Mathematics
Grade 9
15 min
Reflections: graph the image
Reflections: graph the image
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Graph the image of a point or polygon after a reflection across the x-axis or y-axis.
Graph the image of a point or polygon after a reflection across the line y = x or y = -x.
Apply the coordinate rules to determine the vertices of an image algebraically without graphing.
Identify the line of reflection given a pre-image and its reflected image.
Describe how the graph of a simple function, such as a line or a parabola, changes after a reflection across an axis.
Verify that a reflection is a rigid transformation that preserves side lengths and angle measures.
Ever looked in a mirror and seen a perfect 'flip' of yourself? 🤔 That's a reflection, and we can do the exact same thing with shapes and functions on a graph!
This tutorial will teach...
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Key Concepts & Vocabulary
TermDefinitionExample
ReflectionA transformation that 'flips' a figure over a line, creating a mirror image. Each point in the image is the same distance from the line of reflection as the corresponding point in the pre-image.Flipping the point (3, 2) over the y-axis results in the image point (-3, 2).
Pre-imageThe original figure before a transformation is applied.If we are reflecting Triangle ABC, then Triangle ABC is the pre-image.
ImageThe new figure that results from applying a transformation to the pre-image.After reflecting Triangle ABC, the resulting figure, Triangle A'B'C', is the image.
Line of ReflectionThe fixed line over which a figure is flipped. It acts as the 'mirror'.The x-axis, the y-axis, or the line y = x can all be lines of reflectio...
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Core Formulas
Reflection Across the x-axis
(x, y) \rightarrow (x, -y)
To reflect a point across the x-axis, keep the x-coordinate the same and take the opposite of the y-coordinate.
Reflection Across the y-axis
(x, y) \rightarrow (-x, y)
To reflect a point across the y-axis, take the opposite of the x-coordinate and keep the y-coordinate the same.
Reflection Across the line y = x
(x, y) \rightarrow (y, x)
To reflect a point across the line y = x, swap the x- and y-coordinates.
Reflection Across the line y = -x
(x, y) \rightarrow (-y, -x)
To reflect a point across the line y = -x, swap the x- and y-coordinates and take the opposite of both.
4 more steps in this tutorial
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Challenging
A line segment has endpoints P(1, 6) and Q(4, 2). It is reflected to create the image P'Q'. If the image of P is P'(6, 1), what is the line of reflection and what are the coordinates of Q'?
A.y-axis; Q'(-4, 2)
B.x-axis; Q'(4, -2)
C.y = x; Q'(2, 4)
D.y = -x; Q'(-2, -4)
Challenging
The parabola y = (x + 4)² - 2, with its vertex at (-4, -2), is reflected across the y-axis. What is the equation of the new parabola?
A.y = (x - 4)² - 2
B.y = -(x + 4)² + 2
C.y = (x + 4)² + 2
D.y = (-x + 4)² - 2
Challenging
A triangle's vertices are in Quadrant I. After a single reflection, its image is in Quadrant II. The side lengths and angle measures are unchanged, but its left-to-right orientation is reversed. What was the line of reflection?
A.The x-axis
B.The line y = x
C.vertical line x = c where c < 0
D.The y-axis
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