{"id":173,"date":"2026-10-22T22:25:09","date_gmt":"2026-10-22T22:25:09","guid":{"rendered":"https:\/\/quantumleapstar.com\/?p=173"},"modified":"2026-10-22T22:25:09","modified_gmt":"2026-10-22T22:25:09","slug":"four-transformation-techniques","status":"publish","type":"post","link":"https:\/\/quantumleapstar.com\/?p=173","title":{"rendered":"Four Transformation Techniques"},"content":{"rendered":"<p>Transformations change a shape\u2019s position, orientation or size. This guide from Quantum Leap Star Limited explains translation, rotation, reflection and enlargement, using the examples and practice from our Lesson 40 notes.<\/p>\n<p><strong>GCSE Mathematics:<\/strong> learn to draw images accurately on coordinate grids, describe transformations completely, use positive, fractional and negative scale factors, and identify invariant points.<\/p>\n<h2>Essential language<\/h2>\n<ul>\n<li><strong>Object:<\/strong> the starting shape.<\/li>\n<li><strong>Image:<\/strong> the transformed shape.<\/li>\n<li><strong>Corresponding vertices:<\/strong> matching corners, such as A and A\u2032.<\/li>\n<li><strong>Congruent:<\/strong> the same shape and size.<\/li>\n<li><strong>Invariant:<\/strong> unchanged by a transformation.<\/li>\n<\/ul>\n<h2>1. Translation<\/h2>\n<p>A translation slides a shape. Every vertex moves the same horizontal and vertical distance. Give the movement as a <strong>column vector<\/strong>: the top number is horizontal and the bottom number is vertical. Right and up are positive; left and down are negative.<\/p>\n<h3>Worked example: 4 right and 3 up<\/h3>\n<p>Translate A(\u22123, \u22121), B(\u22121, \u22121) and C(\u22123, 1). The vector has 4 above 3. The coordinate rule is <strong>(x, y) \u2192 (x + 4, y + 3)<\/strong>.<\/p>\n<ul>\n<li>A\u2032 = (\u22123 + 4, \u22121 + 3) = <strong>(1, 2)<\/strong><\/li>\n<li>B\u2032 = (\u22121 + 4, \u22121 + 3) = <strong>(3, 2)<\/strong><\/li>\n<li>C\u2032 = (\u22123 + 4, 1 + 3) = <strong>(1, 4)<\/strong><\/li>\n<\/ul>\n<figure><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 530 510\" role=\"img\" aria-label=\"Translation by 4 right and 3 up\" style=\"width:100%;max-width:530px;height:auto;background:white\"><title>Translation by 4 right and 3 up<\/title><path d=\"M 55 35 V 455 M 55 455 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"55\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-5<\/text><text x=\"236.9090909090909\" y=\"459\" font-size=\"12\" text-anchor=\"end\">-5<\/text><path d=\"M 93.18181818181819 35 V 455 M 55 416.8181818181818 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"93.18181818181819\" 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y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-1<\/text><text x=\"236.9090909090909\" y=\"306.27272727272725\" font-size=\"12\" text-anchor=\"end\">-1<\/text><path d=\"M 245.9090909090909 35 V 455 M 55 264.09090909090907 H 475\" fill=\"none\" stroke=\"#68758b\" stroke-width=\"2\"\/><path d=\"M 284.09090909090907 35 V 455 M 55 225.9090909090909 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"284.09090909090907\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">1<\/text><text x=\"236.9090909090909\" y=\"229.9090909090909\" font-size=\"12\" text-anchor=\"end\">1<\/text><path d=\"M 322.27272727272725 35 V 455 M 55 187.72727272727272 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"322.27272727272725\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">2<\/text><text x=\"236.9090909090909\" y=\"191.72727272727272\" font-size=\"12\" text-anchor=\"end\">2<\/text><path d=\"M 360.45454545454544 35 V 455 M 55 149.54545454545453 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"360.45454545454544\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">3<\/text><text x=\"236.9090909090909\" y=\"153.54545454545453\" font-size=\"12\" text-anchor=\"end\">3<\/text><path d=\"M 398.6363636363636 35 V 455 M 55 111.36363636363636 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"398.6363636363636\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">4<\/text><text x=\"236.9090909090909\" y=\"115.36363636363636\" font-size=\"12\" text-anchor=\"end\">4<\/text><path d=\"M 436.8181818181818 35 V 455 M 55 73.18181818181819 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"436.8181818181818\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">5<\/text><text x=\"236.9090909090909\" y=\"77.18181818181819\" font-size=\"12\" text-anchor=\"end\">5<\/text><path d=\"M 475 35 V 455 M 55 35 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"475\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">6<\/text><text x=\"236.9090909090909\" y=\"39\" font-size=\"12\" text-anchor=\"end\">6<\/text><polygon points=\"131.36363636363637,302.27272727272725 207.72727272727272,302.27272727272725 131.36363636363637,225.9090909090909\" fill=\"none\" stroke=\"#1769aa\" stroke-width=\"3\"\/><text x=\"136.36363636363637\" y=\"295.27272727272725\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">A<\/text><text x=\"212.72727272727272\" y=\"295.27272727272725\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">B<\/text><text x=\"136.36363636363637\" y=\"218.9090909090909\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">C<\/text><polygon points=\"284.09090909090907,187.72727272727272 360.45454545454544,187.72727272727272 284.09090909090907,111.36363636363636\" fill=\"none\" stroke=\"#c64d18\" stroke-width=\"3\"\/><text x=\"289.09090909090907\" y=\"180.72727272727272\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">A\u2032<\/text><text x=\"365.45454545454544\" y=\"180.72727272727272\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">B\u2032<\/text><text x=\"289.09090909090907\" y=\"104.36363636363636\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">C\u2032<\/text><text x=\"487\" y=\"268.09090909090907\" font-size=\"17\">x<\/text><text x=\"253.9090909090909\" y=\"24\" font-size=\"17\">y<\/text><text x=\"70\" y=\"493\" fill=\"#1769aa\" font-size=\"16\">Blue: object<\/text><text x=\"280\" y=\"493\" fill=\"#c64d18\" font-size=\"16\">Orange: image<\/text><\/svg><figcaption>Translation by 4 right and 3 up<\/figcaption><\/figure>\n<p><strong>Check:<\/strong> shape, size and orientation stay the same. A nonzero translation has no invariant points.<\/p>\n<p><strong>Try:<\/strong> translate P(2, \u22123) by 5 left and 2 up.<\/p>\n<h2>2. Rotation<\/h2>\n<p>A rotation turns a shape about a fixed point. State the <strong>centre, angle and direction<\/strong>. For 180\u00b0, clockwise and anticlockwise give the same result.<\/p>\n<h3>Worked example: 90\u00b0 anticlockwise about the origin<\/h3>\n<p>Rotate A(1, 1), B(4, 1) and C(1, 3) about O(0, 0). Use <strong>(x, y) \u2192 (\u2212y, x)<\/strong>.<\/p>\n<ul>\n<li>A\u2032 = <strong>(\u22121, 1)<\/strong><\/li>\n<li>B\u2032 = <strong>(\u22121, 4)<\/strong><\/li>\n<li>C\u2032 = <strong>(\u22123, 1)<\/strong><\/li>\n<\/ul>\n<figure><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 530 510\" role=\"img\" aria-label=\"Rotation through 90\u00b0 anticlockwise about the origin\" style=\"width:100%;max-width:530px;height:auto;background:white\"><title>Rotation through 90\u00b0 anticlockwise about the origin<\/title><path d=\"M 55 35 V 455 M 55 455 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"55\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-5<\/text><text x=\"236.9090909090909\" y=\"459\" font-size=\"12\" text-anchor=\"end\">-5<\/text><path d=\"M 93.18181818181819 35 V 455 M 55 416.8181818181818 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"93.18181818181819\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-4<\/text><text x=\"236.9090909090909\" y=\"420.8181818181818\" font-size=\"12\" text-anchor=\"end\">-4<\/text><path d=\"M 131.36363636363637 35 V 455 M 55 378.6363636363636 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"131.36363636363637\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-3<\/text><text x=\"236.9090909090909\" y=\"382.6363636363636\" font-size=\"12\" text-anchor=\"end\">-3<\/text><path d=\"M 169.54545454545453 35 V 455 M 55 340.45454545454544 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"169.54545454545453\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-2<\/text><text x=\"236.9090909090909\" y=\"344.45454545454544\" font-size=\"12\" text-anchor=\"end\">-2<\/text><path d=\"M 207.72727272727272 35 V 455 M 55 302.27272727272725 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"207.72727272727272\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-1<\/text><text x=\"236.9090909090909\" y=\"306.27272727272725\" font-size=\"12\" text-anchor=\"end\">-1<\/text><path d=\"M 245.9090909090909 35 V 455 M 55 264.09090909090907 H 475\" fill=\"none\" stroke=\"#68758b\" stroke-width=\"2\"\/><path d=\"M 284.09090909090907 35 V 455 M 55 225.9090909090909 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"284.09090909090907\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">1<\/text><text x=\"236.9090909090909\" y=\"229.9090909090909\" font-size=\"12\" text-anchor=\"end\">1<\/text><path d=\"M 322.27272727272725 35 V 455 M 55 187.72727272727272 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"322.27272727272725\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">2<\/text><text x=\"236.9090909090909\" y=\"191.72727272727272\" font-size=\"12\" text-anchor=\"end\">2<\/text><path d=\"M 360.45454545454544 35 V 455 M 55 149.54545454545453 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"360.45454545454544\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">3<\/text><text x=\"236.9090909090909\" y=\"153.54545454545453\" font-size=\"12\" text-anchor=\"end\">3<\/text><path d=\"M 398.6363636363636 35 V 455 M 55 111.36363636363636 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"398.6363636363636\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">4<\/text><text x=\"236.9090909090909\" y=\"115.36363636363636\" font-size=\"12\" text-anchor=\"end\">4<\/text><path d=\"M 436.8181818181818 35 V 455 M 55 73.18181818181819 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"436.8181818181818\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">5<\/text><text x=\"236.9090909090909\" y=\"77.18181818181819\" font-size=\"12\" text-anchor=\"end\">5<\/text><path d=\"M 475 35 V 455 M 55 35 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"475\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">6<\/text><text x=\"236.9090909090909\" y=\"39\" font-size=\"12\" text-anchor=\"end\">6<\/text><polygon points=\"284.09090909090907,225.9090909090909 398.6363636363636,225.9090909090909 284.09090909090907,149.54545454545453\" fill=\"none\" stroke=\"#1769aa\" stroke-width=\"3\"\/><text x=\"289.09090909090907\" y=\"218.9090909090909\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">A<\/text><text x=\"403.6363636363636\" y=\"218.9090909090909\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">B<\/text><text x=\"289.09090909090907\" y=\"142.54545454545453\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">C<\/text><polygon points=\"207.72727272727272,225.9090909090909 207.72727272727272,111.36363636363636 131.36363636363637,225.9090909090909\" fill=\"none\" stroke=\"#c64d18\" stroke-width=\"3\"\/><text x=\"212.72727272727272\" y=\"218.9090909090909\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">A\u2032<\/text><text x=\"212.72727272727272\" y=\"104.36363636363636\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">B\u2032<\/text><text x=\"136.36363636363637\" y=\"218.9090909090909\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">C\u2032<\/text><text x=\"487\" y=\"268.09090909090907\" font-size=\"17\">x<\/text><text x=\"253.9090909090909\" y=\"24\" font-size=\"17\">y<\/text><text x=\"70\" y=\"493\" fill=\"#1769aa\" font-size=\"16\">Blue: object<\/text><text x=\"280\" y=\"493\" fill=\"#c64d18\" font-size=\"16\">Orange: image<\/text><\/svg><figcaption>Rotation through 90\u00b0 anticlockwise about the origin<\/figcaption><\/figure>\n<p>Other rules about the origin:<\/p>\n<ul>\n<li>90\u00b0 clockwise: <strong>(x, y) \u2192 (y, \u2212x)<\/strong><\/li>\n<li>180\u00b0: <strong>(x, y) \u2192 (\u2212x, \u2212y)<\/strong><\/li>\n<\/ul>\n<p><strong>Check:<\/strong> each vertex stays the same distance from the centre. The centre is invariant. For a different centre, subtract its coordinates, rotate the displacement, then add the centre back.<\/p>\n<p><strong>Try:<\/strong> rotate P(2, \u22123) through 90\u00b0 clockwise about the origin.<\/p>\n<h2>3. Reflection<\/h2>\n<p>A reflection makes a mirror image. Give the <strong>equation of the mirror line<\/strong>. The line is the perpendicular bisector of the segment joining each point to its image.<\/p>\n<h3>Worked example: reflection in x = 1<\/h3>\n<p>Reflect A(\u22123, 1), B(\u22121, 1) and C(\u22123, 4). In x = a, the rule is <strong>(x, y) \u2192 (2a \u2212 x, y)<\/strong>.<\/p>\n<ul>\n<li>A\u2032 = (2 \u2212 (\u22123), 1) = <strong>(5, 1)<\/strong><\/li>\n<li>B\u2032 = (2 \u2212 (\u22121), 1) = <strong>(3, 1)<\/strong><\/li>\n<li>C\u2032 = (2 \u2212 (\u22123), 4) = <strong>(5, 4)<\/strong><\/li>\n<\/ul>\n<figure><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 530 510\" role=\"img\" aria-label=\"Reflection in the vertical line x = 1\" style=\"width:100%;max-width:530px;height:auto;background:white\"><title>Reflection in the vertical line x = 1<\/title><path d=\"M 55 35 V 455 M 55 455 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"55\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-5<\/text><text x=\"236.9090909090909\" y=\"459\" font-size=\"12\" text-anchor=\"end\">-5<\/text><path d=\"M 93.18181818181819 35 V 455 M 55 416.8181818181818 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"93.18181818181819\" 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455 M 55 149.54545454545453 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"360.45454545454544\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">3<\/text><text x=\"236.9090909090909\" y=\"153.54545454545453\" font-size=\"12\" text-anchor=\"end\">3<\/text><path d=\"M 398.6363636363636 35 V 455 M 55 111.36363636363636 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"398.6363636363636\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">4<\/text><text x=\"236.9090909090909\" y=\"115.36363636363636\" font-size=\"12\" text-anchor=\"end\">4<\/text><path d=\"M 436.8181818181818 35 V 455 M 55 73.18181818181819 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"436.8181818181818\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">5<\/text><text x=\"236.9090909090909\" y=\"77.18181818181819\" font-size=\"12\" text-anchor=\"end\">5<\/text><path d=\"M 475 35 V 455 M 55 35 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"475\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">6<\/text><text x=\"236.9090909090909\" y=\"39\" font-size=\"12\" text-anchor=\"end\">6<\/text><path d=\"M 284.09090909090907 455 L 284.09090909090907 35\" stroke=\"#984a9c\" stroke-width=\"3\"\/><text x=\"289.09090909090907\" y=\"25\" font-size=\"15\">x = 1<\/text><polygon points=\"131.36363636363637,225.9090909090909 207.72727272727272,225.9090909090909 131.36363636363637,111.36363636363636\" fill=\"none\" stroke=\"#1769aa\" stroke-width=\"3\"\/><text x=\"136.36363636363637\" y=\"218.9090909090909\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">A<\/text><text x=\"212.72727272727272\" y=\"218.9090909090909\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">B<\/text><text x=\"136.36363636363637\" y=\"104.36363636363636\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">C<\/text><polygon points=\"436.8181818181818,225.9090909090909 360.45454545454544,225.9090909090909 436.8181818181818,111.36363636363636\" fill=\"none\" stroke=\"#c64d18\" stroke-width=\"3\"\/><text x=\"441.8181818181818\" y=\"218.9090909090909\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">A\u2032<\/text><text x=\"365.45454545454544\" y=\"218.9090909090909\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">B\u2032<\/text><text x=\"441.8181818181818\" y=\"104.36363636363636\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">C\u2032<\/text><text x=\"487\" y=\"268.09090909090907\" font-size=\"17\">x<\/text><text x=\"253.9090909090909\" y=\"24\" font-size=\"17\">y<\/text><text x=\"70\" y=\"493\" fill=\"#1769aa\" font-size=\"16\">Blue: object<\/text><text x=\"280\" y=\"493\" fill=\"#c64d18\" font-size=\"16\">Orange: image<\/text><\/svg><figcaption>Reflection in the vertical line x = 1<\/figcaption><\/figure>\n<ul>\n<li>In y = b: <strong>(x, y) \u2192 (x, 2b \u2212 y)<\/strong><\/li>\n<li>In y = x: swap coordinates.<\/li>\n<li>In y = \u2212x: <strong>(x, y) \u2192 (\u2212y, \u2212x)<\/strong><\/li>\n<\/ul>\n<p><strong>Check:<\/strong> matching points have equal perpendicular distances from the mirror line. Every point on the mirror line is invariant.<\/p>\n<p><strong>Try:<\/strong> reflect P(2, \u22123) in y = x.<\/p>\n<h2>4. Enlargement<\/h2>\n<p>An enlargement scales a shape. Give the <strong>centre and scale factor<\/strong>. Nonzero scale factors preserve angles; lengths multiply by the absolute value of the scale factor.<\/p>\n<h3>Worked example: scale factor 2 about the origin<\/h3>\n<p>Enlarge A(1, 1), B(3, 1) and C(1, 2). Double each displacement from O(0, 0).<\/p>\n<ul>\n<li>A\u2032 = <strong>(2, 2)<\/strong><\/li>\n<li>B\u2032 = <strong>(6, 2)<\/strong><\/li>\n<li>C\u2032 = <strong>(2, 4)<\/strong><\/li>\n<\/ul>\n<figure><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 530 510\" role=\"img\" aria-label=\"Enlargement by scale factor 2 about the origin\" style=\"width:100%;max-width:530px;height:auto;background:white\"><title>Enlargement by scale factor 2 about the origin<\/title><path d=\"M 55 35 V 455 M 55 455 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"55\" y=\"419.5\" font-size=\"12\" text-anchor=\"middle\">-1<\/text><text x=\"98.5\" y=\"459\" font-size=\"12\" text-anchor=\"end\">-1<\/text><path d=\"M 107.5 35 V 455 M 55 402.5 H 475\" fill=\"none\" stroke=\"#68758b\" stroke-width=\"2\"\/><path d=\"M 160 35 V 455 M 55 350 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"160\" y=\"419.5\" font-size=\"12\" text-anchor=\"middle\">1<\/text><text x=\"98.5\" y=\"354\" font-size=\"12\" text-anchor=\"end\">1<\/text><path d=\"M 212.5 35 V 455 M 55 297.5 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"212.5\" y=\"419.5\" font-size=\"12\" text-anchor=\"middle\">2<\/text><text x=\"98.5\" y=\"301.5\" font-size=\"12\" text-anchor=\"end\">2<\/text><path d=\"M 265 35 V 455 M 55 245 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"265\" y=\"419.5\" font-size=\"12\" text-anchor=\"middle\">3<\/text><text x=\"98.5\" y=\"249\" font-size=\"12\" text-anchor=\"end\">3<\/text><path d=\"M 317.5 35 V 455 M 55 192.5 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"317.5\" y=\"419.5\" font-size=\"12\" text-anchor=\"middle\">4<\/text><text x=\"98.5\" y=\"196.5\" font-size=\"12\" text-anchor=\"end\">4<\/text><path d=\"M 370 35 V 455 M 55 140 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"370\" y=\"419.5\" font-size=\"12\" text-anchor=\"middle\">5<\/text><text x=\"98.5\" y=\"144\" font-size=\"12\" text-anchor=\"end\">5<\/text><path d=\"M 422.5 35 V 455 M 55 87.5 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"422.5\" y=\"419.5\" font-size=\"12\" text-anchor=\"middle\">6<\/text><text x=\"98.5\" y=\"91.5\" font-size=\"12\" text-anchor=\"end\">6<\/text><path d=\"M 475 35 V 455 M 55 35 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"475\" y=\"419.5\" font-size=\"12\" text-anchor=\"middle\">7<\/text><text x=\"98.5\" y=\"39\" font-size=\"12\" text-anchor=\"end\">7<\/text><polygon points=\"160,350 265,350 160,297.5\" fill=\"none\" stroke=\"#1769aa\" stroke-width=\"3\"\/><text x=\"165\" y=\"343\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">A<\/text><text x=\"270\" y=\"343\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">B<\/text><text x=\"165\" y=\"290.5\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">C<\/text><polygon points=\"212.5,297.5 422.5,297.5 212.5,192.5\" fill=\"none\" stroke=\"#c64d18\" stroke-width=\"3\"\/><text x=\"217.5\" y=\"290.5\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">A\u2032<\/text><text x=\"427.5\" y=\"290.5\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">B\u2032<\/text><text x=\"217.5\" y=\"185.5\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">C\u2032<\/text><text x=\"487\" y=\"406.5\" font-size=\"17\">x<\/text><text x=\"115.5\" y=\"24\" font-size=\"17\">y<\/text><text x=\"70\" y=\"493\" fill=\"#1769aa\" font-size=\"16\">Blue: object<\/text><text x=\"280\" y=\"493\" fill=\"#c64d18\" font-size=\"16\">Orange: image<\/text><\/svg><figcaption>Enlargement by scale factor 2 about the origin<\/figcaption><\/figure>\n<p>The base changes from 2 units to 4 units, so the scale factor is 4 \u00f7 2 = 2.<\/p>\n<p>For centre (a, b) and scale factor k:<\/p>\n<p><strong>x\u2032 = a + k(x \u2212 a)<br \/>y\u2032 = b + k(y \u2212 b)<\/strong><\/p>\n<h3>Fractional scale factors<\/h3>\n<p>When 0 &lt; k &lt; 1, the image is smaller and lies between the centre and the object. With k = 1, every point is unchanged.<\/p>\n<p><strong>Try:<\/strong> enlarge P(4, 2) by factor \u00bd about C(2, 0).<\/p>\n<h3>Negative scale factors<\/h3>\n<p>A negative scale factor places each image point on the opposite side of the centre.<\/p>\n<p><strong>Example:<\/strong> enlarge A(1, 1), B(2, 1) and C(1, 2) by factor \u22122 about the origin. Double the distances and reverse their directions.<\/p>\n<ul>\n<li>A\u2032 = <strong>(\u22122, \u22122)<\/strong><\/li>\n<li>B\u2032 = <strong>(\u22124, \u22122)<\/strong><\/li>\n<li>C\u2032 = <strong>(\u22122, \u22124)<\/strong><\/li>\n<\/ul>\n<figure><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 530 510\" role=\"img\" aria-label=\"Enlargement by scale factor \u22122 about the origin\" style=\"width:100%;max-width:530px;height:auto;background:white\"><title>Enlargement by scale factor \u22122 about the origin<\/title><path d=\"M 55 35 V 455 M 55 455 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"55\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">-5<\/text><text x=\"236.9090909090909\" y=\"459\" font-size=\"12\" text-anchor=\"end\">-5<\/text><path d=\"M 93.18181818181819 35 V 455 M 55 416.8181818181818 H 475\" fill=\"none\" stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"93.18181818181819\" y=\"281.09090909090907\" 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stroke=\"#dde3eb\" stroke-width=\"1\"\/><text x=\"475\" y=\"281.09090909090907\" font-size=\"12\" text-anchor=\"middle\">6<\/text><text x=\"236.9090909090909\" y=\"39\" font-size=\"12\" text-anchor=\"end\">6<\/text><polygon points=\"284.09090909090907,225.9090909090909 322.27272727272725,225.9090909090909 284.09090909090907,187.72727272727272\" fill=\"none\" stroke=\"#1769aa\" stroke-width=\"3\"\/><text x=\"289.09090909090907\" y=\"218.9090909090909\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">A<\/text><text x=\"327.27272727272725\" y=\"218.9090909090909\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">B<\/text><text x=\"289.09090909090907\" y=\"180.72727272727272\" fill=\"#1769aa\" font-size=\"17\" font-weight=\"bold\">C<\/text><polygon points=\"169.54545454545453,340.45454545454544 93.18181818181819,340.45454545454544 169.54545454545453,416.8181818181818\" fill=\"none\" stroke=\"#c64d18\" stroke-width=\"3\"\/><text x=\"174.54545454545453\" y=\"333.45454545454544\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">A\u2032<\/text><text x=\"98.18181818181819\" y=\"333.45454545454544\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">B\u2032<\/text><text x=\"174.54545454545453\" y=\"409.8181818181818\" fill=\"#c64d18\" font-size=\"17\" font-weight=\"bold\">C\u2032<\/text><text x=\"487\" y=\"268.09090909090907\" font-size=\"17\">x<\/text><text x=\"253.9090909090909\" y=\"24\" font-size=\"17\">y<\/text><text x=\"70\" y=\"493\" fill=\"#1769aa\" font-size=\"16\">Blue: object<\/text><text x=\"280\" y=\"493\" fill=\"#c64d18\" font-size=\"16\">Orange: image<\/text><\/svg><figcaption>Enlargement by scale factor \u22122 about the origin<\/figcaption><\/figure>\n<p>Lengths double; they do not become negative. To find |k|, divide an image side length by its corresponding object side length. Opposite sides of the centre indicate a negative sign.<\/p>\n<p>To find the centre, extend lines through two pairs of corresponding vertices. Choose pairs giving different lines. Their intersection is the centre.<\/p>\n<p>A scale factor of \u22121 gives the same result as a 180\u00b0 rotation about the centre. Corresponding sides are parallel for both positive and negative factors.<\/p>\n<p><strong>Try:<\/strong> enlarge P(\u22122, 3) by factor \u22121 about the origin.<\/p>\n<h2>Combining transformations: order matters<\/h2>\n<p>Start at P(1, 2). Reflect in the y-axis to get (\u22121, 2), then translate 3 right to get <strong>(2, 2)<\/strong>.<\/p>\n<p>Reverse the order: translate to (4, 2), then reflect to get <strong>(\u22124, 2)<\/strong>. The results differ. Work in the stated order and label intermediate images.<\/p>\n<h2>Exam checklist<\/h2>\n<ul>\n<li><strong>Translation:<\/strong> column vector.<\/li>\n<li><strong>Rotation:<\/strong> angle, direction when needed, and centre.<\/li>\n<li><strong>Reflection:<\/strong> mirror-line equation.<\/li>\n<li><strong>Enlargement:<\/strong> signed scale factor and centre.<\/li>\n<\/ul>\n<p>If asked for a single transformation, give one transformation. Translations, rotations and reflections preserve lengths and angles.<\/p>\n<h2>Check your understanding<\/h2>\n<p>Try these six questions before opening the answers.<\/p>\n<ol>\n<li>Translate (3, \u22122) by 4 left and 5 up.<\/li>\n<li>Rotate (2, 5) through 180\u00b0 about (1, 1).<\/li>\n<li>Reflect (\u22122, 4) in x = 3.<\/li>\n<li>Enlarge (3, 2) by factor \u22122 about (1, 1).<\/li>\n<li>State the invariant points of reflection in y = \u22121.<\/li>\n<li>A corresponding side increases from 3 to 9 units. Object and image are on opposite sides of the centre. Find k.<\/li>\n<\/ol>\n<details>\n<summary><strong>Show answers and explanations<\/strong><\/summary>\n<ol>\n<li>(3 \u2212 4, \u22122 + 5) = <strong>(\u22121, 3)<\/strong>.<\/li>\n<li>The displacement from (1, 1) is (1, 4). Reverse it to (\u22121, \u22124), then add the centre: <strong>(0, \u22123)<\/strong>.<\/li>\n<li>(2 \u00d7 3 \u2212 (\u22122), 4) = <strong>(8, 4)<\/strong>.<\/li>\n<li>(1 \u2212 2(3 \u2212 1), 1 \u2212 2(2 \u2212 1)) = <strong>(\u22123, \u22121)<\/strong>.<\/li>\n<li><strong>All points on y = \u22121.<\/strong><\/li>\n<li>|k| = 9 \u00f7 3 = 3. Opposite sides give <strong>k = \u22123<\/strong>.<\/li>\n<\/ol>\n<h3>Answers to the \u201cTry\u201d prompts<\/h3>\n<ul>\n<li>Translation: (2 \u2212 5, \u22123 + 2) = <strong>(\u22123, \u22121)<\/strong>.<\/li>\n<li>Clockwise rotation: (x, y) \u2192 (y, \u2212x), giving <strong>(\u22123, \u22122)<\/strong>.<\/li>\n<li>Reflection in y = x: swap coordinates to get <strong>(\u22123, 2)<\/strong>.<\/li>\n<li>Fractional enlargement: (2 + \u00bd(4 \u2212 2), 0 + \u00bd(2 \u2212 0)) = <strong>(3, 1)<\/strong>.<\/li>\n<li>Factor \u22121 about the origin: <strong>(2, \u22123)<\/strong>.<\/li>\n<\/ul>\n<\/details>\n<p><em>Based on Quantum Leap Star Limited\u2019s revised Lesson 40 notes. Diagrams constructed independently from the example coordinates.<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Transformations change a shape\u2019s position, orientation or size. This guide from Quantum Leap Star Limited explains translation, rotation, reflection and enlargement, using the examples and practice from our Lesson 40<\/p>\n<p><a href=\"https:\/\/quantumleapstar.com\/?p=173\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\">Four Transformation Techniques<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,8,9],"tags":[],"class_list":["post-173","post","type-post","status-publish","format-standard","hentry","category-gcse-mathematics","category-mathematics-year-10","category-mathematics-year-11"],"_links":{"self":[{"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=\/wp\/v2\/posts\/173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=173"}],"version-history":[{"count":1,"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=\/wp\/v2\/posts\/173\/revisions"}],"predecessor-version":[{"id":174,"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=\/wp\/v2\/posts\/173\/revisions\/174"}],"wp:attachment":[{"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/quantumleapstar.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}