Reflections of graphs

Reflections of Graphs

Introduction

• The process of flipping a graph across a line, causing the image to appear as if seen in a mirror, is referred to as reflection.
• Reflections can be produced in relation to the x-axis, y-axis, or the origin.
• The changes produced in the graph largely depend on the axis of reflection.

Reflection in the X-axis

• Reflection of a graph in the x-axis makes the y-coordinates switch signs.
• For instance, if you have the point (2, 3) in the original graph, it will become (2, -3) in the reflected graph.

Reflection in the Y-axis

• Reflecting a graph in the y-axis leads to inversion of the sign of the x-coordinates.
• Therefore, a point having coordinates (4, 5) will transition to (-4, 5) after reflection.

Reflection in the Origin

• When a graph is reflected in the origin, both the x-coordinate and the y-coordinate change signs.
• For example, a point at coordinates (6, -7) will be reflected to (-6, 7).

Interpreting Reflections

• The reflection of a graph does not affect the shape but impacts the location and orientation.
• Despite the change in position, the size and shape of the graph remains constant after reflection.
• Reflections are crucial in understanding the symmetry of mathematical functions.

Examples

• For instance, the reflection in the y-axis of the graph of the function f(x) = x^2 will still be f(x) = x^2, as this is a symmetrical function.
• The reflection in the origin of the function f(x) = x^3 produces the inverted function f(x) = -x^3.

Conclusion

• Reflections of graphs help in understanding the properties of algebraic functions and their graphical representations.
• The recognition and drawing of these reflections are foundational skills in GCSE algebra.