Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the function that is finally graphed after \(y=\sqrt{25-x^{2}}\) is reflected about the \(x\) -axis and shifted right 4 units.

Short answer

\(y = -\sqrt{25-(x-4)^{2}}\).

Step-by-step solution

Title - Reflect the Function about the x-axis

To reflect the function \(y=\sqrt{25-x^{2}}\) about the x-axis, multiply the output (y-values) by -1. This results in the function: \[y = -\sqrt{25-x^{2}}.\]

Title - Shift the Function Right by 4 Units

To shift the function \(y = -\sqrt{25-x^{2}}\) 4 units to the right, replace \(x\) with \(x-4\). This modifies the function to: \[y = -\sqrt{25-(x-4)^{2}}.\]

Key concepts

reflection

Let's start with the concept of reflection in the context of functions.
A reflection flips a function's graph over a specific line.
In most problems, you'll encounter reflections over the x-axis or y-axis.
For our function, we reflect it over the x-axis.
This means we multiply the entire function by -1.
Imagine flipping a piece of paper over a horizontal line; that's what happens to the function.
Given our original function is \(y = \sqrt{25-x^{2}}\), the reflection looks like this: \[y = -\sqrt{25-x^{2}}.\]
Notice how we only changed the sign in front of the square root.
This negative sign affects the y-values, making all positive y-values negative and vice versa.

translation

Next, let's talk about translations.
A translation shifts the entire graph of a function either vertically or horizontally.
For horizontal shifts, we replace \(x\) with \(x-h\), where \text{h}\ is the number of units to shift the graph.
In our case, we want to shift the function right by 4 units.
This means replacing every \(x\) in the function with \(x-4\).
So, if our current function is \[-\sqrt{25-x^{2}},\] the shifted function will be: \[-\sqrt{25-(x-4)^{2}}.\]
Note how we replaced \(x\) with \(x-4\) inside the square root.
This adjustment shifts the graph to the right by 4 units, aligning with our translation requirement.

function graphing

Finally, let's put it all together by talking about function graphing.
Graphing a function means plotting its points on a coordinate plane to visualize its shape.
For our problem, we started with the function \[y = \sqrt{25-x^{2}}.\]
This function represents a semicircle above the x-axis with a radius of 5.
After reflecting around the x-axis, the semicircle flips downward.
Finally, by shifting this reflected semicircle right by 4 units, we get the function \[-\sqrt{25-(x-4)^{2}}.\]
Imagine taking the semicircle, flipping it down, and sliding it four units to the right.
It's like moving a picture frame to a new spot on the wall.
Each transformation—reflection and translation—is applied step by step to get the final function graph.