Coordinate Reflections GuideGeometry Transformations 8th Grade & HS Geometry • Coordinate Plane Reflecting Over the Axes: Double Reflection of Point H Target Ordered Pair H″ = (4, −2) (x,y) Official Solution The coordinates of point H″ are (4, −2) Quadrant Progression Quad II → Quad III → Quad IV Visual Coordinate Grid Scale: 1 square = 1 unit QI QII QIII QIV x y -4 4 2 -2 Flip over x-axis Flip over y-axis H (−4, 2) H′ (−4, −2) H″ (4, −2) (0,0) Notice the perpendicular distance to each axis is preserved during reflection. 0 Starting Position: Point H H = (−4, 2) Located in Quadrant II: 4 units left of the y-axis, 2 units above the x-axis. 1 Step 1: Reflect over the x-axis → H′ H′ = (−4, −2) Algebraic Rule: \((x, y) \to (x, -y)\) The x-value stays −4. The y-value changes sign from \(+2\) to \(-2\). The point moves 2 units below the x-axis into Quadrant III. 2 Step 2: Reflect over the y-axis → H″ H″ = (4, −2) Algebraic Rule: \((x, y) \to (-x, y)\) The y-value stays −2. The x-value changes sign from \(-(-4) = +4\). The point lands 4 units right of the y-axis in Quadrant IV. Coordinate Plane Reflection Rules Cheat Sheet Reflection Axis x-axis \((x, y) \to (x, -y)\) y-coordinate negates Reflection Axis y-axis \((x, y) \to (-x, y)\) x-coordinate negates Double Reflection Origin / Both Axes \((x, y) \to (-x, -y)\) Both coordinates negate Diagonal Line Line y = x \((x, y) \to (y, x)\) Coordinates swap Mental Math Shortcut: The 180° Origin Rotation Equivalence Reflecting a figure across the x-axis and then across the y-axis is geometrically equivalent to a 180° rotation around the origin (0,0). Simply negate both coordinates: \((-4, 2) \to (-(-4), -(2)) = (4, -2)\). Try Another P(3, −5) → P″ (−3, 5) Key Takeaway: Distance to each axis of reflection is invariant (preserved). \(H(-4, 2) \to H'(-4, -2) \to H''(4, -2)\). CCSS.MATH.CONTENT.8.G.A.3
Reflections Practice WorksheetGeometry Skills Practice Coordinate Plane Reflections: x-Axis and y-Axis Classwork / Homework CCSS 8.G.A.3 Name: Date: Period: Core Reflection Rules: Over x-axis: (x, y) → (x, −y) Over y-axis: (x, y) → (−x, y) 1 Problem 1: Reflecting over x-axis Core Fact Point T is located at (−1, −3) on the coordinate plane. Point T is reflected over the x-axis to create point T′. T(−1,−3) Plot T′ on the grid and record its coordinates below: T′ = ( , ) Quadrant: ____________ Rule: Keep x-value the same; negate y-value. 2 Problem 2: Reflecting over y-axis Axis Shift Point M is located at (4, −2) on the coordinate plane. Point M is reflected over the y-axis to create point M′. M(4,−2) Plot M′ on the grid and record its coordinates below: M′ = ( , ) Quadrant: ____________ Rule: Negate x-value; keep y-value the same. 3 Problem 3: Double Reflection Composite Point R is located at (−5, 2). Part A: Reflect over x-axis: R′ = ( ____ , ____ ) Part B: Reflect R′ over y-axis: R″ = ( ____ , ____ ) Bonus Question: What single rotation around the origin maps point R directly to R″? ____________________ 4 Problem 4: Algebraic Rules Fluency Quick Fill Apply the correct algebraic rule to find the coordinates of each image: Original PointReflection LineImage CoordinatesA (5, −6)x-axisA′ ( ____ , ____ )B (−6, −1)y-axisB′ ( ____ , ____ )C (0, −4)x-axisC′ ( ____ , ____ ) Error Analysis: Identify and Fix the Mistake Critical Thinking A classmate claims: “Reflecting point P(2, −6) over the x-axis gives P′(−2, −6) because reflection means making the number negative.” Explain the mistake and write the correct coordinates for P′: Self-Check Tip: When folding your paper along the axis of reflection, the pre-image and image points must overlap perfectly! Coordinate Geometry
Reflections Answer KeyTeacher Answer Key & Guide Coordinate Reflections • 8th Grade / HS Geometry Solutions: Coordinate Plane Reflections (x-Axis & y-Axis) Answer Status Verified Complete A′ Target Problem Spotlight Point A(5, −6) reflected over the x-axis: A′ = (5, 6) Rule: \((x, y) \to (x, -y)\) \((5, -(-6)) = (5, 6)\) 1 Problem 1: Point T(−1, −3) T′ = (−1, 3) Reflection Line: x-axis \(\to\) Rule: \((x, y) \to (x, -y)\) Work: \(x = -1\), \(y = -(-3) = +3\) Quadrant: Point moves from Quadrant III to Quadrant II. Distance to x-axis is 3 units before and after reflection. 2 Problem 2: Point M(4, −2) M′ = (−4, −2) Reflection Line: y-axis \(\to\) Rule: \((x, y) \to (-x, y)\) Work: \(x = -(4) = -4\), \(y = -2\) Quadrant: Point moves from Quadrant IV to Quadrant III. Distance to y-axis is 4 units before and after reflection. 3 Problem 3: Point R(−5, 2) Composite Double reflection breakdown: Part A (over x-axis): R′ = (−5, −2) y-coordinate negates: \(-5\) stays, \(2 \to -2\). Part B (R′ over y-axis): R″ = (5, −2) x-coordinate negates: \(-(-5) = 5\), \(-2\) stays. Bonus Answer: 180° rotation around origin \((x, y) \to (-x, -y)\). 4 Problem 4: Fluency Solutions Key Table Completed coordinate table: OriginalLineCorrect ImageA (5, −6)x-axisA′ (5, 6)B (−6, −1)y-axisB′ (6, −1)C (0, −4)x-axisC′ (0, 4) Error Analysis: Model Exemplar Student Explanation Scoring Anchor Exemplar Response: “The classmate made two critical errors. First, reflecting over the x-axis alters the y-coordinate, not the x-coordinate. Second, 'negating' does not mean making a number negative; it means taking the opposite sign. Since the original y-coordinate was \(-6\), its opposite is \(+6\). The correct location of P′ is .”
