Reflections and Translations on a Coordinate Plane

Welcome back to math class! Today, we're diving into the fascinating world of reflections and translations on the coordinate plane. You might remember these concepts from geometry, where you worked with shapes and points. This year, we're taking it a step further by exploring how these transformations affect equations and graphs of functions.

Key Vocabulary

  • Transformation: A change in the position, size, or shape of a figure.
  • Translation: A transformation that "slides" each point in a figure the same distance in the same direction.
  • Reflection: A transformation that "flips" a figure across a line, called the line of reflection. Each reflected point is the same distance from the line of reflection, but on the opposite side.

Translations

A translation, or slide, moves every point of a figure the same distance in the same direction. We can represent translations using coordinate notation. Think about how the coordinates change when we shift a point horizontally or vertically.

  • Horizontal Translation: Shifts each point right or left by a number of units. If we shift a point $(x, y)$ horizontally by $h$ units, the new point is $(x + h, y)$. If $h > 0$, the shift is to the right. If $h < 0$, the shift is to the left.
  • Vertical Translation: Shifts each point up or down by a number of units. If we shift a point $(x, y)$ vertically by $k$ units, the new point is $(x, y + k)$. If $k > 0$, the shift is upwards. If $k < 0$, the shift is downwards.

For example, translating the point $(-3, 4)$ five units to the right results in the point $(2, 4)$. Translating the point $(-3, 4)$ two units left and two units down results in the point $(-5, 2)$.

Reflections

A reflection "flips" a figure across a line, known as the line of reflection. Here's how coordinates change for reflections across the x and y axes:

  • Reflection Across the x-axis: Each point flips across the x-axis. The x-coordinate stays the same, and the y-coordinate changes its sign. $(x, y) ightarrow (x, -y)$.
  • Reflection Across the y-axis: Each point flips across the y-axis. The y-coordinate stays the same, and the x-coordinate changes its sign. $(x, y) ightarrow (-x, y)$.

When a function is translated or reflected, the original graph and the graph of the transformation are congruent because the size and shape of the graphs are the same.

Transforming Functions

You can transform a function by transforming its ordered pairs! Transformations can involve translations and reflections.

For example, to translate a function $y = f(x)$ two units up, add 2 to each y-coordinate. To reflect a function $y = f(x)$ across the x-axis, multiply each y-coordinate by $-1$.

Discussion Question

What are some examples of transformations outside of math class, and how do they relate to the translations and reflections we looked at today? Think about everyday scenarios where objects or images are moved or flipped!

Remember: Keep practicing, and don't hesitate to ask questions. You've got this!

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Homework: Page 11-14 # 14-20, 28, 29, 52