Reflection

Definition

Reflection is a mathematical transformation that flips or mirrors a function or graph about a line, either the x-axis or the y-axis. This concept is essential in understanding the behavior and properties of various functions and their graphs.

5 Must Know Facts For Your Next Test

1. Reflection about the x-axis flips the graph vertically, while reflection about the y-axis flips the graph horizontally.
2. Absolute value functions exhibit reflection symmetry about the y-axis, as the graph is a mirror image on either side of the y-axis.
3. Inverse functions are reflections of the original function about the line $y = x$, effectively swapping the x and y coordinates.
4. Exponential and logarithmic functions exhibit reflection symmetry when graphed on a semi-log plot, where the x-axis represents the independent variable, and the y-axis represents the logarithm of the dependent variable.
5. Inverse trigonometric functions, such as $\sin^{-1}(x)$ and $\cos^{-1}(x)$, are reflections of the original trigonometric functions about the line $y = x$.

Review Questions

• Explain how reflection affects the graph of a function and describe the visual changes that occur.
  • Reflection is a transformation that flips or mirrors the graph of a function about a line, either the x-axis or the y-axis. When a function is reflected about the x-axis, the graph is flipped vertically, and all y-coordinates are multiplied by -1. Conversely, when a function is reflected about the y-axis, the graph is flipped horizontally, and all x-coordinates are multiplied by -1. These reflections result in a mirrored image of the original graph, with the line of reflection acting as the axis of symmetry.
• Analyze the role of reflection in the properties and behavior of absolute value functions, inverse functions, and inverse trigonometric functions.
  • Absolute value functions exhibit reflection symmetry about the y-axis, as the graph is a mirror image on either side of the y-axis. Inverse functions, such as exponential, logarithmic, and inverse trigonometric functions, are reflections of the original function about the line $y = x$, effectively swapping the x and y coordinates. This reflection property is crucial in understanding the behavior and properties of these functions, as it allows for the determination of the domain, range, and transformations of the inverse function based on the original function.
• Evaluate the significance of reflection in the graphical representation of exponential, logarithmic, and inverse trigonometric functions, and explain how this concept is applied in the analysis of these functions.
  • Exponential and logarithmic functions exhibit reflection symmetry when graphed on a semi-log plot, where the x-axis represents the independent variable, and the y-axis represents the logarithm of the dependent variable. This reflection property is essential in understanding the relationship between exponential and logarithmic functions, as they are inverse functions of each other. Similarly, inverse trigonometric functions, such as $\sin^{-1}(x)$ and $\cos^{-1}(x)$, are reflections of the original trigonometric functions about the line $y = x$. This reflection symmetry allows for the analysis of the domain, range, and transformations of these inverse trigonometric functions, which are crucial in various applications, including the study of periodic phenomena and the solution of trigonometric equations.

Related terms

Transformation: A change in the appearance or position of a function or graph, such as translation, reflection, dilation, or rotation.

Symmetry:

A property where a function or graph appears the same on both sides of a line of reflection, either the x-axis or y-axis.

Inverse Function:

A function that undoes the operation of another function, often represented by flipping the graph about the line $y = x$.