رياضيات الصف العاشر - كتاب الطالب PDF
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6th of October University
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Summary
هذا الكتاب هو كتاب الطالب في مادة الرياضيات للصف العاشر. يغطي الكتاب العديد من المفاهيم الرياضية بما في ذلك الجبر، والحساب المثلثي، والتغير، والهندسة المستوية. يقدم الكتاب شرحاً و تماريناً لمساعدة الطلاب على فهم هذه المفاهيم.
Full Transcript
ﻭﺯﺍﺭﺓ ﺍﻟﺘﺮﺑﻴﺔ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻭﺗﺆﻣﻦ ﻓﺮﺹ ﺗﻌﻠﱡﻢ ّ ﺍﻟﺮﻳﺎﺿﻴّﺎﺕ ﻣﻮﺍﻗﻒ ﺣﻴﺎﺗﻴّﺔ ﻳﻮﻣﻴّﺔ،ﺗﻄﺮﺡ ﺳﻠﺴﻠﺔ ﱢ...
ﻭﺯﺍﺭﺓ ﺍﻟﺘﺮﺑﻴﺔ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻭﺗﺆﻣﻦ ﻓﺮﺹ ﺗﻌﻠﱡﻢ ّ ﺍﻟﺮﻳﺎﺿﻴّﺎﺕ ﻣﻮﺍﻗﻒ ﺣﻴﺎﺗﻴّﺔ ﻳﻮﻣﻴّﺔ،ﺗﻄﺮﺡ ﺳﻠﺴﻠﺔ ﱢ ﺍﻟﻌﺪﺩﻱ ،ﻭﺣ ّﻞ ﺍﻟﻤﺴﺎﺋﻞ، ّ ﻭﺍﻟﺤﺲ ّ ﺗﻌﺰﺯ ﺍﻟﻤﻬﺎﺭﺍﺕ ﺍﻷﺳﺎﺳﻴّﺔ، ﻛﺜﻴﺮﺓ.ﻓﻬﻲ ّ ﻔﻬﻲ ﺍﻟﺸ ّﻭﺍﻟﺠﻬﻮﺯﻳّﺔ ﻟﺪﺭﺍﺳﺔ ﺍﻟﺠﺒﺮ ،ﻭﺍﻟﻬﻨﺪﺳﺔ ،ﻭﺗﻨﻤّﻲ ﻣﻬﺎﺭﺗ َ ِﻲ ﺍﻟﺘّﻌﺒﻴﺮ ّ ﻛﺘﺎﺏ ﺍﻟﻄﺎﻟﺐ ﺍﻟﺮﻳﺎﺿﻴﱠﺎﺕ.ﻭﻫﻲ ﺗﺘﻜﺎﻣﻞ ﻣﻊ ﺍﻟﻤﻮﺍﺩّ ﻭﺍﻟﻜﺘﺎﺑﻲ ﻭﻣﻬﺎﺭﺍﺕ ﺍﻟﺘﻔﻜﻴﺮ ﻓﻲ ﱢ ّ ﺟﺰﺀﺍ ﻣﻦ ﺛﻘﺎﻓﺔ ﺷﺎﻣﻠﺔ ﻣﺘﻤﺎﺳﻜﺔ ﺗﺤﻔّﺰ ّ ﺍﻟﻄﻼﺏ ً ﻓﺘﻜﻮﻥ ﺍﻷﺧﺮﻯ ﺔ ﺍﻟﺪﺭﺍﺳﻴّ ﺣﺐ ﺍﻟﻤﻌﺮﻓﺔ. ﻋﻠﻰ ﺍﺧﺘﻼﻑ ﻗﺪﺭﺍﺗﻬﻢ ﻭﺗﺸ ّﺠﻌﻬﻢ ﻋﻠﻰ ّ ﺗﺘﻜﻮﻥ ﺍﻟﺴﻠﺴﻠﺔ ﻣﻦ: ّ ﻛﺘﺎﺏ ﺍﻟﻄﺎﻟﺐ ﻛﺘﺎﺏ ﺍﻟﻤﻌﻠّﻢ ﺍﻟﺼﻒ ﺍﻟﻌﺎﺷﺮ ﻛﺮﺍﺳﺔ ﺍﻟﺘﻤﺎﺭﻳﻦ ّ ﻛﺮﺍﺳﺔ ﺍﻟﺘﻤﺎﺭﻳﻦ ﻣﻊ ﺍﻹﺟﺎﺑﺎﺕ ّ ﺍﻟﻔﺼﻞ ﺍﻟﺪﺭﺍﺳﻲ ﺍﻷﻭ ﹼﻝ ﺍﻟﻄﺒﻌﺔ ﺍﻟﺜﺎﻧﻴﺔ ISBN 978-614-406-291-3 á«fÉãdG á©Ñ£dG 10 9 786144 062913 ١٤٤٥ھـ ٢٠٢٣ ٢٠٢٢م á```````eó`≤e ﺍﳊﻤﺪﷲ ﺭﺏ ﺍﻟﻌﺎﳌﲔ ،ﻭﺍﻟﺼﻼﺓ ﻭﺍﻟﺴﻼﻡ ﻋﻠﻰ ﺳﻴﺪ ﺍﳌﺮﺳﻠﲔ ،ﻣﺤﻤﺪ ﺑﻦ ﻋﺒﺪﺍﷲ ﻭﺻﺤﺒﻪ ﺃﺟﻤﻌﲔ. ﻋﻨﺪﻣﺎ ﺷﺮﻋﺖ ﻭﺯﺍﺭﺓ ﺍﻟﺘﺮﺑﻴﺔ ﻓﻲ ﻋﻤﻠﻴﺔ ﺗﻄﻮﻳﺮ ﺍﳌﻨﺎﻫﺞ ،ﺍﺳﺘﻨﺪﺕ ﻓﻲ ﺫﻟﻚ ﺇﻟﻰ ﺟﻤﻠﺔ ﻣﻦ ﺍﻷﺳﺲ ﻭﺍﳌﺮﺗﻜﺰﺍﺕ ﺍﻟﻌﻠﻤﻴﺔ ﻭﺍﻟﻔﻨﻴﺔ ﻭﺍﳌﻬﻨﻴﺔ ،ﺣﻴﺚ ﺭﺍﻋﺖ ﻣﺘﻄﻠﺒﺎﺕ ﺍﻟﺪﻭﻟﺔ ﻭﺍﺭﺗﺒﺎﻁ ﺫﻟﻚ ﺑﺴﻮﻕ ﺍﻟﻌﻤﻞ ،ﻭﺣﺎﺟﺎﺕ ﺍﳌﺘﻌﻠﻤﲔ ﻭﺍﻟﺘﻄﻮﺭ ﺍﳌﻌﺮﻓﻲ ﻭﺍﻟﻌﻠﻤﻲ ،ﺑﺎﻹﺿﺎﻓﺔ ﺇﻟﻰ ﺟﻤﻠﺔ ﻣﻦ ﺍﻟﺘﺤﺪﻳﺎﺕ ﺍﻟﺘﻲ ﲤﺜﻠﺖ ﺑﺎﻟﺘﺤﺪﻱ ﺍﻟﻘﻴﻤﻲ ﻭﺍﻻﺟﺘﻤﺎﻋﻲ ﻭﺍﻻﻗﺘﺼﺎﺩﻱ ﻭﺍﻟﺘﻜﻨﻮﻟﻮﺟﻲ ﻭﻏﻴﺮﻫﺎ، ﻭﺇﻥ ﻛﻨﺎ ﻧﺪﺭﻙ ﺃﻥ ﻫﺬﻩ ﺍﳉﻮﺍﻧﺐ ﻟﻬﺎ ﺻﻠﺔ ﻭﺛﻴﻘﺔ ﺑﺎﻟﻨﻈﺎﻡ ﺍﻟﺘﻌﻠﻴﻤﻲ ﺑﺸﻜﻞ ﻋﺎﻡ ﻭﻟﻴﺲ ﺍﳌﻨﺎﻫﺞ ﺑﺸﻜﻞ ﺧﺎﺹ. ﻭﳑﺎ ﻳﺠﺐ ﺍﻟﺘﺄﻛﻴﺪ ﻋﻠﻴﻪ ،ﺃﻥ ﺍﳌﻨﻬﺞ ﻋﺒﺎﺭﺓ ﻋﻦ ﻛﻢ ﺍﳋﺒﺮﺍﺕ ﺍﻟﺘﺮﺑﻮﻳﺔ ﻭﺍﻟﺘﻌﻠﻴﻤﻴﺔ ﺍﻟﺘﻲ ﺗﹸﻘﺪﻡ ﺃﻳﻀﺎ ﺑﻌﻤﻠﻴﺎﺕ ﺍﻟﺘﺨﻄﻂ ﻭﺍﻟﺘﻨﻔﻴﺬ ،ﻭﺍﻟﺘﻲ ﻓﻲ ﻣﺤﺼﻠﺘﻬﺎ ﺍﻟﻨﻬﺎﺋﻴﺔ ﻟﻠﻤﺘﻌﻠﻢ ،ﻭﻫﺬﺍ ﻳﺮﺗﺒﻂ ﹰ ﺗﺄﺗﻲ ﻟﺘﺤﻘﻴﻖ ﺍﻷﻫﺪﺍﻑ ﺍﻟﺘﺮﺑﻮﻳﺔ ،ﻭﻋﻠﻴﻪ ﺃﺻﺒﺤﺖ ﻋﻤﻠﻴﺔ ﺑﻨﺎﺀ ﺍﳌﻨﺎﻫﺞ ﺍﻟﺪﺭﺍﺳﻴﺔ ﻣﻦ ﺃﻫﻢ ﻣﻜﻮﻧﺎﺕ ﺍﻟﻨﻈﺎﻡ ﺍﻟﺘﻌﻠﻴﻤﻲ ،ﻷﻧﻬﺎ ﺗﺄﺗﻲ ﻓﻲ ﺟﺎﻧﺒﲔ ﻣﻬﻤﲔ ﻟﻘﻴﺎﺱ ﻛﻔﺎﺀﺓ ﺍﻟﻨﻈﺎﻡ ﺍﻟﺘﻌﻠﻴﻤﻲ، ﻭﻣﻘﻴﺎﺳﺎ ﺃﻭ ﻣﻌﻴﺎﺭﹰﺍ ﻣﻦ ﻣﻌﺎﻳﻴﺮ ﻛﻔﺎﺀﺗﻪ ﻣﻦ ﹰ ﻓﻬﻲ ﻣﻦ ﺟﻬﺔ ﲤﺜﻞ ﺃﺣﺪ ﺍﳌﺪﺧﻼﺕ ﺍﻷﺳﺎﺳﻴﺔ ﺟﻬﺔ ﺃﺧﺮﻯ ،ﻋﺪﺍ ﺃﻥ ﺍﳌﻨﺎﻫﺞ ﺗﺪﺧﻞ ﻓﻲ ﻋﻤﻠﻴﺔ ﺇﳕﺎﺀ ﺷﺨﺼﻴﺔ ﺍﳌﺘﻌﻠﻢ ﻓﻲ ﺟﻤﻴﻊ ﺟﻮﺍﻧﺒﻬﺎ ﺍﳉﺴﻤﻴﺔ ﻭﺍﻟﻌﻘﻠﻴﺔ ﻭﺍﻟﻮﺟﺪﺍﻧﻴﺔ ﻭﺍﻟﺮﻭﺣﻴﺔ ﻭﺍﻻﺟﺘﻤﺎﻋﻴﺔ. ﻣﻦ ﺟﺎﻧﺐ ﺁﺧﺮ ،ﻓﻨﺤﻦ ﻓﻲ ﻗﻄﺎﻉ ﺍﻟﺒﺤﻮﺙ ﺍﻟﺘﺮﺑﻮﻳﺔ ﻭﺍﳌﻨﺎﻫﺞ ،ﻋﻨﺪﻣﺎ ﻧﺒﺪﺃ ﻓﻲ ﻋﻤﻠﻴﺔ ﺗﻄﻮﻳﺮ ﺍﳌﻨﺎﻫﺞ ﺍﻟﺪﺭﺍﺳﻴﺔ ،ﻧﻨﻄﻠﻖ ﻣﻦ ﻛﻞ ﺍﻷﺳﺲ ﻭﺍﳌﺮﺗﻜﺰﺍﺕ ﺍﻟﺘﻲ ﺳﺒﻖ ﺫﻛﺮﻫﺎ ،ﺑﻞ ﺇﻧﻨﺎ ﻧﺮﺍﻫﺎ ﻣﺤﻔﺰﺍﺕ ﻭﺍﻗﻌﻴﺔ ﺗﺪﻓﻌﻨﺎ ﻟﺒﺬﻝ ﻗﺼﺎﺭﻯ ﺟﻬﺪﻧﺎ ﻭﺍﳌﻀﻲ ﻗﺪ ﹰﻣﺎ ﻓﻲ ﺍﻟﺒﺤﺚ ﻓﻲ ﺍﳌﺴﺘﺠﺪﺍﺕ ﺍﻟﺘﺮﺑﻮﻳﺔ ﺳﻮﺍﺀ ﻓﻲ ﺷﻜﻞ ﺍﳌﻨﺎﻫﺞ ﺃﻡ ﻓﻲ ﻣﻀﺎﻣﻴﻨﻬﺎ ،ﻭﻫﺬﺍ ﻣﺎ ﻗﺎﻡ ﺑﻪ ﺍﻟﻘﻄﺎﻉ ﺧﻼﻝ ﺍﻟﺴﻨﻮﺍﺕ ﺍﳌﺎﺿﻴﺔ ،ﺣﻴﺚ ﺍﻟﺒﺤﺚ ﻋﻦ ﺃﻓﻀﻞ ﻣﺎ ﺗﻮﺻﻠﺖ ﺇﻟﻴﻪ ﻋﻤﻠﻴﺔ ﺻﻨﺎﻋﺔ ﺍﳌﻨﺎﻫﺞ ﺍﻟﺪﺭﺍﺳﻴﺔ ،ﻭﻣﻦ ﺛﻢ ﺇﻋﺪﺍﺩﻫﺎ ﻭﺗﺄﻟﻴﻔﻬﺎ ﻭﻓﻖ ﻣﻌﺎﻳﻴﺮ ﻋﺎﳌﻴﺔ ﺍﺳﺘﻌﺪﺍﺩﹰﺍ ﻟﺘﻄﺒﻴﻘﻬﺎ ﻓﻲ ﺍﻟﺒﻴﺌﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ. ﻭﻟﻘﺪ ﻛﺎﻧﺖ ﻣﻨﺎﻫﺞ ﺍﻟﻌﻠﻮﻡ ﻭﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻣﻦ ﺃﻭﻝ ﺍﳌﻨﺎﻫﺞ ﺍﻟﺘﻲ ﺑﺪﺃﻧﺎ ﺑﻬﺎ ﻋﻤﻠﻴﺔ ﺍﻟﺘﻄﻮﻳﺮ ،ﺇﳝﺎﻧ ﹰﺎ ﺑﺄﻫﻤﻴﺘﻬﺎ ﻭﺍﻧﻄﻼ ﹰﻗﺎ ﻣﻦ ﺃﻧﻬﺎ ﺫﺍﺕ ﺻﻔﺔ ﻋﺎﳌﻴﺔ ،ﻣﻊ ﺍﻷﺧﺬ ﺑﺎﳊﺴﺒﺎﻥ ﺧﺼﻮﺻﻴﺔ ﺍﺘﻤﻊ ﺍﻟﻜﻮﻳﺘﻲ ﻭﺑﻴﺌﺘﻪ ﺍﶈﻠﻴﺔ ،ﻭﻋﻨﺪﻣﺎ ﺃﺩﺭﻛﻨﺎ ﺃﻧﻬﺎ ﺗﺘﻀﻤﻦ ﺟﻮﺍﻧﺐ ﻋﻤﻠﻴﺔ ﺍﻟﺘﻌﻠﻢ ﻭﻧﻌﻨﻲ ﺑﺬﻟﻚ ﺍﳌﻌﺮﻓﺔ ﻭﺍﻟﻘﻴﻢ ﻭﺍﳌﻬﺎﺭﺍﺕ ،ﻗﻤﻨﺎ ﺑﺪﺭﺍﺳﺘﻬﺎ ﻭﺟﻌﻠﻬﺎ ﺗﺘﻮﺍﻓﻖ ﻣﻊ ﻧﻈﺎﻡ ﺍﻟﺘﻌﻠﻴﻢ ﻓﻲ ﺩﻭﻟﺔ ﺍﻟﻜﻮﻳﺖ، ﻣﺮﻛﺰﻳﻦ ﻟﻴﺲ ﻓﻘﻂ ﻋﻠﻰ ﺍﻟﻜﺘﺎﺏ ﺍﳌﻘﺮﺭ ﻭﻟﻜﻦ ﺷﻤﻞ ﺫﻟﻚ ﻃﺮﺍﺋﻖ ﻭﺃﺳﺎﻟﻴﺐ ﺍﻟﺘﺪﺭﻳﺲ ﻭﺍﻟﺒﻴﺌﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ ﻭﺩﻭﺭ ﺍﳌﺘﻌﻠﻢ ،ﻣﺆﻛﺪﻳﻦ ﻋﻠﻰ ﺃﻫﻤﻴﺔ ﺍﻟﺘﻜﺎﻣﻞ ﺑﲔ ﺍﳉﻮﺍﻧﺐ ﺍﻟﻌﻠﻤﻴﺔ ﻭﺍﻟﺘﻄﺒﻴﻘﻴﺔ ﺣﺘﻰ ﺗﻜﻮﻥ ﺫﺍﺕ ﻃﺒﻴﻌﺔ ﻭﻇﻴﻔﻴﺔ ﻣﺮﺗﺒﻄﺔ ﺑﺤﻴﺎﺓ ﺍﳌﺘﻌﻠﻢ. ﻭﻓﻲ ﺿﻮﺀ ﻣﺎ ﺳﺒﻖ ﻣﻦ ﻣﻌﻄﻴﺎﺕ ﻭﻏﻴﺮﻫﺎ ﻣﻦ ﺍﳉﻮﺍﻧﺐ ﺫﺍﺕ ﺍﻟﺼﻔﺔ ﺍﻟﺘﻌﻠﻴﻤﻴﺔ ﻭﺍﻟﺘﺮﺑﻮﻳﺔ ﰎ ﺍﺧﺘﻴﺎﺭ ﺳﻠﺴﻠﺔ ﻣﻨﺎﻫﺞ ﺍﻟﻌﻠﻮﻡ ﻭﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﺍﻟﺘﻲ ﺃﻛﻤﻠﻨﺎﻫﺎ ﺑﺸﻜﻞ ﻭﻭﻗﺖ ﻣﻨﺎﺳﺒﲔ ،ﻭﻟﻨﺤﻘﻖ ﻧﻘﻠﺔ ﻧﻮﻋﻴﺔ ﻓﻲ ﻣﻨﺎﻫﺞ ﺗﻠﻚ ﺍﳌﻮﺍﺩ ،ﻭﻫﺬﺍ ﻛﻠﻪ ﺗﺰﺍﻣﻦ ﻣﻊ ﻋﻤﻠﻴﺔ ﺍﻟﺘﻘﻮﱘ ﻭﺍﻟﻘﻴﺎﺱ ﻟﻸﺛﺮ ﺍﻟﺬﻱ ﺗﺮﻛﺘﻪ ﺗﻠﻚ ﺍﳌﻨﺎﻫﺞ ،ﻭﻣﻦ ﺛﻢ ﻋﻤﻠﻴﺎﺕ ﺍﻟﺘﻌﺪﻳﻞ ﺍﻟﺘﻲ ﻃﺮﺃﺕ ﺃﺛﻨﺎﺀ ﻭﺑﻌﺪ ﺗﻨﻔﻴﺬﻫﺎ ،ﻣﻊ ﺍﻟﺘﺄﻛﻴﺪ ﻋﻠﻰ ﺍﻻﺳﺘﻤﺮﺍﺭ ﻓﻲ ﺍﻟﻘﻴﺎﺱ ﺍﳌﺴﺘﻤﺮ ﻭﺍﳌﺘﺎﺑﻌﺔ ﺍﻟﺪﺍﺋﻤﺔ ﺣﺘﻰ ﺗﻜﻮﻥ ﻣﻨﺎﻫﺠﻨﺎ ﺃﻛﺜﺮ ﺗﻔﺎﻋﻠﻴﺔ. ﺩ.ﺳﻌﻮﺩ ﻫﻼﻝ ﺍﳊﺮﺑﻲ ﺍﻟﻮﻛﻴﻞ ﺍﳌﺴﺎﻋﺪ ﻟﻘﻄﺎﻉ ﺍﻟﺒﺤﻮﺙ ﺍﻟﺘﺮﺑﻮﻳﺔ ﻭﺍﳌﻨﺎﻫﺞ äÉj o ƒn àëŸG ١٠ ﺍﻟﻮﺣﺪﺓ ﺍﻷﻭﻟﻰ :ﺍﳉﺒﺮ -ﺍﻷﻋﺪﺍﺩ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ ١ - ١ﺧﻮﺍﺹ ﻧﻈﺎﻡ ﺍﻷﻋﺪﺍﺩ ﺍﳊﻘﻴﻘﻴﺔ ١٢.................................................................................................................................................................................................................... ٢ - ١ﺗﻘﺪﻳﺮ ﺍﳉﺬﺭ ﺍﻟﺘﺮﺑﻴﻌﻲ ١٨.......................................................................................................................................................................................................................................... ٣ - ١ﺣﻞ ﺍﳌﺘﺒﺎﻳﻨﺎﺕ ٢٢........................................................................................................................................................................................................................................................... ٤ - ١ﺍﻟﻘﻴﻤﺔ ﺍﳌﻄﻠﻘﺔ ٢٨..................................................................................................................................................................................................................................................... ٥ - ١ﺩﺍﻟﺔ ﺍﻟﻘﻴﻤﺔ ﺍﳌﻄﻠﻘﺔ ٣٦.......................................................................................................................................................................................................................................... ٦ - ١ﺣﻞ ﻧﻈﺎﻡ ﻣﻌﺎﺩﻟﺘﲔ ﺧﻄﻴﺘﲔ ٤٣....................................................................................................................................................................................................................... ٧ - ١ﺣﻞ ﻣﻌﺎﺩﻻﺕ ﻣﻦ ﺍﻟﺪﺭﺟﺔ ﺍﻟﺜﺎﻧﻴﺔ ﻓﻲ ﻣﺘﻐﻴﺮ ﻭﺍﺣﺪ ٤٨......................................................................................................................................................................... ٦٠ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺜﺎﻧﻴﺔ :ﻭﺣﺪﺓ ﺣﺴﺎﺏ ﺍﳌﺜﻠﺜﺎﺕ ١ - ٢ﺍﻟﺰﻭﺍﻳﺎ ﻭﻗﻴﺎﺳﺎﺗﻬﺎ ٦٢................................................................................................................................................................................................................................................... ٢ - ٢ﺍﻟﻨﺴﺐ ﺍﳌﺜﻠﺜﻴﺔ :ﺍﳉﻴﺐ ﻭﺟﻴﺐ ﺍﻟﺘﻤﺎﻡ ﻭﻣﻘﻠﻮﺑﺎﺗﻬﻤﺎ ٦٩.................................................................................................................................................................... ٣ - ٢ﻇﻞ ﺍﻟﺰﺍﻭﻳﺔ ﻭﻣﻘﻠﻮﺑﻪ ٧٥........................................................................................................................................................................................................................................... ٤ - ٢ﺍﻟﻨﺴﺐ ﺍﳌﺜﻠﺜﻴﺔ ﻟﺒﻌﺾ ﺍﻟﺰﻭﺍﻳﺎ ﺍﳋﺎﺻﺔ ٨٠.................................................................................................................................................................................................. ٥ - ٢ﺣﻞ ﺍﳌﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ٨٤..................................................................................................................................................................................................................................... ٦ - ٢ﺯﻭﺍﻳﺎ ﺍﻻﺭﺗﻔﺎﻉ ﻭﺍﻻﻧﺨﻔﺎﺽ ٨٧................................................................................................................................................................................................................................. ٧ - ٢ﺍﻟﻘﻄﺎﻉ ﺍﻟﺪﺍﺋﺮﻱ ﻭﺍﻟﻘﻄﻌﺔ ﺍﻟﺪﺍﺋﺮﻳﺔ ٩٠.......................................................................................................................................................................................................... ٩٨ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺜﺎﻟﺜﺔ :ﺍﳉﺒﺮ -ﺍﻟﺘﻐﻴﺮ ١ - ٣ﺍﻟﻨﺴﺒﺔ ﻭﺍﻟﺘﻨﺎﺳﺐ ١٠٠........................................................................................................................................................................................................................................... ٢ - ٣ﺍﻟﺘﻐﻴﺮ ﺍﻟﻄﺮﺩﻱ ١١٠..................................................................................................................................................................................................................................................... ٣ - ٣ﺍﻟﺘﻐﻴﺮ ﺍﻟﻌﻜﺴﻲ ١١٨................................................................................................................................................................................................................................................ ١٢٦ ﺍﻟﻮﺣﺪﺓ ﺍﻟﺮﺍﺑﻌﺔ :ﺍﻟﻬﻨﺪﺳﺔ ﺍﳌﺴﺘﻮﻳﺔ ١ - ٤ﺍﳌﻀﻠﻌﺎﺕ ﺍﳌﺘﺸﺎﺑﻬﺔ ١٢٨..................................................................................................................................................................................................................................... ٢ - ٤ﺗﺸﺎﺑﻪ ﺍﳌﺜﻠﺜﺎﺕ ١٣٥.................................................................................................................................................................................................................................................... ٣ - ٤ﺍﻟﺘﺸﺎﺑﻪ ﻓﻲ ﺍﳌﺜﻠﺜﺎﺕ ﻗﺎﺋﻤﺔ ﺍﻟﺰﺍﻭﻳﺔ ١٤٧...................................................................................................................................................................................................... ٤ - ٤ﺍﻟﺘﻨﺎﺳﺒﺎﺕ ﻭﺍﳌﺜﻠﺜﺎﺕ ﺍﳌﺘﺸﺎﺑﻬﺔ ١٥٢............................................................................................................................................................................................................. ﺍﻟﺮﺑﻂ ﺑﺎﻟﺘﻌﻠﻢ ﺍﻟﺴﺎﺑﻖ :ﺍﻟﻌﻼﻗﺔ ﺑﲔ ﻣﺤﻴﻄﻲ ﺷﻜﻠﲔ ﻣﺘﺸﺎﺑﻬﲔ ﻭﺍﻟﻌﻼﻗﺔ ﺑﲔ ﻣﺴﺎﺣﺘﻴﻬﻤﺎ ١٦٠......................................................................... ١٦٨ ﺍﻟﻮﺣﺪﺓ ﺍﳋﺎﻣﺴﺔ :ﺍﳌﺘﺘﺎﻟﻴﺎﺕ )ﺍﳌﺘﺘﺎﺑﻌﺎﺕ( ١ - ٥ﺍﻷﳕﺎﻁ ﺍﻟﺮﻳﺎﺿﻴﺔ ﻭﺍﳌﺘﺘﺎﻟﻴﺎﺕ )ﺍﳌﺘﺘﺎﺑﻌﺎﺕ( ١٧٠................................................................................................................... ٢ - ٥ﺍﳌﺘﺘﺎﻟﻴﺔ ﺍﳊﺴﺎﺑﻴﺔ ١٧٧.................................................................................................................................................... ٣ - ٥ﺍﳌﺘﺘﺎﻟﻴﺔ ﺍﻟﻬﻨﺪﺳﻴﺔ ١٨٦................................................................................................................................................. IóMƒdG IóMƒdG É¡«∏Y äÉ«∏ª©dGh OGóYC’G - ôÑédG á°ùeÉÿG ¤hC’G Algebra - Numbers and Operations º¡°S’C G AGô°T :IóMƒdG ´hô°ûe ! "#$ % &' ( ) *+,- :% ./ 0/ 1 .!/2 3 4. 2 #$ $ 56 78+ 9;+.- ? 9, 2 >@ 95$- A- @ . 95$B. >#@C DE ># F= @ G 95H :" 5 I J 9$= K =- −./.- _H g4 95$B h) ,% &' V JH #$ C ijie 8 N 95 , * )$ @ 5V J. kU ?c=V 2l - = ` h00 ' ? .L .R$B . 4m % & *4 g 4K6E & +`6 E & *R `> 45 +6 *R 2a.I g 4K6E & +`6 E & π +, *R N 4a , π 8VXJ 45 6I +, F.P ( i *R2 4I ; .B6E L5 K.. > L Z 2a HH- 2a 2.c> 1.E 2.c> !KR* Z 2a +62.c> -E $ = L , L L5 k5 $ 2a ( 2.c> ( [ + 6 FMD. l)* m.00 C B.2+ ^ C B.2+ C B :.d/- − ,VX , :.d/- 8 8 8F π. C B ,I ,1 ,f ,1− ,I− , JF8 1jOSOOS :.l .)2M C B ,O ,I ,1 ,f $ 2a FD HH- 2a 2.c> !m.E $ HH g ; 2 !.E FD n 1/2 MHP !I FD n 1/2 MH !.* +HH 2 !I g4 [ S− O− I− 1− f 1 I O S 1 3d/ J2 ^ C` 5@- x 2 C` . C B / @x - C S1F [ 1 − - e 1jf1ff1fff1 C fjOOO g4 : J2 C #' 1 − - e J2 ^ C #' S1F [ J2 C #' 1 = fjO = fjOOO g4 O J2 ^ C #' 1jf1ff1fff1 C >- 3 πe ,1jS , SF :J2 ^ C` 5@- x 2 C` . C B / @x - C 1 O 8 á«≤«≤ëdG OGóY’C G ≈∏Y Üô°†dGh ™ªédG »à«∏ªY ¢UGƒN - 2 Properties of Addition and Multiplication of Real Numbers :Lo, p " GC ,e , !K [w !.L ^e=e^ +e=e+ ] GC ^ e ^ = GC ^ e ^ ) GC + e + = GC + e + .c6 = ^=^ = +=+ * -. ≠ = ^ = ^ = + − = − + @ 3K . GC ^ + e ^ = GC + e ^ *r6 ^ GC + ^ e = ^ GC + e Order of Real Numbers á«≤«≤ëdG OGóYC’G Ö«JôJ - 3 ( [2 r>4 N D6O HH * 2 k5 P4 H> AM6P P5 + * n ,.2/.# !/ J'.00 C B.# !/ $0' sk *t 5 s> ":5t 5 s> I5t > ; 2 L9 H $ = 5 > 5.5 L E L 4nC uC> +HH 2 !K 7V< $7V .6 * $V7− = 7V- 3 1 2) F i Ie F C IeF ± g4 1riF− [ 1F - b .2 ^ C - V 5w) ,.2 C - J'.) =&! z) 1f 1ff F i ± = 1 ± ,1j1− = 1jI1F− ,11 = 1I1F :.2+ =&! / ]d H ` F fjreSs O ,IjIOr eF :.2+ ^ =&! / s I 3d/ J2 ^ - x 2 C` J@ / C > / C b J2 C = rSF - J2 ^ C IjI1OeiSO…− SjiF −.2$. $ [ :»°VÉjQ í∏£°üe J2 ^ C fjOssirSSsO sF 1.2$. $ g4 :.@).q JH >- 3 C #' J2+ C ) J2 ^ - 2 C̀ J@ / C > > / C b I x A=C =l/ - o+/ 1OF - C #' J2+ ^ C ) = l >C o+/ ^ rIeF − [ 1fffF − g4 I 1e F C Estimating Square Roots á«©«HôàdG QhòédG ôjó≤J - 1 $Perfect Squares /> I ( > 1.E M 2a ( > 1I 11 1f i s r e S O I 1 J)2M C ) 1SS 1I1 1ff 1 rS Si Or Ie 1r i S 1 /l $ O - ' N D6O L 6 4nc . *H6 /> K ( . N D6O K.* < :á«°VÉjQ áeƒ∏©e O 3d/ 4nC C> 25 ka o D = b D 9, ,1ejS1F 4#@ / )2I @C A- C b $YvP uE6 C. 6 : 1r ,i /l ) #' 1ejS1 1r > 1ejS1 > i C l J) =&! $ 1rF > 1ejS1F > iF F2 S > 1ejS1F > O #' S / 2` @D >#l@ 1ejS1F >H 1r ( [D- 1ejS1 C ) > T S ,O #' 1ejS1F ` Oji - Oj 2` @0 A@ >- 3 o D = b D 9, ,OfjF− C ) 4#@ / L @C A- C b O .2$. $ .) =&!.2@0. D C!@ l @ S 3d/ .2$. /̀ / c / *y4 [DB o D 4- 9, ,IjrOF 0@ / @C A- C b : Or ,Ie /l ) #' IjrO Or > IjrO > Ie C l J) =&! $ OrF > IjrOF > IeF F2 r > IjrOF > e r ,e #' IjrOF ` F 28$63 = 5$350 700 889 :.2$. $ ejS 2` @0 A@ IjrOF >- A- >- 3 .2$. /̀ / c / *y4 [DB o D 4- 9, ,1OjsF 0@ / @C A- C b S e 3d/ 9$ s ,9$ e '. 80 o@ N J)6 E#I Td/ 3#I 4- : =#^dH.@; sS = Si + Ie = Is + Ie 1 ,rS /l ) 0@ sS :ôcòJ 9$ i , #' Td 3#I ∴ ,.@ y 98D Td JH jrfIO s + eF.2$. $ = # 3#I / 9$ jr Td 3#I J#I J) / %# !/ . 80.@ y J)6 >- 3 9$ 1O ,9$ i '. 80 o@ N J)6 E#I ,.@ y 98D Td/ 3#I 4- e á«JÉ«M äÉ≤«Ñ£J r 3d/ Kq J #d > /y =/B % %Sji = %.C) 2 =/- i %Sji = % i = I> Sji.6#H/ Sji F i − = > - i => Sji F.2$. $ . , 1jOee > _=B ( 9! G . , ( # y@ A- >- 3 ` 1S % + , K6. ( * 6. n# $L 6, K6> L 6 * 6> 8 > 3 , 7 > X + 3 6 * 6. L5 6 L Q. ND6O / +@2/.4& (B.4= q/ +@2/ HM - JH 5 - O>F s>S+F :IóYÉ°ùe äÉë∏£°üe Solving Inequalities äÉæjÉÑàªdG πM C >- "J85 E". J+) "6. m ABE +K ,K m ( /. m N D6O I ; 5 .C ; .B6E m m P5 * 6> !-E l @ E C / ^ 3# ` > mH-6 Y > !-6*.> m /- > + m `P& 2 * 6. LK* P; 5 $ %I +, $uE6 [2 w cEm TE -: m > mH-E ,T i > & ; $ .C ; äÉæjÉÑàªdG πM »a »©ªédG ¢Sƒμ©ªdG á«°UÉN ΩGóîà°SG 1 3d/ .L / P0 9, ,C B FV ( x 3# d/ I− > s − F.+@2 .# !/ 4- I− > s − F : HM ( s− C ) J) ! F#l) .H6 ? , "I JH q 6 s + I − > s + s − F F e>F e ,∞− :.# !/ − 8 X J = 7 b :P0 x e = F h / P0 :1 c#M .M2 .C) ].M2 .C) v I− = s − F F e g _# I − =? s − e ✓ I− = I− .+@2 JH z@#) v.D].L / P0 :I c#M :ôcòJ I− > s − F d (.# s − S t C ) >- J+) ,J ✓ I− > O− JH `+ w/ I− > s − F.+@2 #' e > F ?& ,P0 I ,1 #M / d (.0q c8 w/ C ) >- J+) ,J >- 3 JH :J@ / l C B FV ( .# !/ d/.+@2 .# !/ 4- 1 e − F 1I [ 1≤S−n - I 3d/ /̀ 4# Se .)/B >N @y@ E-b > M U U :v80 .)/B ?. d.20 l >N# H ,/̀ 4# 1s ?20 >N > v : 9! Se @y@ E. d.20 >N 8N (B.20 >N N = N l Se ≥ N + 1s.+@2 1s− J) ! F#l) .H6 ? "I JH q 6 Se + 1s − ≥ N + 1s + 1s− F I ≥ N 9! I (. d ?20 >N @ >- l @ >- 3 Y _ JH )0/ Off g F= JH.8.0 I ?.$= 3# H.0 / 9'=#w l @ @& []M C 9lH ,2I ` i U) +@2 JH J w F#l) .LV $ v.D] tl ,v$ C (.+@2/ JHI 90 - v$ C JH.+@2/ JHI [w / + 8 O 3d/ F C B FV ( x 3# d/ ,1 >.+@2 .# !/ 4- I− F :Ió«Øe áeƒ∏©e 1> : x [6 I− ^ 1 < m m ^ I− I− F >H ,f < g4 ,[ > > J w F#l) JH HM / ] [ I− > , g4 [ > g4 v.D] tl I− g4 g4 F I− < F >H ,f > g4 ,[ > > [ X− − X x d/ $ < , g4 [ < g4 ∞ ,I− = .# !/ g4 g4 >- 3 [ C B FV ( x 3# d/ ,1 ≤.+@2 .# !/ 4- O S @ ! .LH S 3d/ C >C / h m A5 L#..L- y@ ` @ 4 / 9 @5U x : =+@C S eff DB ( >#l@ C e g ` w/ C ! C :.+@2 C ! C = > l S eff ≤ >e e (.+@2 JHI 9D S eff ≤ >e e e F iff ≤ > DB ( @ 4 / iff [&! >- y@ 2- C` 7+@ e g ` w/ iff / 2- C A- ,S eff #' iff ^ e >B.#0)/. 4Y :. 4Y.#0)/ / P0 S eff / >- 3 ` 9lH ,9! f @y+ >N F$#/ >- _H 9! 1 fff K +H JH ) /.# ( DB S ) l @ ]@y ?>/ 95 @ >- X :äGƒ£îdG IOó©àe äÉæjÉÑàe πM 16 1 a C4 > ( * 6> Cn.P +, c ( I[ ND6 O $M0 > I5 N D6O P; 5 ( * 6. ! u/M6* + !.E ^ vM I[ ,− !.E ^ .- I[ ,3 `c. !.E L5 62 $ `/ ?`vP O > I − F.+@2 - I]2 $ .+@2 .4& > O>I−F.+@2 JHI ( I+.H6 > I+O>I+I−F I]2 N- "& F2 > e>F.@< e > F ` ,* .+@2 JHI ( O−.H6 O − 1− < O − O + FI > I]2 N- "& F2 S− < FI >.@< # !/ ( "I 9D S− < FI > @/ I I I− I− < F ` @ I] / 2- * wV.I] J e 3d/ C B FV ( .# !/ db / 1 ≤ O − I + I :.+@2 .# !/ 4- : 1 ≤ O − I + I @N#.b LV 1≤ O−S+ I 3 Y.b LV 1≤S+ O− I F2 1≤S+ −.+@2 JHI / S kI S−1≤S−S+ − O−≤ − v.D] tl)+ O≥ − − 8 X O ,∞− = .# !/ >- 3 I ≥ F e + S + F O - :C B FV ( bd/ 9,.+@2 .# !/ 4- e O > FI −1 ≥ O− [ á«JÉ«M äÉ≤«Ñ£J r 3d/ .w2 9.N - A@ >-. / ,.w2 9.y vL (.H+/.U h6 ?L &' ' V=B J' '=)$- : 5 JH >y 5@@ J #2) C F l Ie + FO :* . FO − e + FS F2 Ie > e + F.+@2 JHI / e kI e − Ie > e − e + F If > F = JH c#2 Of / d- >y A@ + c#2 If / D- #2) C >#l@ / + wH- >#l.H+ . >- A- 5 (B. / o/) J02@ >- >y vL ( & ,_ wH- t.H+ . o6) / ∴ >- 3 ?.w2 9. k E H/M. Y6. Lo, F C ) J) ! á«≤«≤ëdG OGóYÓC d á≤∏£ªdG ᪫≤dG ¢UGƒN ¢†©H " e , K |e| × | | = |e × | O | |=| −| I ≤| | 1 | | | − e| = |e − | r ≤| | e ≠ e f , = S |e| e 1 3d/ .0M . 0 y/= $ >C |S − F| G@) - : f < S − F T S−F f = S − F T f > S − F T S − F−. f = |S − F| S≤F :S−F = S>F :F−S. >- 3 .0M . 0 y/= b G@) - 1 $ >C J@ / |O + F| - |FI − S| [ b á≤∏£e ᪫b øª°†àJ ä’OÉ©e πM $H/M> . .B6E (& > !- 25 F0 N D6O K.* $8 = |3| . / ! H. +P !.6 m * O O O A #l − = F - = F :#' = |F|.C) >H 24#/ ` 00 x C̀ > 1 5+ 2) . .# ! { , −} .# !/ ∅ - { } @y/ ∅ 5.# !/ = |F|.C) >H 2$ ` 00 x C̀ > I.{٠} 5.# !/ = |F| >Hf = > O I 3d/ .L / P0 9, ,s = |O − nI| :.C) .# !/ 4- s = |O − nI| : s− - s >#l >- l @ O − nI. D s − = O − nI - s = O − nI .C) JHI ( O+.H6 S− = nI 1f = nI I ( "I . D I− = n e=n {e ,I−} = .# !/ I− = n / + e = n / + :P0 s = |O − nI| s = |O − nI| s =? |O − I−I| s =? |O − eI| ✓ s = |s−| ✓ s = |s| >- 3 b .# !/ 4- I .L / P0 9, ,C) / = |O + Fe| - f = |1 − FI| [ < "I JH.0M . 0 w@ A& 2) 6# - , #M cC )/ 8/ + O 3d/ f = O + |1 + FI| :.C) .# !/ 4- f = O + |1 + FI| : O− = |1 + FI| v$ C f > O− > T ∅ = .# !/ ∴ >- 3 f = |S + FI−| + e :.C) .# !/ 4- O S 3d/ 11 = e − |O + FI|S.C) .# !/ 4- 11 = e − |O + FI|S :.C) JHI ( e.H6 1r = |O + FI|S S ( "I . D S = |O + FI| S− = O + FI - S = O + FI.C) JHI ( O−.H6 s− = FI 1 = FI I ( "I . D s− = F 1=F I I I I/ * s− , 1 = .# !/ >- 3 b .# !/ 4- S :C) / f = O + |S − Fe| [ f = r− |S + FI|8 - 8.0@I - EC) n− = F - n = F w ,c .0@I |n| = |F|.C) + .# !/ @ 3#5! z@#) 90 / P0 .!+.C) 9, HM e 3d/ |1 + | = |O − I| :.C) .# !/ 4- : c .0@I :È- V¡ J) ! F#l) #' 5+/ - ,>@/ ' ` 5- E 1 − − = O − I - 1+ =O− I O + 1− = + I O+1= − I I= O S= I= O O / * I , S = %# !/ HM .0@I :` , `|1 + |j = `|O − I|j I I I = I| | I 1 + = IO − I :∂JÉeƒ∏©e ≈dGE ∞°VCG 1 + I + I = i + 1I − I S f = + 1S − I O F = |F| = | F| I I I f = I − OS − I = - S = O |1 + | = |O − I| :P0 :á«°VÉjQ áeƒ∏©e I = / + S= / + >H |n| = |F| > O n− = F - n = F 1 + I =? |O − I × I| |1 + S| =? |O − S × I| O O `|n|j = `|F|j e = e− I I b >E#20/ >] P P |e| = |e| O O O/ *S , I = .# !/ 8 >- 3 : C) / .# !/ 4- e |s − F| = |e − F| [ |O + nI| = |e − n| - .0@I 9, c .0@I $ :+ 6 - 1/2 `,I 5 +, H/M> . .B6E (& > ! T nI K.* r 3d/ :∂JÉeƒ∏©e ≈dEG ∞°VCG I − FO = |O + FI| :.C) .# !/ 4- |F| = IFF I − FO = |O + FI| : 3 f F = IFF v!@ ` ,.0M . 0 C#4.! v$ ^.C) @B "M >- 9) :U Gw ?& v$ ^.C) @B "M >#l@ >- a I A - / 2- F 9D 20 I ≤ F A- f ≤ I − FO O O I i ∞ , O E J' z@#).# !/ >- A- I + FO− = O + FI - I − FO = O + FI O − I = FO + FI O − I− = FO − FI 1− = Fe e− = F− 1− e= F :Ió«Øe áeƒ∏©e = F e.# !/ J' .# !/ I I i ∞ , O E ∈ 1e − i ∞ , OE ∈ e ∴ ∴ z@#).# !/ /.8y4 _#H/ 1 − = F ∴ 3#20/ e = F ∴ e {e} = .# !/ >- 3 I + F = |1 − FS| :.C) .# !/ 4- r 8 :ôcòJ á≤∏£e ᪫b øª°†àJ äÉæjÉÑàe πM 1 ≥ |F| $ 2a F0 N D6O H/M>. ; K.* ; .B6E ( * 6> ! B*5 ) >- J+) $ ≥ |3| * 6. / a +P !.6 m * 8− − − 8 < F $ ≤ |3| * 6. / +P +P !.6 m * / qL- #' 8− − − 8 A@ - 9 ) FV ( 1 2` 4#/ ` 00 C` l C B ≥ F ≥ − aHl ≥ |F| 1 J+) 1 ≤ |F| F ) >- − ≥ F - ≤ F aHl ≤ |F| I 2- #' < A@ - / FV ( 1 C B s 3d/ C B FV ( .# !/ db / ,1I ≥ S + |1 + FI|S.+@2 .# !/ 4- 1I ≥ S + |1 + FI|S :.+@2 JHI ( S−.H6 ≥ |1 + FI|S S ( "I . D I ≥ |1 + FI|.RHl .+@2 . I ≥ 1 + FI ≥ I− 1−.H6 1 ≥ FI ≥ O− I (. 0 1 ≥ F ≥ O− 8− I I 1 , O BI I B − = .# !/ − − >- 3 C - FV ( .# !/ bd/ fjr > S − F 1.+@2 .# !/ 4- s e I 88 3d/ C B FV ( db / ,e < 1 − |S − O|I :.+@2 .# !/ 4- e < 1 − |S − O|I :.+@2 JHI ( 1.H6 r < |S − O|I I ( "I . D O < |S − O|.RHl .+@2 . O− > S − O - O O s< O O ( "I . D 1> s< O O − − 7 1 s i O ,∞−i ∪ i∞ , O i = .# !/ 8 8 >- 3 C B FV ( bd/ s ≤ F − OS :.+@2 .# !/ 4- (»FGôKEG) á«JÉ«M äÉ≤«Ñ£J i 3d/ 9$ 1 ( @y@ E MV /' / 9$ Se. c (// c8 C MD 3#I 2@ :.6@= U &' P0 (// c8 C MD 2).0M/. D w.+@2/ v - C - FV ( 5db /.#20 M0 3#I 9D 4- [ : 9$ Se 0+@ - @y@ E F > T ,.$ c (// c8 C MD 3#I F l 1 ≥ |Se − F| P0 F 9D >H ,9$ 1 / d 1 ≥ Se − F ≥ 1− Sr ≥ F ≥ SS Sr ,SS ( J +.#20 M0 3#I 9D >- A- Sr ,SS = .# !/ X X8 XX XJ X= X7 >- 3 4=C 2).0M/. D w.+@2/ v fjI k $ /' / S J' 9I M .6# .4=C i C - FV ( b 9, 5 .#20 .6# 8X á«JÉ«M äÉ≤«Ñ£J 1f 3d/ 5 N / P0 #2) z) cC#! vD / = @ /̀ 4 Sef c=& P8D= c#2 >N 2@ 94 e c=& c#2 >N 5 N K N- b 2.+@2/ v : 94 e / d 2 >N# 0@ - 5 N @y@ J J'.#20 ^ #2) c#2) >N F l e < |Sef − F| A- e− > Sef − F - e < Sef − F SSe > F - See < F X8J XX XXJ XJ XJJ X= >- 3 KN# / P0 + /̀ 4 sef >y #2 JH !d ] - _)@ 1f /̀ 4 Sf ) c#2) >N 5 N C B FV ( db /.#20 #2) > N- b 2.0M/. D w.+@2/ v 8J á≤∏£ªdG ᪫≤dG ádGO 5-1 Absolute Value Function º∏©àJ ±ƒ°S D+ l- A- I− = S− = − S O I I (++ F-= w@ n ,F.2 N¢ 9D 3 4 w 1 S− O− I− 1− f F r− e− S− O− I− 1− f 1 I n S I f I S >- 3 x 9$= 1 |O + FI|− = n :. 8= I 3d/ .0M . 0 y/= $ >C 5 ) I + |O − F| = n. x 9$= : .0M . 0 y/= $ >C. l ) I + |O − F| = n f≤O−F: I+O−F =n f > O − F : I + O − F− O≤F: 1−F =n r O>F: e + F− e [− S I ,O = g4 , . (++/ F-= O h I x x 9$ :/ ] 1 O > F T e + F− = n ,O ≤ F T 1 − F = n I− 1− f 1 I O S e r 1− 1 I O F e S O F I− S O I n S O I n >- 3 .0M . 0 y/= $ >C 5 ) O − 1 + F 1 = n :. x 9$= I I = C b.2l O 3d/.$= N #!./) ,l JH ./).2l o$= / 9' 3y+/ 0@ I− 1− f 1 I O S e r s N #! = b @ 90/ FV (.,]d D # &' 4# = b 9' 3y+/ )2@ 9 t) !E JH 9 s o+./).2l )2 9 I = b .$= )2 ?.$= 3y+ )2 Jd/./).2l 3y+ )2 > : 90 F ( 3y+ D#/ F l .2l 3y+ )2 |F − s| ,.$= 3y+ )2 |I− − F| ∴ |I + F|I = |F − s|∴.0M . 0 n #V |S + FI| = |s − F| 87 S − FI− = s − F - S + FI = s − F S − s = FI + F S −s − = F − FI O = FO ? ,.6#H/ 11− = F 1=F ./).2l .5! = b 9 1 9' 3y+/ )2@ >- 3 = b 9 S )2./).2l h . D ,O 3d/ JH O á«°Sóæ¡dG äÓjƒëàdG ¢†©H ΩGóîà°SÉH ≥∏£ªdG ∫GhO ¿É«H º°SQ Graph of Absolute Value Functions Using some Geometric Transformations .0M . 0 3 C z) 9$= JH )` / +,E - x $-= - x 0H- [ E . NY "#$ S 3d/ I − |F| = n ,|F| = n : / > 9$= |F| = n. J 2 9$ I− |F| = n. J 2 9$ F2@ G GL : x 9$= 9, ,9D 3 4 +L e S − |3| = |3| = 3 |F| = n O I I S S− 1 f I I− r− e− S− O− I− 1− f 1 I O S e r 1− I− f f I− I − |F| = n O− f I I I S S |F| = n. D / I g qL- I− |F| = n. D >#l ,F q . D l H-6/ U H *-E ?H/ C !.2 T K.* :.;]/ 8 z)2 J$- [ E / J0HB [ E =@ ( c 3 ,|F| = n. J 2 9$ [ #' v4#/ J00 C 3 T |3 + F| = n. J 2 9$ .54 ( c 3 , |F| = n 4 . [ #' |3 − F| = n. J 2 9$ ?& =.54 r 3d/ |F| = n 4 . C 2)/ ` ,. NY /̀ /. C x 9$= 9, 3 [ E.H/. D C b , / l |O − F| = n [ |I + F| = n - : : O = 3 ,|F| = n J' 4 . C I = 3 ,|F| = n J' 4 . C (. NY J+) − c=UY = (. NY J+) + c=UY ( ], |F| = n 9$= kN- = ( |F| = n 9$= kN- r r e e n S |= S |F n O O |= n |I F I +F |= I − |F |O |= 1 1 n e− S− O− I− 1− f 1 I O S e r s s− r− e− S− O− I− 1− f 1 I O S e 1− 1− >- 3 e + F = n. 9$ [ E 4 . C $ r I X =.54 ( c 3 ,|F|− = n. J 2 9$ [ #' +k ∋ 3 T |3 + F| − = n :. J 2 b 9$ .54 ( c 3 ,|F| − = n 4 . [ #' |3 − F| − = n. J 2 9$ ?& s 3d/ |F| − = n 4 . C 2)/ ` ,. NY /̀ b /. C x 9$= 9, ,3 [ E.H/. D C , / l |S − F|− = n [ |S + F| − = n - : : S = 3 ,|F|− = n : 4 . C S = 3 ,|F| − = n 4 . C .54 ( .) =- [ E J+) S− =.54 ( .) =- [ E J+) S+ . x 9$= 9, f ,S F- 6 . x 9$= 9, f ,S− F- 6 1 r− e− S− O− I− 1− 1 I O S e r s i i− − s− r− e− S− O− I− 1− 1 I O S e 1− |S 1− I− −F I− =n n |F = O− |− O− |− |− |− =n +F =n | F S− S− |S e− e− r− r− >- 3 [ E /̀ /. C b x 9$= 9, ,3 [ E.H/. D 4 . C C , / l s |I − F| − = n- |O + F|− = n [ X l @ .0M . 0 3 C z)2 J 9$= ( 3# 4 3 J$- - J0HB [ E $. 1 :.;]/ F #DB ax + by = c , GC = e + 3. 2D. c2l C . lM * T n \05 1 9 O U > ( P 0 9 ": /6DE I EC) ; $ O U !K ,. ! . 2 #$ :"l$ + 3 = g4 + 3 = [ 8 + 3 = - 1 = X + − 3X. 3 = . + 3− = . :+ 6 x 2 uC5 (& . > r !K I ? 2 > ?I6R> U HP P NO/ ! J 2 9$ > 5 !I H-E +6 ( "6. ? c*9 (& > N @P ! $ Pg .`E& > !.6 O g 6M0 6 > N @P ! K.* X8 3# / J85 E C - , E - o >#l@ >- MV C)/ ;+ l @ X 8 X 8 8 − − 8 −− 8 X− 8− − − 8 X − − − − 02M+/ ^ >@N #/ > 0 >02M+/ > 0 >)I0/ > 0 ;+ E 3# / J85 E C ;+ ;+ 1 3d/ / P0 x 1 = nO − FI ;+ .# !/ 4- 1f = nS + FO. : .C)/ d @ A& 90 x 9$= 1f = nS + FO 1 = nO − FI I 1 f F I 1 f F 8 n − n 1 1 I 1 X 8 8 1 ,I 0 I0.M0 = C) P0@ 1 ,I v y > > / P0 :P0 J 3 1f = nS + FO 1 = nO − FI X 8− 8 = 1f =? 1S + IO 1 =? 1O − II 1f =? S + r 1 =? O − S 1jI ✓ 1f = 1f ✓ 1= 1 − 8 X J = 7 − {1 ,I} = ;+ .# !/ ∴ = − X+ 38 8− >- 3 / P0 e = n + FI ;+ .# !/ 4- 1 x 1− = n + F−. XX EC) JH [w !.LV "&.0@M @x 24 MV C)/ ; l @ I 3d/ 1O = n − FI ;+ .# !/ C!@Y "&.0@I $ s = n + FO. : J) ! F#l) #'. d.C) JH n /)/ 1 1O = n − FI C) ! ?& (B.C) JH n /) I s = n + FO. If = Fe S= F C) V s = n + FO I.C) JH S g F _# s = n + SO b F s = n + 1I e− = n {e− ,S} = .# !/ >- 3 11 = nO + FI ;+ .# !/ C!@Y "&.0@I $ I 1f = nS + FI−..b LV $ V¡ J) ! F#l) 5+/ F - n ]/)/ 2 @ T ;+ JC)/ L 3# b >- l @ EC) JH [w O 3d/ O = nO + FI ;+ .# !/ C!@Y "&.0@I $ 1S = ne − FO. 1 O = nO + FI : I 1S = ne − FO 1e = n1e + F1f e JH 1.C) [6 O = nO + FI SI = n1e − Fi O JH I.C) [6 1S = ne − FO 4 es = F1i O= F XJ C) V O= nO + FI 1.C) JH O g F _# b O= nO + OI O= nO + r O− = nO 1− = n {1− ,O} = .# !/ >- 3 1I = nO + FI ;+ .# !/ C!@Y "&.0@I $ O 1O = n − Fe. z@#).0@M @x 24 C)/ ; ẁ@- l @ . d.C) JH o 0 o+ _# b ,C) JH V.E @q -. D C b S 3d/ 1 = 3 − O ;+ z@#).0@I $ e = 3I − O. .E 3. D C b ,5$- 5 B '=V 9 (B.C) JH : 1=3− O 1− O=3 :5 0 3 _# b. d.C) JH e = 1 − OI − O F e= I+ r− O O= O− 1− = 1 − O = 3 JH 1− g _# b 1 − 1−O = 3 S− = 3 S− = 3 ,1− = :#' ;+ >- 3 z@#).0@I /̀ / O + =I = ;+ S r = S − =e. X= á«JÉ«M äÉ≤«Ñ£J e 3d/ / # , C ejIff ol JH 9$ HC ,# J)MD AU [ #- r , =+@C Ijff / HC ?#.)MD )$ / A [# )$ / # MD r A : #.)MD )$ g ,A [# )$ £ l £^r =+@C Ijff # )MD AU [ #- r g ^ I HC / Ijff = = + 9$ g ^ r £^I C ejIff HC # MD r A / > # ejIff = = + Ijff = gI + £r : ;+ =)$B.H) ejIff = gr + £I. fjff = g ,fjIff = £ :( 56 P2$ J P8 M / A- $ =+@C fjff = #.)MD )$ ,=+@C fjIff = A [# )$ >- A- >- 3 94 Ose ( V 5w) 94 eff ( A#@ #2) z) ,c#2 1S JH ( / 9! r hN e ?%# / #2) C / X7 óMGh ô«¨àe »a á«fÉãdG áLQódG øe ä’OÉ©e πM 7-1 Solving Quadratic Equations in One Variable º∏©àJ ±ƒ°S D+ l# D :+ 6 . +,.I ,!/-6 P C4 (& > !- m . L5 O. d.4= / b f = 1f + Fs − IF :.C) y $ :.C) . =0 = + 37 − 3 J 2 l = J − 3 − 3 .4= /. = J − 3 5 = − 3 ∴ $ . d J = 3 5 = 3 k5 [6 7 %# !/.C) A=&4 J = 3 5 = 3 . ! m i; 9 /.C)/ C!@ $!/-6 `/ K.* & 5 u M* (& . K 9 . d.4= :+ 6 . +,.I ,A. .Io + \05 H*I 2 f- P T n ' =&4 b f = e − Fr + IF :.C) + n FI + 3 = + 3 :!> K A. n0yP : 3= = 3 . A> P4 H. < = ,8 = 1/2 !M-P $!> I A> 1/2 !M- < = ,M/ BP . !- ,Y/2 J = 3= + 3 Fr + IF = 0 3 < + J = < + 3= + 3 X = 8 + 3 XF! = 8 + 3 XF − 8− = 3 5 XF + 8− = 3 $P C4 > > k5 !- l/ME A. .I9 H*I L9 :™HôªdG ∫ɪcEÉH óMGh ô«¨àe »a á«fÉãdG áLQódG øe ádOÉ©e πM -1 Solving Quadratic Equation by Completing the Square :OÉ°TQGE 1 3d/ ( Gw 3 Y b.# !/ 4- 3 1r− = F1f + IF :.C) I F /)/ 1 i HM : I ` )` / 2 F1f + F l ,]/ I Xb :>- ! .C) JHI ( Ie.H6 Ie + 1r− = Ie + F1f + IF 1r − Ie = Ie + F i = Ie + F O!=e+F {− ,I−} :.# !/ − = F - I− = F A- O ! e− = F >- 3 b 3 1e− = F − IF :.C) 1 óMGh ô«¨àe »a á«fÉãdG áLQódG ä’OÉ©e πëd ¿ƒfÉ≤dG ΩGóîà°SG -2 Solving Quadratic Equations by Using the Quadratic Formula , = GC + 3 e + 3 :4M 1/2 P C4 > > k5 !- N 2 LP 6 6O& A. .I9 H*I ND6E m :k 2 > n0y T i = + 3= + 3 : . ! :./) c=# :AC ) 3d = GC + 3 e + 3 = + 3= + 3 GC e f ≠ T (. 0 = + 3 + 3 ? I (. 0 = + 3 = + 3 GC e − = 38 + 3 − = 3 + 3 GC e e e − c 8 m = c 8 m + 3 c 8 m + 3 − c m = c m + 3 c m + 3 GC e e − < = c 8 + 3 m − = c + 3m ( X GC X − e
