Finding Reflection Points Across Axis of Symmetry
Finding reflection points across the axis of symmetry is a Grade 11 Algebra 1 skill from enVision Chapter 8 that uses the formula x₂ = 2h - x₁ to locate a parabola point mirrored across the line x = h. For f(x) = x² - 4x + 3 with axis x = 2, the point (0, 3) reflects to (2·2 - 0, 3) = (4, 3). For f(x) = 2x² + 8x + 5 with axis x = -2, the point (-1, -1) reflects to (2(-2) - (-1), -1) = (-3, -1). Both original and reflection points share the same y-coordinate and lie equidistant from the axis.
Key Concepts
For any point $(x 1, y 1)$ on a parabola with axis of symmetry $x = h$, its reflection point is $(x 2, y 1)$ where $x 2 = 2h x 1$. The distance from each point to the axis of symmetry is equal: $|x 1 h| = |x 2 h|$.
Common Questions
What is the formula to find a reflection point across an axis of symmetry?
x₂ = 2h - x₁, where h is the axis of symmetry and x₁ is the x-coordinate of the original point. The y-coordinate stays the same.
For axis x = 2 and point (0, 3), what is the reflection point?
x₂ = 2(2) - 0 = 4, so the reflection point is (4, 3).
Why do reflected points on a parabola share the same y-coordinate?
Because the axis of symmetry acts like a mirror. The parabola has equal height on both sides at any given distance from the axis.
What is the reflection of (-1, -1) across axis x = -2?
x₂ = 2(-2) - (-1) = -4 + 1 = -3, so the reflection point is (-3, -1).
How does the distance property confirm a reflection?
Each point is the same distance from the axis: |x₁ - h| = |x₂ - h|. For (0, 3) and (4, 3) across x = 2: |0-2| = |4-2| = 2. ✓
When is finding reflection points useful for graphing parabolas?
When you already know one point on the parabola, you can quickly find its mirror image without computing f(x) again, which is especially useful when plotting the y-intercept and its symmetric partner.