Problem 71 - College Algebra

Question

Assume that \((a, b)\) is a point on the graph of \(f .\) What is the corresponding point on the graph of each of the following functions? $$ y=f(-x) $$

Step-by-Step Solution

Answer

The corresponding point on the graph of y=f(-x) to the point (a, b) on the graph of f(x) is (-a, b).

Step 1: Reflection Over the Y-Axis

When negating the argument of a function, such as f(-x), the effect is a reflection over the y-axis. Therefore, the x-value of the corresponding point will be the negative of the original x-value.

Step 2: Apply Reflection to Point

To find the corresponding point on the graph of y=f(-x), we negate the x-coordinate. So, the x-coordinate becomes -a instead of a, while the y-coordinate remains the same b.

Step 3: Find the Corresponding Point

Then, the corresponding point on the graph of y=f(-x) to the point (a, b) on the function y=f(x) is (-a,b).

Key Concepts

Reflection Over the Y-Axis

Imagine flipping a picture horizontally. That's similar to what happens when a function experiences a reflection over the y-axis.
When you adjust a function to become f(-x), every point's x-coordinate on the graph of y = f(x) is flipped to its opposite.

• If you have a point (a, b) on y = f(x), this transformation alters it to (-a, b) on y = f(-x).
• This is because only the x-value changes sign. The y-value remains constant.

This effect is purely visual and greatly impacts the symmetry of the graph. Often, reflections are used when solving problems or proving symmetry in functions, making them a crucial graphing technique.

Function Notation

Function notation is like a special shorthand in mathematics that helps describe the relationship between variables clearly. For any function f, you can express it as y = f(x). Here's what that means:

• The letter "f" represents the function. It's the rule or operation applied to x.
• The variable "x" is the input or the starting value we give the function.
• "y" is often the output or result after applying f to x.
• In the question, f(-x) means applying the function rule f to -x instead of x.

Function notation makes handling complex equations simpler and better organized. It's particularly helpful when talking about transformations or shifts in a function's graph.

Graphing Techniques

Graphing a function accurately is vital to understanding its behavior. Besides simply plotting points, several techniques help make this process easier and more insightful:

• Reflections: Changes to the graph like f(-x) demonstrate reflections. As described, this results in mirroring the graph about the y-axis.
• Shifting: Moving graphs up, down, left, or right. For example, y = f(x) + k shifts up if k is positive.
• Stretching/Compressing: Involves adjusting the graph's "shape." Multiplying f(x) by a coefficient changes the slope, affecting how steep or flat the graph appears.

Graphing techniques do more than just plot a graph; they help in understanding symmetry, patterns, and connections between different functions.