Problem 60 - 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 stretched by a factor of 2, reflected in the \(x\) -axis, and shifted three units upward.

Step-by-Step Solution

Answer

The new function after applying the transformations is \(-2*\sqrt{x} + 3\)

Step 1: Vertical Stretch

To vertically stretch the function by a factor of 2, we multiply the function by 2. The new function becomes \(2*g(x) = 2*\sqrt{x}\)

Step 2: Reflection in the x-axis

To reflect the function in the x-axis, we change the sign of the function. The result of the transformation is \(-2*\sqrt{x}\)

Step 3: Vertical Shift

To shift the function three units upward, add 3 to the function. The final function becomes \(-2*\sqrt{x} + 3\)

Step 4: Validation

To validate the transformation, plot both the original function and transformed function on the same graphing utility. If all the transformations were applied correctly, the graph of the transformed function should correspond to a vertically stretched, reflected, and shifted version of the original function '\(\sqrt{x}\)'.

Key Concepts

Vertical Stretch

A vertical stretch occurs when we multiply a function by a constant factor greater than 1. In the context of our function transformation, the vertical stretch by a factor of 2 means that each point on the graph of the original function, \(g(x) = \sqrt{x}\), becomes twice as far from the \(x\)-axis. This adjustment modifies the function to \(2 \cdot g(x) = 2 \cdot \sqrt{x}\). Here's how you can think about it:

• Every output value (or \(y\)-value) is doubled, making the graph appear taller.
• This transformation makes the curve steeper, affecting how it rises as \(x\) increases.

By visualizing or plotting this change, you should see that the graph elongates vertically without shifting to the left or right.

Reflection in the x-axis

A reflection in the \(x\)-axis changes the sign of the output values of a function. Essentially, every point on the graph is flipped to its opposite position on the y-axis. For our function, after a vertical stretch, we have \(2\sqrt{x}\). Reflecting this function requires changing its sign, resulting in \(-2\sqrt{x}\).

• If a point is above the x-axis on the original graph, it will be moved directly below the x-axis at the same distance.
• This does not change the shape of the graph, only its vertical orientation.
• Think of it as looking at the graph in a mirror placed along the x-axis.

This reflection flips the graph upside down, but keeps its stretched nature due to the previous transformation.

Vertical Shift

A vertical shift alters the position of a graph up or down along the y-axis. To apply a vertical shift of three units upward, we're simply adding 3 to every \(y\)-value of our already stretched and reflected function, \(-2\sqrt{x}\). As a result, the function becomes \(-2\sqrt{x} + 3\).

• A shift upwards moves the entire graph three units higher, without changing its shape or orientation.
• This step is like moving a physical piece of the graph upwards on graph paper.
• This ensures all transformations are in place: the graph is now stretched, reflected, and vertically shifted from its original form \(\sqrt{x}\).

The series of these transformations should result in a graph of \(-2\sqrt{x} + 3\) that looks significantly different from the original function, \(\sqrt{x}\), in its final placement.