Problem 53 - College Algebra

Question

describe how the graph of each function is a transformation of the graph of the original function \(f.\) $$g(x)=-f(x)$$

Step-by-Step Solution

Answer

The function \(g(x) = -f(x)\) is a vertical reflection of \(f(x)\) over the x-axis.

Step 1: Identify the Basic Function

The original function is given as \(f(x)\). We need to compare \(g(x) = -f(x)\) with this function.

Step 2: Determine the Transformation Type

The given function \(g(x) = -f(x)\) represents a transformation of \(f(x)\) by multiplying it by \(-1\). This changes the sign of all the output values of \(f(x)\).

Step 3: Describe the Reflection

Multiplying the function by \(-1\) reflects the graph of \(f(x)\) across the x-axis. Therefore, every point \((x, y)\) on the graph of \(f(x)\) is transformed to \((x, -y)\) on the graph of \(g(x)\).

Step 4: Analyze the Effect on Graph

This reflection means that peaks (maxima) in the graph of \(f(x)\) become troughs (minima) in \(g(x)\), and troughs become peaks. This transformation does not affect the x-coordinates of the function values.

Step 5: Confirm Graph Characteristics

Ensure that you can recognize the graph visually. Reflecting over the x-axis doesn't shift or stretch the graph; it merely flips it vertically.

Key Concepts

Reflection Over X-Axis

When discussing graph transformations, reflection over the x-axis is a fundamental concept. It occurs when any function, such as a given function \(f(x)\), is multiplied by \(-1\). This change results in the transformation of every point on the graph of \(f(x)\) to a new corresponding point on the graph of \(-f(x)\). This is because each output \(y\) value from the original function is negated.

Imagine you have a point \((x, y)\) on the graph of \(f(x)\). When you apply the reflection, you transform this to \((x, -y)\).

• The x-coordinate remains unchanged, signifying no horizontal movement.
• Only the y-coordinate is altered, flipping the graph over the x-axis.

Visualize this by considering the highs and lows of a graphed curve: peaks (or maxima) are inverted into troughs (or minima), and vice versa. This is a neat, straightforward flip that can be easily spotted in graphs.

Function Multiplication

Function multiplication by a constant is a key operation in modifying function outputs. In the context of reflections, multiplying by \(-1\) plays a special role. This multiplication does not modify the basic shape of the graph, such as stretching or compressing it. Instead, it serves to entirely reverse the sign of the resulting outputs.

Here's how it works with a given function \(f(x)\):

• When you multiply \(f(x)\) by \(-1\), you get \(g(x) = -f(x)\).
• This changes the signs of all output values. If \(f(x)\) is positive at a point, \(g(x)\) becomes negative, and vice versa.

This operation is purely vertical. Unlike other transformations, such as shifts or stretches, function multiplication leaves the x-values untouched. This means the x-intercepts remain the same, providing an easy checkpoint for double-checking your work.

Negative Function Transformation

Negative function transformation is the underlying mechanism in reflections over the x-axis. It is the act of converting a positive function value to a negative one or the opposite. Understanding this concept helps in visualizing the end result on a graph quickly.

Suppose the base function \(f(x)\) represents values as increasing inputs lead to growing outputs:

• Applying a negative transformation with \(g(x) = -f(x)\) inverts this behavior.
• Originally positive outputs become negative, and originally negative outputs become positive.

This transformation entirely flips the orientation of all output values around the x-axis. Consequently, it's a vital tool in graph analysis whenever a stark visual change is needed without altering the behavior of inputs themselves.