Problem 92 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=-f(x)$$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is the reflection of the graph of \(f(x)\) across the x-axis. In other words, every point on the graph of \(f(x)\) is flipped over the x-axis to produce the graph of \(g(x)\).

Step 1: Understand the given functions

Here, two functions are given: \(g(x)\) and \(f(x)\). The relationship between \(g(x)\) and \(f(x)\) is given by the equation \(g(x) = -f(x)\). This means that the graph of \(g(x)\) is just the graph of \(f(x)\), but with all the \(y\)-values negated.

Step 2: Visualize the effect on the graph

To help understand this, imagine a function \(f(x)\) and its graph. Now, take every point on that graph and flip it over the x-axis. The \(y\)-coordinate of every point will be its opposite (i.e. positive becomes negative and vice-versa). This is the effect of the minus sign before \(f(x)\) in the definition of \(g(x)\). This process creates the graph of \(g(x)\).

Key Concepts

Graph Reflection

When we talk about graph reflection in mathematical functions, we often mean transforming the graph of a function in such a way that it mirrors over a specific axis. For functions, reflection commonly occurs over either the x-axis or the y-axis.

• If reflecting over the x-axis, all the y-values of the function are negated, flipping the graph upside down.
• For reflection over the y-axis, which is not covered in this context, we would negate the x-values instead.

Reflection transforms the graph into its mirror image, which can help visualize relationships between functions like how changing a function into its negative might affect its appearance. This is particularly useful in understanding transformations like those between the graphs of \(f(x)\) and \(g(x)=-f(x)\).

Negation of Y-values

Negating y-values is a key part of transforming functions, specifically when creating a reflection over the x-axis. In simple terms:

• Every positive y-value of the original function \(f(x)\) becomes negative in the transformed function \(g(x)\).
• Likewise, every negative y-value of \(f(x)\) becomes positive in \(g(x)\).
• If the y-value is zero, it remains unchanged.

This negation process affects each point's vertical placement on the graph. For instance, the point \((x, y)\) in \(f(x)\) shifts to \((x, -y)\) in \(g(x)\). Thus, negating y-values results in a graph that has been flipped over the x-axis, maintaining the same shape and size but with inverted vertical orientation.

X-axis Reflection

The x-axis reflection is a specific type of graph reflection. It is achieved when the entire graph of a function is flipped over the x-axis, changing every point’s position in relation to this axis.

• This transformation involves taking the function \(f(x)\) and converting it to \(g(x)=-f(x)\), as in our exercise.
• The operation impacts the graph by inverting all its y-values, so every peak becomes a valley and every valley a peak.
• The horizontal symmetry of the graph remains unchanged.

In visual terms, think of it like placing a piece of paper on a table and flipping it upside down. The top becomes the bottom while sides stay the same, mirroring the function over the x-axis without altering its x-values. Understanding this concept is crucial as it forms the basis of recognizing transformation behaviors in equations and graphical representations.