Reflection

(noun)

A mirror image of a function across a given line.

Related Terms

Examples of Reflection in the following topics:

  • Reflections

    • Reflections produce a mirror image of a function.  
    • A vertical reflection is a reflection across the $x$-axis, given by the equation:
    • A horizontal reflection is a reflection across the $y$-axis, given by the equation:
    • The third type of reflection is a reflection across a line.  
    • The reflected equation, as reflected across the line $y=x$, would then be:

  • Symmetry of Functions

    • They can have symmetry after a reflection.  
    • The image below shows an example of a function and its symmetry over the $x$-axis (vertical reflection) and over the $y$-axis (horizontal reflection).  
    • The points given, $(1,3)$ and $(-1,-3)$ are reflected across the origin.
    • The graph above has symmetry since the points labeled are reflected over the origin.
    • A function can have symmetry by reflecting its graph horizontally or vertically.  

  • Transformations of Functions

    • The four main types of transformations are translations, reflections, rotations, and scaling.
    • A reflection of a function causes the graph to appear as a mirror image of the original function.  
    • Therefore, we can say that $f(-x)$ is a reflection of $f(x)$ across the $y$-axis.
    • Differentiate between three common types of transformations: reflections, rotations, and scaling

  • Parabolas As Conic Sections

    • Parabolas have the property that, if they are made of material that reflects light, then light which enters a parabola traveling parallel to its axis of symmetry is reflected to its focus.
    • This happens regardless of where on the parabola the reflection occurs.
    • Conversely, light that originates from a point source at the focus is reflected, or collimated, into a parallel beam.
    • This reflective property is the basis of many practical uses of parabolas.

  • Inverse Trigonometric Functions

    • Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line $y = x$.
    • The arctangent function is a reflection of the tangent function about the line $y = x$.
    • The arcsine function is a reflection of the sine function about the line $y = x$.
    • The arccosine function is a reflection of the cosine function about the line $y = x$.

  • Introduction to Inverse Functions

    • This is equivalent to reflecting the graph across the line $y=x$, an increasing diagonal line through the origin.
    • The black line represents the line of reflection, in which is $y=x$.
    • The function graph (red) and its inverse function graph (blue) are reflected about the line $y=x$ (dotted black line).
    • For example, $(0,1)$ on the red (function) curve is reflected over the line $y=x$ and becomes $(1,0)$ on the blue (inverse function) curve.

  • Complex Conjugates and Division

    • Geometrically, z* is the "reflection" of z about the real axis (as shown in the figure below).

  • Introduction to Exponential and Logarithmic Functions

    • The graph of the logarithm function $log_b(x)$ (blue) is obtained by reflecting the graph of the function $b(x)$ (red) at the diagonal line ($x=y$).

  • Applications of the Parabola

    • One well-known example is the parabolic reflector—a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point.

  • Fitting a Curve

    • Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a greater degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.