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Chapter 2: Problem 14

The graph of \(y=-f(x)\) is the graph of \(y=f(x)\) reflected across the _____ -axis.

Short Answer

Expert verified
x-axis.

Step by step solution

01

Understanding the Reflection

To determine the reflection axis for the graph of a function, it's important to understand the sign change in the function. Here, the transformation is from the function \( y = f(x) \) to \( y = -f(x) \).
02

Significance of \( -f(x) \)

When we change \( y = f(x) \) to \( y = -f(x) \), we are negating the output values of the function. This implies that every point \( (x, y) \) on the graph of \( y = f(x) \) will be transformed to the point \( (x, -y) \) on the graph of \( y = -f(x) \).
03

Reflection Across the Axis

Points that are transformed from \( (x, y) \) to \( (x, -y) \) indicate a reflection over a specific axis. Since the x-coordinate remains the same and only the y-coordinate is inverted, the reflection is across the x-axis.
04

Conclusion

Therefore, the graph of \( y = -f(x) \) is the graph of \( y = f(x) \) reflected across the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection Across X-Axis
The concept of reflection across the x-axis is essentially flipping a graph upside down. Imagine you have a function graph, say, a parabola pointing upwards. Reflecting this graph across the x-axis would mean that every point on the graph swaps its y-coordinate to its negative counterpart.
This means if a point on the original function is \( (3, 5) \), after the reflection, the point will become \( (3, -5) \).
The x-coordinate remains unchanged, but the y-coordinate becomes negative, causing a 'mirror' effect across the x-axis. This transformation is not restricted to parabolas; it applies to any function.
Function Transformation
Function transformation refers to changes made to the basic graph of a function resulting in shifts, reflections, stretches, or compressions. The transformation the exercise focuses on is through multiplicative changes.
When we take a function \( y = f(x) \) and transform it to \( y = -f(x) \), we are simply multiplying the entire function by -1. This multiplication by -1 only affects the output or y-values of the function, leaving the x-values unchanged.
This type of transformation specifically flips the graph over the x-axis, leaving the overall shape and x-intersections unchanged, while changing the direction in which the graph opens.
Coordinate Geometry
Coordinate geometry helps us understand the spatial relationship and positions of points on a graph. In the case of reflections, knowing the coordinates of points before and after the transformation is crucial.
For instance, consider a point \( (x, y) \) on the graph of the function \( y = f(x) \). When reflecting across the x-axis to generate the graph of \( y = -f(x) \), this point becomes \( (x, -y) \).
This relationship shows the original point has been flipped over the x-axis, reinforcing our understanding through practical application and visualization of coordinate geometry principles.
Negative Function Values
Understanding negative function values is critical when dealing with reflections across the x-axis. In essence, a negative function value flips the output of each point.
For example, if the original function \( f(x) \) yields a positive y-value, then -f(x) will yield the negative of that y-value. Similarly, if the original function returns a negative y-value, -f(x) will return a positive y-value.
Hence, reflecting a function across the x-axis changes the signs of all the y-values but retains their magnitude. This forms the foundation for understanding many other transformations and operations on functions.

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Most popular questions from this chapter

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