different factors and functions to see how vertical shrinking affects their graphs.

Horizontal Shift

The horizontal shift is another transformation that moves the graph of a function along the x-axis. When we say 'shift to the right', it means that every point on the graph moves a certain number of units in the positive x-direction. For the square root function \(h(x) = \frac{1}{2}\sqrt{x}\), shifting the graph three units to the right is done by replacing every \(x\) in the function with \(x-3\). It results in the new function \(h(x) = \frac{1}{2}\sqrt{x-3}\).

This maneuver can be thought of as delaying the start of the function by three units. If a graph starts rising at an x-value of zero, after the shift, it will start rising at three. Being able to identify and apply horizontal shifts is crucial, especially when graphing functions by hand or predicting the effects of certain transformations without a calculator.

Graphing Utility

A graphing utility, such as a calculator or computer software, can drastically improve understanding function transformations. It allows for quick visual representations and checks whether the transformations applied match the expected results. After applying a vertical shrink and horizontal shift to our function \(g(x) = \sqrt{x}\), we could use a graphing utility to plot \(h(x) = \frac{1}{2}\sqrt{x-3}\).

By comparing the original and transformed functions' graphs, the effects of our changes become clear. If the graphing utility shows that \(h(x)\) is narrower and shifted to the right of \(g(x)\), our transformations are correct. These tools offer immediate feedback, which is invaluable for learning and confirming mathematical concepts.

Square Root Function

The square root function, written as \(g(x) = \sqrt{x}\), is a fundamental function with a unique shape that starts at the point (0,0) and curves gently upwards as x increases. Its graph is part of a parabola that extends infinitely to the right. The square root function is often the subject of transformations in exercises because it helps students understand the changes in curvature and steepness caused by these adjustments.

When applying transformations like vertical shrink or horizontal shift to the square root function, it's helpful to plot key points and observe how they move. This basic function is the building block for understanding more complex transformations and is a critical part of any students' mathematical toolkit.