Equations & Inequalities

𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑳𝒊𝒏𝒆𝒂𝒓 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔 & 𝒊𝒏𝒆𝒒𝒖𝒂𝒍𝒊𝒕𝒊𝒆𝒔

𝑰𝒔𝒐𝒍𝒂𝒕𝒆 𝒕𝒉𝒆 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆

Equations & Inequalities 𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑸𝒖𝒂𝒅𝒓𝒂𝒕𝒊𝒄 Equations & Inequalities

Isolate Zero

Equations & Inequalities 𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒂𝒊𝒍 Equations & Inequalities

Isolate Zero

Equations & Inequalities 𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑹𝒂𝒕𝒊𝒐𝒏𝒂𝒍 Equations & Inequalities

Multiply by the LCD

Equations & Inequalities 𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑨𝒃𝒔𝒐𝒍𝒖𝒕𝒆 𝑽𝒂𝒍𝒖𝒆 Equations & Inequalities

Isolate the absolute value

𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑨𝒃𝒔𝒐𝒍𝒖𝒕𝒆 𝑽𝒂𝒍𝒖𝒆 Equations 𝒂 =

a, if a > 0 -a, if a < 0

𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑨𝒃𝒔𝒐𝒍𝒖𝒕𝒆 𝑽𝒂𝒍𝒖𝒆 Inequalities 𝒇(𝒙) > c means….

f(x) > c OR f(x)< -c

𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑬𝒙𝒑𝒐𝒏𝒆𝒏𝒕𝒊𝒂𝒍 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔

Isolate the power

𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑹𝒂𝒅𝒊𝒄𝒂𝒍 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔

Isolate the radical

𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑳𝒐𝒈𝒂𝒓𝒊𝒕𝒉𝒎𝒊𝒄 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔

Write as the log of one expression and then isolate the log

𝑺𝒐𝒍𝒗𝒊𝒏𝒈 𝑻𝒓𝒊𝒈 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔 (3)

Use zero product property Isolate the trig ratio Use zero product property Use a trig identity

Properties of Functions

𝑻𝒐 𝒕𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒆 𝒂 𝒈𝒓𝒂𝒑𝒉 𝒖𝒑…

…𝒂𝒅𝒅 𝒂 𝒏𝒖𝒎𝒃𝒆𝒓

𝑻𝒐 𝒕𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒆 𝒂 𝒈𝒓𝒂𝒑𝒉 𝒅𝒐𝒘𝒏…

…𝒔𝒖𝒃𝒕𝒓𝒂𝒄𝒕 𝒂 𝒏𝒖𝒎𝒃𝒆𝒓

𝑻𝒐 𝒕𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒆 𝒂 𝒈𝒓𝒂𝒑𝒉 𝒓𝒊𝒈𝒉𝒕…

…𝒔𝒖𝒃𝒕𝒓𝒂𝒄𝒕 𝒂 𝒏𝒖𝒎𝒃𝒆𝒓 "𝒘𝒊𝒕𝒉𝒊𝒏"

𝑻𝒐 𝒕𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒆 𝒂 𝒈𝒓𝒂𝒑𝒉 𝒍𝒆𝒇𝒕…

…𝒂𝒅𝒅 𝒂 𝒏𝒖𝒎𝒃𝒆𝒓

𝑻𝒐 𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍𝒍𝒚 𝒔𝒕𝒓𝒆𝒕𝒄𝒉 𝒂 𝒈𝒓𝒂𝒑𝒉

…𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒚 𝒃𝒚 𝒂 𝒏𝒖𝒎𝒃𝒆𝒓 c (c>1)

𝑻𝒐 𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍𝒍𝒚 𝒔𝒉𝒓𝒊𝒏𝒌 𝒂 𝒈𝒓𝒂𝒑𝒉

…𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒚 𝒃𝒚 𝒂 𝒏𝒖𝒎𝒃𝒆𝒓 c (0<c<1)

𝒇 𝒙 +𝒌

𝒕𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒆𝒔 𝒈𝒓𝒂𝒑𝒉 𝒖𝒑 𝒌 𝒖𝒏𝒊𝒕𝒔

𝒇 𝒙 −𝒌

𝒕𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒆𝒔 𝒈𝒓𝒂𝒑𝒉 𝒅𝒐𝒘𝒏 𝒌 𝒖𝒏𝒊𝒕𝒔

𝒌∗𝒇 𝒙 , 𝒘𝒉𝒆𝒓𝒆 𝒌>𝟏

𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒔𝒕𝒓𝒆𝒕𝒄𝒉

𝒌∗𝒇 𝒙 , 𝒘𝒉𝒆𝒓𝒆 𝟎<𝒌<𝟏

𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒔𝒉𝒓𝒊𝒏𝒌

𝒇 𝒙+𝒉

𝒕𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒆𝒔 𝒈𝒓𝒂𝒑𝒉 𝒍𝒆𝒇𝒕 𝒉 𝒖𝒏𝒊𝒕𝒔

𝒇 𝒙−𝒉

𝒕𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒆𝒔 𝒈𝒓𝒂𝒑𝒉 𝒓𝒊𝒈𝒉𝒕 𝒉 𝒖𝒏𝒊𝒕𝒔

𝑻𝒐 𝒇𝒊𝒏𝒅 𝒙−𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕𝒔

𝒔𝒖𝒃𝒔𝒕𝒊𝒕𝒖𝒕𝒆 𝟎 𝒇𝒐𝒓 𝒚

𝑻𝒐 𝒇𝒊𝒏𝒅 𝒚−𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕

𝑺𝒖𝒃𝒔𝒕𝒊𝒕𝒖𝒕𝒆 𝟎 𝒇𝒐𝒓 𝒙

𝑯𝒐𝒘 𝒕𝒐 𝒇𝒊𝒏𝒅 𝒊𝒏𝒗𝒆𝒓𝒔𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 (𝟒 𝒔𝒕𝒆𝒑𝒔)

𝟏) 𝑹𝒆𝒑𝒍𝒂𝒄𝒆 𝒇 𝒙 𝒘𝒊𝒕𝒉 𝒚 𝟐) 𝑺𝒘𝒊𝒕𝒄𝒉 𝒙 & 𝒚 𝟑) 𝑺𝒐𝒍𝒗𝒆 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒏𝒆𝒘 𝒚 𝟒) 𝑹𝒆𝒑𝒍𝒂𝒄𝒆 𝒈 𝒙 𝒇𝒐𝒓 𝒕𝒉𝒆 𝒏𝒆𝒘 𝒚

𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔 𝒐𝒇 𝒊𝒏𝒗𝒆𝒓𝒔𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

𝟏) 𝑺𝒚𝒎𝒎𝒆𝒕𝒓𝒊𝒄 𝒘𝒊𝒕𝒉 𝒚=𝒙 𝟐) 𝒇 𝒈 𝒙 =𝒈 𝒇 𝒙 =𝒙 𝟑) 𝒐𝒏𝒆−𝒕𝒐−𝒐𝒏𝒆 𝟒) 𝑫𝒐𝒎𝒂𝒊𝒏 & 𝑹𝒂𝒏𝒈𝒆 𝒂𝒓𝒆 𝒊𝒏𝒕𝒆𝒓𝒄𝒉𝒂𝒏𝒈𝒆𝒅

𝒇 𝒌𝒙 , 𝒘𝒉𝒆𝒓𝒆 𝒌>𝟏

𝑯𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒔𝒉𝒓𝒊𝒏𝒌

𝒇 𝒌𝒙 , 𝒘𝒉𝒆𝒓𝒆 𝟎<𝒌<𝟏

𝑯𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒔𝒕𝒓𝒆𝒕𝒄𝒉

𝒇 −𝒙

𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒂𝒄𝒓𝒐𝒔𝒔 𝒕𝒉𝒆 𝒚−𝒂𝒙𝒊𝒔

−𝒇 𝒙

𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒂𝒄𝒓𝒐𝒔𝒔 𝒙−𝒂𝒙𝒊𝒔

−𝒇 −𝒙

𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒕𝒉𝒓𝒐𝒖𝒈𝒉 𝒐𝒓𝒊𝒈𝒊𝒏

𝒇 𝒙

𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝑸𝑰 𝒂𝒏𝒅 𝑸𝑰𝑽 𝒕𝒉𝒓𝒐𝒖𝒈𝒉 𝒚−𝒂𝒙𝒊𝒔 (𝒍𝒐𝒔𝒆 𝑸𝑰𝑰 & 𝑸𝑰𝑰𝑰)

𝒇(𝒙)

𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝑸𝑰𝑰𝑰 𝒂𝒏𝒅 𝑸𝑰𝑽 𝒕𝒉𝒓𝒐𝒖𝒈𝒉 𝒙−𝒂𝒙𝒊𝒔 (𝒍𝒐𝒔𝒆 𝑸𝑰 & 𝑸𝑰𝑰)

𝟏 𝒇 𝒙

𝒚→ 𝟎 + ↔ 𝒚→+∞ 𝒚→ 𝟎 − ↔ 𝒚→−∞\ 𝒚=𝟎↔ 𝒚 𝒊𝒔 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅

𝒇 𝒉−𝒙

=𝒇 𝒙+𝒉 𝒕𝒉𝒆𝒏 𝒓𝒆𝒑𝒍𝒂𝒄𝒆 𝒙 𝒃𝒚 −𝒙 (𝐫𝐞𝐟𝐥𝐞𝐜𝐭𝐢𝐧 𝐨𝐟 𝐟 𝐱+𝐡 𝐭𝐡𝐫𝐨𝐮𝐠𝐡 𝐲−𝐚𝐱𝐢𝐬)

𝒇(𝒙) 𝒅𝒆𝒇𝒊𝒏𝒆𝒅 𝒂𝒔 𝒂 𝒑𝒊𝒆𝒄𝒆𝒘𝒊𝒔𝒆 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

𝒇 𝒙 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝒘𝒉𝒆𝒓𝒆 𝒇 𝒙 ≥𝟎 −𝒇 𝒙 𝒇𝒐𝒓 𝒂𝒍𝒍 𝒙 𝒘𝒉𝒆𝒓𝒆 𝒇 𝒙 <𝟎

𝒆𝒗𝒆𝒏 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

𝒂 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒕𝒉𝒂𝒕 𝒊𝒔 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒊𝒄 𝒕𝒐 𝒊𝒕𝒔𝒆𝒍𝒇 𝒕𝒉𝒓𝒐𝒖𝒈𝒉 𝒕𝒉𝒆 𝒚−𝒂𝒙𝒊𝒔 𝒇 −𝒙 =𝒇(𝒙) 𝒕𝒐 𝒊𝒕𝒔𝒆𝒍𝒇 𝒕𝒉𝒓𝒐𝒖𝒈𝒉 𝒕𝒉𝒆 𝒚−𝒂𝒙𝒊𝒔 𝒇 −𝒙 =𝒇(𝒙)

𝒐𝒅𝒅 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

𝒂 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝒕𝒉𝒂𝒕 𝒊𝒔 𝒔𝒚𝒎𝒎𝒆𝒕𝒓𝒊𝒄 𝒕𝒐 𝒊𝒕𝒔𝒆𝒍𝒇 𝒕𝒉𝒓𝒐𝒖𝒈𝒉 𝒕𝒉𝒆 𝒐𝒓𝒊𝒈𝒊𝒏 −𝒇 −𝒙 =𝒇(𝒙)

Algebraic Functions

𝒇 𝒙 =𝒙

𝑳𝒊𝒏𝒆𝒂𝒓 𝑭𝒂𝒎𝒊𝒍𝒚

𝒇 𝒙 = 𝒙 𝟐 , 𝒙 𝟒 , 𝒙 𝟔 …

𝑷𝒂𝒓𝒂𝒃𝒐𝒍𝒊𝒄 𝑭𝒂𝒎𝒊𝒍𝒚

𝒇 𝒙 = 𝒙 𝟑 , 𝒙 𝟓 , 𝒙 𝟕 …

𝑪𝒖𝒃𝒊𝒄 𝑭𝒂𝒎𝒊𝒍𝒚

𝒇 𝒙 = 𝒙 𝟏 𝟐 , 𝒙 𝟏 𝟒 , 𝒙 𝟏 𝟔 …

𝑺𝒒𝒖𝒂𝒓𝒆 𝑹𝒐𝒐𝒕 𝑭𝒂𝒎𝒊𝒍𝒚

𝒇 𝒙 = 𝒙 𝟏 𝟑 , 𝒙 𝟏 𝟓 , 𝒙 𝟏 𝟕 …

𝑪𝒖𝒃𝒊𝒄 𝑹𝒐𝒐𝒕 𝑭𝒂𝒎𝒊𝒍𝒚

𝒇 𝒙 = 𝒙 −𝟏 , 𝒙 −𝟑 , 𝒙 −𝟓 …

𝑹𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏

𝒇 𝒙 = 𝒙 −𝟐 , 𝒙 −𝟒 , 𝒙 −𝟔 …

𝑩𝒆𝒍𝒍 𝑪𝒖𝒓𝒗𝒆 𝑭𝒂𝒎𝒊𝒍𝒚

𝒇 𝒙 = 𝒙 𝟐 𝟑 , 𝒙 𝟒 𝟓 , 𝒙 𝟔 𝟕 …

𝑩𝒊𝒓𝒅 𝑭𝒂𝒎𝒊𝒍𝒚

𝒇 𝒙 = 𝒙

𝑮𝒓𝒆𝒂𝒕𝒆𝒔𝒕 𝒊𝒏𝒕𝒆𝒈𝒆𝒓 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

𝒇 𝒙 = 𝒙

𝑨𝒃𝒔𝒐𝒍𝒖𝒕𝒆 𝑽𝒂𝒍𝒖𝒆

𝒇 𝒙 =𝒂 𝒙 𝟐 +𝒃𝒙+𝒄

𝑷𝒂𝒓𝒂𝒃𝒐𝒍𝒂 𝑭𝒂𝒎𝒊𝒍𝒚

𝒇 𝒙 =𝒂 𝒙 𝟐 +𝒃𝒙+𝒄 𝑽𝒆𝒓𝒕𝒆𝒙?

𝑽𝒆𝒓𝒕𝒆𝒙 → 𝒉, 𝒌 𝒉= −𝒃 𝟐𝒂 𝒌=𝒇( −𝒃 𝟐𝒂 )

𝒇 𝒙 = 𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 +…+ 𝒂 𝟎 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏

𝟏) 𝒆𝒏𝒅 𝒃𝒆𝒉𝒂𝒗𝒊𝒐𝒓→𝒑𝒂𝒓𝒂𝒃𝒐𝒍𝒊𝒄 𝟐) 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭𝐬 𝟑) 𝐫𝐞𝐥𝐚𝐭𝐢𝐯𝐞 𝐞𝐱𝐭𝐫𝐞𝐦𝐚 𝐧−𝟏 𝟒)𝐬𝐲𝐦𝐦𝐞𝐭𝐫𝐲

𝒇 𝒙 = 𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 +…+ 𝒂 𝟎 𝒏 𝒊𝒔 𝒐𝒅𝒅

𝟏) 𝒐𝒖𝒕𝒔𝒊𝒅𝒆 𝒃𝒆𝒉𝒂𝒗𝒊𝒐𝒓→𝒄𝒖𝒃𝒊𝒄 𝟐) 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭𝐬 𝟑) 𝐫𝐞𝐥𝐚𝐭𝐢𝐯𝐞 𝐞𝐱𝐭𝐫𝐞𝐦𝐚 𝐧−𝟏 𝟒)𝐬𝐲𝐦𝐦𝐞𝐭𝐫𝐲

𝒇 𝒙 = 𝒂 𝒏 𝒙 𝒏 + 𝒂 𝒏−𝟏 𝒙 𝒏−𝟏 +…+ 𝒂 𝟎 𝒃 𝒎 𝒙 𝒎 + 𝒃 𝒎−𝟏 𝒙 𝒎−𝟏 +…+ 𝒃 𝟎

𝟏) 𝒂𝒔𝒚𝒎𝒑𝒕𝒐𝒕𝒆𝒔 𝟐) 𝐢𝐧𝐭𝐞𝐫𝐜𝐞𝐩𝐭𝐬 𝟑) 𝐬𝐲𝐦𝐦𝐞𝐭𝐫𝐲 𝟒) 𝐩𝐥𝐨𝐭 𝐩𝐨𝐢𝐧𝐭𝐬, 𝐢𝐟 𝐧𝐞𝐞𝐝𝐞𝐝

𝑻𝒐 𝒇𝒊𝒏𝒅 𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒂𝒔𝒚𝒎𝒑𝒕𝒐𝒕𝒆𝒔…

𝑺𝒆𝒕 𝒅𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓 𝒐𝒇 𝒔𝒊𝒎𝒑𝒍𝒊𝒇𝒊𝒆𝒅 𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝒆𝒙𝒑𝒓𝒆𝒔𝒔𝒊𝒐𝒏 𝒆𝒒𝒖𝒂𝒍 𝒕𝒐 𝟎

𝑻𝒐 𝒇𝒊𝒏𝒅 𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒂𝒔𝒚𝒎𝒑𝒕𝒐𝒕𝒆𝒔 𝒐𝒇 𝒓𝒂𝒕𝒊𝒐𝒏𝒂𝒍 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔…

𝒏=𝒎, 𝒉𝒐𝒓𝒊𝒛. 𝒂𝒔𝒚𝒎 @ 𝒚= 𝒂 𝒃 𝒏<𝒎, 𝒉𝒐𝒓𝒊𝒛. 𝒂𝒔𝒚𝒎 @ 𝒚=𝟎 𝒏>𝒎, 𝒏𝒐 𝒉𝒐𝒓𝒊𝒛. 𝒂𝒔𝒚𝒎

𝒇 𝒙 = 𝒄− 𝒙 𝟐 , 𝒄>𝟎

𝑪𝒊𝒓𝒄𝒖𝒍𝒂𝒓 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

𝒇 𝒙 = 𝒙 𝟐 −𝒄 , 𝒄>𝟎

𝑯𝒚𝒑𝒆𝒓𝒃𝒐𝒍𝒊𝒄 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

𝒇 𝒙 = 𝒙 𝟐 +𝒄

𝑯𝒚𝒑𝒆𝒓𝒃𝒐𝒍𝒊𝒄 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏

Trigonometric Functions

𝑳𝒂𝒘 𝒐𝒇 𝑺𝒊𝒏𝒆𝒔

𝒂 𝒔𝒊𝒏 𝑨 = 𝒃 𝒔𝒊𝒏 𝑩 = 𝒄 𝒔𝒊𝒏 𝑪

𝑳𝒂𝒘 𝒐𝒇 𝑪𝒐𝒔𝒊𝒏𝒆𝒔

𝒂 𝟐 = 𝒃 𝟐 + 𝒄 𝟐 −𝟐𝒃𝒄∙𝑪𝒐𝒔𝑨 𝒃 𝟐 = 𝒂 𝟐 + 𝒄 𝟐 −𝟐𝒂𝒄∙𝑪𝒐𝒔𝑨 𝒄 𝟐 = 𝒂 𝟐 + 𝒃 𝟐 −𝟐𝒂𝒃∙𝑪𝒐𝒔𝑨

𝑳𝒂𝒘 𝒐𝒇 𝑺𝒊𝒏𝒆𝒔 𝒊𝒔 𝒖𝒔𝒆𝒅 𝒇𝒐𝒓…

𝑨𝑨𝑺, 𝑨𝑺𝑨, 𝑺𝑺𝑨

𝑳𝒂𝒘 𝒐𝒇 𝑪𝒐𝒔𝒊𝒏𝒆𝒔 𝒊𝒔 𝒖𝒔𝒆𝒅 𝒇𝒐𝒓…

𝑺𝑺𝑺, 𝑺𝑨𝑺

𝑨𝒓𝒆𝒂 𝒐𝒇 𝒂 𝒕𝒓𝒊𝒂𝒏𝒈𝒍𝒆 (SAS)

𝑨= 𝟏 𝟐 𝒂𝒃∙𝑺𝒊𝒏𝑪 𝑨= 𝟏 𝟐 𝒃𝒄∙𝑺𝒊𝒏𝑨 𝑨= 𝟏 𝟐 𝒂𝒄∙𝑺𝒊𝒏𝑩

𝑨𝒓𝒆𝒂 𝒐𝒇 𝒂 𝑪𝒊𝒓𝒄𝒍𝒆 𝑺𝒆𝒄𝒕𝒐𝒓

𝑨= 𝟏 𝟐 𝒓 𝟐 𝜽

𝑨𝒓𝒄 𝑳𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒂 𝒄𝒊𝒓𝒄𝒍𝒆

𝑺=𝒓𝜽

Signs of Trig Ratios

All Students Take Calculus

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒔𝒊𝒏 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒄𝒐𝒔 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒕𝒂𝒏 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒄𝒔𝒄 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒔𝒆𝒄 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒄𝒐𝒕 𝒙

𝑷𝒆𝒓𝒊𝒐𝒅 𝒐𝒇 𝒔𝒊𝒏 𝒙, 𝒄𝒐𝒔 𝒙, 𝒔𝒆𝒄 𝒙 & 𝒄𝒔𝒄 𝒙

𝟐𝝅

𝑷𝒆𝒓𝒊𝒐𝒅 𝒐𝒇 𝒕𝒂𝒏 𝒙 & 𝒄𝒐𝒕 𝒙

𝝅

𝑨𝒎𝒑𝒍𝒊𝒕𝒖𝒅𝒆 𝒐𝒇 𝑻𝒓𝒊𝒈 𝑭𝒖𝒏𝒄𝒕𝒊𝒐𝒏𝒔

𝒔𝒊𝒏 𝒙 →𝟏 𝒄𝒐𝒔 𝒙→𝟏 𝒕𝒂𝒏 𝒙 →𝑵𝒐𝒏𝒆 𝒐𝒓 𝑵𝑨 𝒄𝒔𝒄 𝒙→𝟏 𝒔𝒆𝒄 𝒙→𝟏 𝒄𝒐𝒕 𝒙→𝑵𝒐𝒏𝒆 𝒐𝒓 𝑵𝑨

𝒇 𝒙 =𝒂∙𝒔𝒊𝒏 𝒃 𝒙−𝒄 +𝒅

𝒂=𝒂𝒎𝒑𝒍𝒊𝒕𝒖𝒅𝒆 𝒃→ 𝟐𝝅 𝒃 𝒐𝒓 𝝅 𝒃 𝒊𝒔 𝒑𝒆𝒓𝒊𝒐𝒅 𝒄=𝑷𝒉𝒂𝒔𝒆 𝒔𝒉𝒊𝒇𝒕 (𝑯𝒐𝒓𝒊𝒛) 𝒅=𝑽𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝑺𝒉𝒊𝒇𝒕

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒙 ∙𝒔𝒊𝒏 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒆 𝒙 ∙𝒔𝒊𝒏 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒙∙𝒔𝒊𝒏 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝟏+𝒔𝒊𝒏 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒙+𝒔𝒊𝒏 𝒙

𝑮𝒓𝒂𝒑𝒉 𝒐𝒇 𝒙 +𝒔𝒊𝒏 𝒙

𝑭𝒖𝒏𝒅𝒂𝒎𝒆𝒏𝒕𝒂𝒍 𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒊𝒆𝒔

𝒕𝒂𝒏 𝒙= 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒙 𝒄𝒐𝒕 𝒙= 𝒄𝒐𝒔 𝒙 𝒔𝒊𝒏 𝒙

𝑷𝒚𝒕𝒉𝒂𝒈𝒐𝒓𝒆𝒂𝒏 𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒊𝒆𝒔

𝒔𝒊𝒏 𝟐 𝒙+ 𝒄𝒐𝒔 𝟐 𝒙=𝟏 𝟏+ 𝒄𝒐𝒕 𝟐 𝒙= 𝒄𝒔𝒄 𝟐 𝒙 𝒕𝒂𝒏 𝟐 𝒙+𝟏= 𝒔𝒆𝒄 𝟐 𝒙

𝒔𝒊𝒏 𝟐 𝒙+ 𝒄𝒐𝒔 𝟐 𝒙=𝟏 𝑻𝒘𝒐 𝒐𝒕𝒉𝒆𝒓 𝒇𝒐𝒓𝒎𝒔

𝟏− 𝒔𝒊𝒏 𝟐 𝒙= 𝒄𝒐𝒔 𝟐 𝒙 𝟏− 𝒄𝒐𝒔 𝟐 𝒙= 𝒔𝒊𝒏 𝟐 𝒙

𝑹𝒆𝒄𝒊𝒑𝒓𝒐𝒄𝒂𝒍 𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒊𝒆𝒔

𝒄𝒔𝒄 𝒙= 𝟏 𝒔𝒊𝒏 𝒙 𝒔𝒆𝒄 𝒙= 𝟏 𝒄𝒐𝒔 𝒙 𝒄𝒐𝒕 𝒙= 𝟏 𝒕𝒂𝒏 𝒙

𝑪𝒐𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒊𝒆𝒔

𝒔𝒊𝒏 ( 𝟏 𝟐 𝝅−𝒙)=𝒄𝒐𝒔 𝒙 𝒄𝒐𝒔 ( 𝟏 𝟐 𝝅−𝒙)=𝒔𝒊𝒏 𝒙 𝒕𝒂𝒏 ( 𝟏 𝟐 𝝅−𝒙)=𝒄𝒐𝒕 𝒙

𝑷𝒐𝒘𝒆𝒓 𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒊𝒆𝒔

𝒔𝒊𝒏 𝟐 𝒙= 𝟏−𝒄𝒐𝒔 𝟐𝒙 𝟐 𝒄𝒐𝒔 𝟐 𝒙= 𝟏+𝒄𝒐𝒔 𝟐𝒙 𝟐

𝑫𝒐𝒖𝒃𝒍𝒆 𝑨𝒏𝒈𝒍𝒆 𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒊𝒆𝒔

𝒔𝒊𝒏 𝟐𝒙=𝟐 𝒔𝒊𝒏𝒙 𝒄𝒐𝒔𝒙 𝒄𝒐𝒔 𝟐𝒙= 𝒔𝒊𝒏 𝟐 𝒙− 𝒄𝒐𝒔 𝟐 𝒙

𝑺𝒖𝒎 & 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒊𝒆𝒔

𝒔𝒊𝒏 𝒙+𝒚 = 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒚+𝒄𝒐𝒔 𝒙 𝒔𝒊𝒏 𝒚 𝒔𝒊𝒏 𝒙−𝒚 = 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒚−𝒄𝒐𝒔 𝒙 𝒔𝒊𝒏 𝒚 𝒔𝒊𝒏 𝒙+𝒚 = 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒚+𝒄𝒐𝒔 𝒙 𝒔𝒊𝒏 𝒚 𝒔𝒊𝒏 𝒙−𝒚 = 𝒔𝒊𝒏 𝒙 𝒄𝒐𝒔 𝒚−𝒄𝒐𝒔 𝒙 𝒔𝒊𝒏 𝒚 𝒄𝒐𝒔 𝒙+𝒚 = 𝒄𝒐𝒔 𝒙 𝒄𝒐𝒔 𝒚−𝒔𝒊𝒏 𝒙 𝒔𝒊𝒏 𝒚 𝒄𝒐𝒔 𝒙−𝒚 = 𝒄𝒐𝒔 𝒙 𝒄𝒐𝒔 𝒚+𝒔𝒊𝒏 𝒙 𝒔𝒊𝒏 𝒚

Conics

𝑺𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝑭𝒐𝒓𝒎 𝒐𝒇 𝒂 𝑪𝒐𝒏𝒊𝒄

𝑨 𝒙 𝟐 +𝑩 𝒚 𝟐 +𝑪𝒙+𝑫𝒚+𝑬+𝟎 A & B cannot both = 0

𝑷𝑨𝑹𝑨𝑩𝑶𝑳𝑨 𝐲=𝐚 𝒙−𝒉 𝟐 +𝒌 Focus = ??? Directrix = ???

Directrix---> 𝐲=𝐤− 𝟏 𝟒𝒂 Focus---> 𝒉, 𝒌+ 𝟏 𝟒𝒂 Directrix---> 𝐲=𝐤− 𝟏 𝟒𝒂

𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝒇𝒐𝒓𝒎 𝒐𝒇 𝒂 𝒄𝒊𝒓𝒄𝒍𝒆

(𝒙−𝒉) 𝟐 𝒂 𝟐 + (𝒚−𝒌) 𝟐 𝒃 𝟐 =𝟏 when a = b OR (𝒙−𝒉) 𝟐 + (𝒚−𝒌) 𝟐 = 𝒓 𝟐

𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝒇𝒐𝒓𝒎 𝒐𝒇 𝒂 𝒆𝒍𝒍𝒊𝒑𝒔𝒆

(𝒙−𝒉) 𝟐 𝒂 𝟐 + (𝒚−𝒌) 𝟐 𝒃 𝟐 =𝟏 (𝒙−𝒉) 𝟐 𝒃 𝟐 + (𝒚−𝒌) 𝟐 𝒂 𝟐 =𝟏 where a > b

𝑰𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝒇𝒐𝒓𝒎 𝒐𝒇 𝒂 𝒉𝒚𝒑𝒆𝒓𝒃𝒐𝒍𝒂

(𝒙−𝒉) 𝟐 𝒂 𝟐 − (𝒚−𝒌) 𝟐 𝒃 𝟐 =𝟏 (𝒚−𝒌) 𝟐 𝒂 𝟐 − (𝒙−𝒉) 𝟐 𝒃 𝟐 =𝟏

𝑰𝒏 𝒂 𝑪𝒐𝒏𝒊𝒄 𝒔𝒆𝒄𝒕𝒊𝒐𝒏, 𝒄= ?

𝑻𝒉𝒆 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒇𝒓𝒐𝒎 𝒕𝒉𝒆 𝒄𝒆𝒏𝒕𝒆𝒓 𝒕𝒐 𝒕𝒉𝒆 𝒇𝒐𝒄𝒖𝒔/𝒇𝒐𝒄𝒊

To find the foci…

𝑬𝑳𝑳𝑰𝑷𝑺𝑬 𝒄 𝟐 = 𝒂 𝟐 − 𝒃 𝟐 𝑯𝒀𝑷𝑬𝑹𝑩𝑶𝑳𝑨 𝒄 𝟐 = 𝒂 𝟐 + 𝒃 𝟐

To find the asymptotes for a hyperbola…

(𝒚−𝒌)=± 𝒃 𝒂 (𝒙−𝒉)

(𝒙−𝒉) 𝟐 𝒂 𝟐 + (𝒚−𝒌) 𝟐 𝒃 𝟐 =𝟎 Is the graph of….

Point (h, k)

(𝒙−𝒉) 𝟐 𝒂 𝟐 + (𝒚−𝒌) 𝟐 𝒃 𝟐 =−𝑪 Is the graph of….

∅ The empty set

(𝒙−𝒉) 𝟐 𝒂 𝟐 − (𝒚−𝒌) 𝟐 𝒃 𝟐 =𝟎 Is the graph of….

Two lines (asymptotes)

(𝒙−𝒉) 𝟐 𝒂 𝟐 − 𝒚−𝒌 𝟐 𝒃 𝟐 =−𝑪 Is the graph of….

A hyperbola (still!!!)

Logarithmic & Exponential Functions

𝒂 𝒙 X is a positive integer

=𝒂∗𝒂∗𝒂…. (𝒙 𝒕𝒊𝒎𝒆𝒔)

𝒂 −𝒙 X is a positive integer

𝟏 𝒂 𝒙

𝒂 𝒙 ∗ 𝒂 𝒚

𝒂 𝒙+𝒚

𝒂 𝒙 𝒂 𝒚

𝒂 𝒙−𝒚

( 𝒂 𝒙 ) 𝒚

𝒂 𝒙𝒚

𝒂 𝟎

𝟏

𝒂 𝒙 𝒚

X is the power, y is the root

Radical form for… 𝒂 𝒎 𝒏

𝒏 𝒂 𝒎 or ( 𝒏 𝒂 ) 𝒎

𝒏 𝒙 𝒏 , where n is even

𝒙

e = (2 diff expressions)

𝟏+ 𝟏 𝒙 𝒙 as 𝒙→∞ or 𝟏+𝒙 𝟏 𝒙 as 𝒙→𝟎

e is approximately…

2.718…

𝒇 𝒙 = 𝒃 𝒙 , 𝒃>𝟏

𝒇 𝒙 = 𝒃 𝒙 , 𝟎<𝒃<𝟏

𝒇 𝒙 = 𝒆 𝒙

𝒍𝒐𝒈 𝒂 (𝒙𝒚)

𝒍𝒐𝒈 𝒂 𝒙 + 𝒍𝒐𝒈 𝒂 (𝒚)

𝒍𝒐𝒈 𝒂 𝒙 𝒚

𝒍𝒐𝒈 𝒂 𝒙 − 𝒍𝒐𝒈 𝒂 (𝒚)

𝒍𝒐𝒈 𝒂 𝒙 𝒚

𝒚∗ 𝒍𝒐𝒈 𝒂 𝒙

Change of Base Formula 𝒍𝒐𝒈 𝒃 𝒂

𝒍𝒐𝒈 𝒂 𝒍𝒐𝒈 𝒃 = 𝒍𝒏 𝒂 𝒍𝒏 𝒃

𝒍𝒏(𝟏)

𝟎

𝒍𝒏(𝒆)

𝟏

𝒇 𝒙 = 𝒍𝒐𝒈 𝒃 (𝒙), 𝒃>𝟏

𝒇 𝒙 = 𝒍𝒐𝒈 𝒃 𝒙 , 𝟎<𝒃<𝟏

𝒇 𝒙 =𝒍𝒏 𝒙

𝑰𝒏𝒗𝒆𝒔𝒕𝒎𝒆𝒏𝒕 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 Simple interest A=

r = interest rate (decimal) 𝑨=𝑷 𝟏+𝒓𝒕 P=principle r = interest rate (decimal) t = # of years

𝑰𝒏𝒗𝒆𝒔𝒕𝒎𝒆𝒏𝒕 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 Compound interest A=

r = interest rate (decimal) 𝑨=𝑷 𝟏+ 𝒓 𝒏 𝒏𝒕 P=principle r = interest rate (decimal) t = # of years

𝑰𝒏𝒗𝒆𝒔𝒕𝒎𝒆𝒏𝒕 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 Continuous interest A=

r = interest rate (decimal) 𝑨=𝑷 𝒆 𝒓𝒕 P=principle r = interest rate (decimal) t = # of years

𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑮𝒓𝒐𝒘𝒕𝒉 y =

𝐲=𝑩 𝒆 𝒌𝒕 , 𝒌>𝟎

𝑹𝒂𝒅𝒊𝒐𝒂𝒄𝒕𝒊𝒗𝒆 𝑫𝒆𝒄𝒂𝒚 y =

𝐲=𝑩 𝒆 𝒌𝒕 , 𝒌<𝟎

𝑷𝒐𝒑𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝑮𝒓𝒐𝒘𝒕𝒉 Model

𝑹𝒂𝒅𝒊𝒐𝒂𝒄𝒕𝒊𝒗𝒆 𝑫𝒆𝒄𝒂𝒚 Model

𝑩𝒐𝒖𝒏𝒅𝒆𝒅 𝑮𝒓𝒐𝒘𝒕𝒉 y =

𝐲=𝑨−𝑩 𝒆 −𝒌𝒕 , 𝒕≥𝟎

𝑳𝒐𝒈𝒊𝒔𝒕𝒊𝒄 𝑮𝒓𝒐𝒘𝒕𝒉 y =

𝐲= 𝑨 𝟏+𝑩 𝒆 −𝑨𝒌𝒕 𝒕≥𝟎

𝑩𝒐𝒖𝒏𝒅𝒆𝒅 𝑮𝒓𝒐𝒘𝒕𝒉 Graph

A A-B

𝑳𝒐𝒈𝒊𝒔𝒕𝒊𝒄 𝑮𝒓𝒐𝒘𝒕𝒉 Graph

A A/(1+B)

Sequence & Series

𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆

𝒂 𝒔𝒆𝒕 𝒐𝒇 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 𝒕𝒉𝒂𝒕 𝒔𝒉𝒂𝒓𝒆 𝒂 𝒄𝒐𝒎𝒎𝒐𝒏 𝒑𝒓𝒐𝒑𝒆𝒓𝒕𝒚 𝒂𝒏𝒅 𝒘𝒉𝒐𝒔𝒆 𝒅𝒐𝒎𝒂𝒊𝒏 𝒊𝒔 𝒕𝒉𝒆 𝒔𝒆𝒕 𝒐𝒇 𝒂𝒍𝒍 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 𝒊𝒏𝒕𝒆𝒈𝒆𝒓𝒔

𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆𝒏𝒕 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆

𝒂 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆 𝒕𝒉𝒂𝒕 𝒉𝒂𝒔 𝒂 𝒍𝒊𝒎𝒊𝒕 𝐚𝐬 𝐧→∞

𝒅𝒊𝒗𝒆𝒓𝒈𝒆𝒏𝒕 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆

𝒂 𝒔𝒆𝒒𝒖𝒆𝒏𝒄𝒆 𝒕𝒉𝒂𝒕 𝐢𝐬 𝐧𝐨𝐭 𝐜𝐨𝐧𝐯𝐞𝐫𝐠𝐞𝐧𝐭

𝑮𝒆𝒏𝒆𝒓𝒂𝒍 𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝒔𝒆𝒓𝒊𝒆𝒔

𝒂 𝟏 + 𝒂 𝟏 +𝒅 +…+ ( 𝒂 𝟏 + 𝒏−𝟏 𝒅)

𝑺𝒖𝒎 𝒐𝒇 𝒂𝒏 𝑨𝒓𝒊𝒕𝒉𝒎𝒆𝒕𝒊𝒄 𝒔𝒆𝒓𝒊𝒆𝒔

𝑺 𝒏 = 𝒏 𝟐 ( 𝒂 𝟏 + 𝒂 𝒏 )

𝑮𝒆𝒏𝒆𝒓𝒂𝒍 𝑮𝒆𝒐𝒎𝒆𝒕𝒊𝒄 𝒔𝒆𝒓𝒊𝒆𝒔

𝒂 𝟏 + 𝒂 𝟏 𝒓 + 𝒂 𝟏 𝒓 𝟐 +…+ ( 𝒂 𝟏 𝒓 𝒏−𝟏 )

𝑺𝒖𝒎 𝒐𝒇 𝒂𝒏 𝑮𝒆𝒐𝒎𝒆𝒕𝒊𝒄 𝒔𝒆𝒓𝒊𝒆𝒔

𝑺 𝒏 = 𝒂 𝟏 (𝟏− 𝒓 𝒏 ) 𝟏−𝒓

𝑷𝒂𝒔𝒄𝒂 𝒍 ′ 𝒔 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆

1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

𝑩𝒊𝒏𝒐𝒎𝒊𝒂𝒍 𝑪𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒏 𝒓

Combination of "n" things chosen "r" at a time

𝑩𝒊𝒏𝒐𝒎𝒊𝒂𝒍 𝑻𝒉𝒆𝒐𝒓𝒆𝒎 (𝒂+𝒃) 𝒏

𝒏 𝟎 𝒂 𝒏 + 𝒏 𝟏 𝒂 𝒏−𝟏 𝒃 𝟏 + 𝒏 𝟐 𝒂 𝒏−𝟐 𝒃 𝟐 +…+ 𝒏 𝒏 𝒃 𝒏

Polars

𝑻𝒐 𝒄𝒐𝒏𝒗𝒆𝒓𝒕 𝒇𝒓𝒐𝒎 𝒓𝒆𝒄𝒕𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒕𝒐 𝒑𝒐𝒍𝒂𝒓 (𝟑 𝒘𝒂𝒚𝒔)

𝟏) 𝒙=𝒓∗𝒄𝒐𝒔𝜽 𝟐) 𝒚=𝒓∗𝒔𝒊𝒏𝜽 𝟑) 𝒙 𝟐 + 𝒚 𝟐 = 𝒓 𝟐

𝑰𝒎𝒑𝒐𝒓𝒕𝒂𝒏𝒕 𝒄𝒉𝒂𝒓𝒂𝒄𝒕𝒆𝒓𝒊𝒔𝒕𝒊𝒄𝒔 𝒐𝒇 𝒑𝒐𝒊𝒏𝒕𝒔 𝒊𝒏 𝒑𝒐𝒍𝒂𝒓 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔

𝑬𝒂𝒄𝒉 𝒐𝒓𝒅𝒆𝒓𝒆𝒅 𝒑𝒂𝒊𝒓 𝒊𝒔 𝒏𝒐𝒕 𝒖𝒏𝒊𝒒𝒖𝒆

𝒏𝒂𝒎𝒆 𝒐𝒇 𝒈𝒓𝒂𝒑𝒉?

𝒍𝒊𝒎𝒂𝒄𝒐𝒏 𝒘𝒊𝒕𝒉 𝒊𝒏𝒏𝒆𝒓 𝒍𝒐𝒐𝒑

𝒏𝒂𝒎𝒆 𝒐𝒇 𝒈𝒓𝒂𝒑𝒉?

𝒄𝒂𝒓𝒅𝒊𝒐𝒊𝒅

𝒏𝒂𝒎𝒆 𝒐𝒇 𝒈𝒓𝒂𝒑𝒉?

𝒍𝒊𝒎𝒂𝒄𝒐𝒏 𝒘𝒊𝒕𝒉 𝒅𝒆𝒏𝒕/𝒅𝒊𝒎𝒑𝒍𝒆

𝒏𝒂𝒎𝒆 𝒐𝒇 𝒈𝒓𝒂𝒑𝒉?

𝒄𝒐𝒏𝒗𝒆𝒙 𝒍𝒊𝒎𝒂𝒄𝒐𝒏

𝒏𝒂𝒎𝒆 𝒐𝒇 𝒈𝒓𝒂𝒑𝒉?

𝒔𝒑𝒊𝒓𝒂𝒍

𝒏𝒂𝒎𝒆 𝒐𝒇 𝒈𝒓𝒂𝒑𝒉?

𝒑𝒆𝒕𝒂𝒍 𝒄𝒖𝒓𝒗𝒆

𝒏𝒂𝒎𝒆 𝒐𝒇 𝒈𝒓𝒂𝒑𝒉?

𝒍𝒆𝒎𝒏𝒊𝒔𝒄𝒂𝒕𝒆

𝒓= 𝒄 𝒄𝒐𝒔𝜽

𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒍𝒊𝒏𝒆

𝒓= 𝒄 𝒔𝒊𝒏𝜽

𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒍𝒊𝒏𝒆

𝜽=𝒄

𝒂 𝒍𝒊𝒏𝒆 𝒕𝒉𝒂𝒕 𝒎𝒂𝒌𝒆𝒔 𝒂𝒏 𝒂𝒏𝒈𝒍𝒆 𝒐𝒇 𝒄 𝒓𝒂𝒅𝒊𝒂𝒏𝒔 𝒘𝒊𝒕𝒉 𝒕𝒉𝒆 𝒑𝒐𝒍𝒂𝒓 𝒂𝒙𝒊𝒔

𝒓=𝒄

𝒂 𝒄𝒊𝒓𝒄𝒍𝒆 𝒘𝒊𝒕𝒉 𝒄𝒆𝒏𝒕𝒆𝒓 (𝟎,𝟎) 𝒓𝒂𝒅𝒊𝒖𝒔=𝒄

𝒓=𝟐𝒂𝒄𝒐𝒔𝜽+𝟐𝒃𝒔𝒊𝒏𝜽

𝒄𝒊𝒓𝒄𝒍𝒆 𝒘𝒊𝒕𝒉 𝒄𝒆𝒏𝒕𝒆𝒓 𝒂,𝒃 𝒓𝒂𝒅𝒊𝒖𝒔= 𝒂 𝟐 + 𝒃 𝟐

𝒓=𝜽

𝒔𝒑𝒊𝒓𝒂𝒍

𝒓=𝒂±𝒃𝒔𝒊𝒏𝜽 𝒓=𝒂±𝒃𝒄𝒐𝒔𝜽

𝒍𝒊𝒎𝒂𝒄𝒐𝒏/𝒄𝒂𝒓𝒅𝒊𝒐𝒅

𝒓=𝒂+𝒃𝒔𝒊𝒏𝜽 𝒑𝒐𝒊𝒏𝒕𝒔?

𝑷𝒐𝒊𝒏𝒕𝒔 𝒖𝒑

𝒓=𝒂−𝒃𝒔𝒊𝒏𝜽 𝒑𝒐𝒊𝒏𝒕𝒔?

𝑷𝒐𝒊𝒏𝒕𝒔 𝒅𝒐𝒘𝒏

𝒓=𝒂+𝒃𝒄𝒐𝒔𝜽 𝒑𝒐𝒊𝒏𝒕𝒔?

𝑷𝒐𝒊𝒏𝒕𝒔 𝒓𝒊𝒈𝒉𝒕

𝒓=𝒂−𝒃𝒄𝒐𝒔𝜽 𝒑𝒐𝒊𝒏𝒕𝒔?

𝑷𝒐𝒊𝒏𝒕𝒔 𝒍𝒆𝒇𝒕

𝒓=𝒂±𝒃𝒔𝒊𝒏𝜽 𝒓=𝒂±𝒃𝒄𝒐𝒔𝜽 𝟎< 𝒂 𝒃 <𝟏

𝒍𝒊𝒎𝒂𝒄𝒐𝒏 𝒘𝒊𝒕𝒉 𝒂 𝒍𝒐𝒐𝒑

𝒓=𝒂±𝒃𝒔𝒊𝒏𝜽 𝒓=𝒂±𝒃𝒄𝒐𝒔𝜽 𝒂 𝒃 =𝟏

𝒄𝒂𝒓𝒅𝒊𝒐𝒊𝒅

𝒓=𝒂±𝒃𝒔𝒊𝒏𝜽 𝒓=𝒂±𝒃𝒄𝒐𝒔𝜽 𝟏< 𝒂 𝒃 <𝟐

𝒍𝒊𝒎𝒂𝒄𝒐𝒏 𝒘𝒊𝒕𝒉 𝒂 𝒅𝒆𝒏𝒕/𝒅𝒊𝒎𝒑𝒍𝒆

𝒓=𝒂±𝒃𝒔𝒊𝒏𝜽 𝒓=𝒂±𝒃𝒄𝒐𝒔𝜽 𝒂 𝒃 >𝟐

𝒄𝒐𝒏𝒗𝒆𝒙 𝒍𝒊𝒎𝒂𝒄𝒐𝒏

𝒆𝒙𝒕𝒓𝒆𝒎𝒆 𝒗𝒂𝒍𝒖𝒆𝒔 𝒐𝒇 𝒂 𝒍𝒊𝒎𝒂𝒄𝒐𝒏/𝒄𝒂𝒓𝒅𝒊𝒐𝒊𝒅

𝒂 + 𝒃 𝒃 − 𝒂

𝒓=𝒂𝒄𝒐𝒔(𝒏𝜽) 𝒏 𝒊𝒔 𝒐𝒅𝒅

𝒑𝒆𝒕𝒂𝒍 𝒄𝒖𝒓𝒗𝒆 𝒘𝒊𝒕𝒉 𝒏 𝒑𝒆𝒕𝒂𝒍𝒔

𝒓=𝒂𝒔𝒊𝒏(𝒏𝜽) 𝒏 𝒊𝒔 𝒐𝒅𝒅

𝒑𝒆𝒕𝒂𝒍 𝒄𝒖𝒓𝒗𝒆 𝒘𝒊𝒕𝒉 𝒏 𝒑𝒆𝒕𝒂𝒍𝒔

𝒓=𝒂𝒄𝒐𝒔(𝒏𝜽) 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏

𝒑𝒆𝒕𝒂𝒍 𝒄𝒖𝒓𝒗𝒆 𝒘𝒊𝒕𝒉 𝟐𝒏 𝒑𝒆𝒕𝒂𝒍𝒔

𝒓=𝒂𝒔𝒊𝒏(𝒏𝜽) 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏

𝒑𝒆𝒕𝒂𝒍 𝒄𝒖𝒓𝒗𝒆 𝒘𝒊𝒕𝒉 𝟐𝒏 𝒑𝒆𝒕𝒂𝒍𝒔

𝒓=𝒂𝒔𝒊𝒏 𝒏𝜽 𝒓=𝒂𝒄𝒐𝒔(𝒏𝜽) 𝒂=?

𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒆𝒂𝒄𝒉 𝒑𝒆𝒕𝒂𝒍

𝒓 𝟐 =𝒂𝒄𝒐𝒔(𝟐𝜽) 𝒓 𝟐 =𝒂𝒔𝒊𝒏(𝟐𝜽)

𝒍𝒆𝒎𝒏𝒊𝒔𝒄𝒂𝒕𝒆

𝒓 𝟐 =𝒂𝒄𝒐𝒔(𝟐𝜽) 𝒓 𝟐 =𝒂𝒔𝒊𝒏 𝟐𝜽 𝒂= ?

𝒂 =𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒆𝒂𝒄𝒉 𝒑𝒆𝒕𝒂𝒍

Vectors

𝑴𝒂𝒈𝒏𝒊𝒕𝒖𝒅𝒆 𝒐𝒇 𝒂 𝒗𝒆𝒄𝒕𝒐𝒓< 𝒂 𝟏 , 𝒂 𝟐 > 𝑨 =

= ( 𝒂 𝟏 ) 𝟐 + ( 𝒂 𝟐 ) 𝟐

𝑫𝒊𝒓𝒆𝒄𝒕𝒊𝒐𝒏 𝒂𝒏𝒈𝒍𝒆 𝒐𝒇 𝒂 𝒗𝒆𝒄𝒕𝒐𝒓< 𝒂 𝟏 , 𝒂 𝟐 > 𝜽=

𝒕𝒂𝒏𝜽= 𝒂 𝟐 𝒂 𝟏

𝑻𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒘𝒐 𝒗𝒆𝒄𝒕𝒐𝒓𝒔 𝒊𝒔…

𝑻𝒉𝒆 𝒔𝒖𝒎 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒐𝒓𝒓𝒆𝒔𝒑𝒐𝒏𝒅𝒊𝒏𝒈 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕𝒔

𝑻𝒉𝒆 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝒕𝒘𝒐 𝒗𝒆𝒄𝒕𝒐𝒓𝒔 𝒊𝒔…

𝑻𝒉𝒆 𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒐𝒓𝒓𝒆𝒔𝒑𝒐𝒏𝒅𝒊𝒏𝒈 𝒄𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕𝒔

𝑺𝒄𝒂𝒍𝒂𝒓 𝑴𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝒊𝒇 𝒄 𝒊𝒔 𝒂 𝒔𝒄𝒂𝒍𝒂𝒓 𝒂𝒏𝒅 𝑨 𝒊𝒔 𝒕𝒉𝒆 𝒗𝒆𝒄𝒕𝒐𝒓< 𝒂 𝟏 , 𝒂 𝟐 > 𝒕𝒉𝒆𝒏 𝒄∗𝑨=

< 𝒄∗𝒂 𝟏 , 𝒄∗ 𝒂 𝟐 >

𝒊

<𝟏, 𝟎>

𝒋

<𝟎, 𝟏>

𝒊𝒏 𝒕𝒆𝒓𝒎𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒖𝒏𝒊𝒕 𝒗𝒆𝒄𝒕𝒐𝒓𝒔 <𝒂,𝒃> 𝒊𝒏 𝒕𝒆𝒓𝒎𝒔 𝒐𝒇 𝒕𝒉𝒆 𝒖𝒏𝒊𝒕 𝒗𝒆𝒄𝒕𝒐𝒓𝒔

𝒂𝒊+𝒃𝒋

𝑨< 𝒂 𝟏 , 𝒂 𝟐 >, 𝑩< 𝒃 𝟏 , 𝒃 𝟐 > 𝑽𝒆𝒄𝒕𝒐𝒓 𝑨𝑩 𝒊𝒏 𝒔𝒕𝒂𝒏𝒅𝒂𝒓𝒅 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏 𝑨< 𝒂 𝟏 , 𝒂 𝟐 >, 𝑩< 𝒃 𝟏 , 𝒃 𝟐 >

< 𝒃 𝟏 − 𝒂 𝟏 , 𝒃 𝟐 − 𝒂 𝟐 >

𝑼𝒏𝒊𝒕 𝒗𝒆𝒄𝒕𝒐𝒓 𝒊𝒏 𝒅𝒊𝒓𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒗 𝒆 𝒗 𝑼𝒏𝒊𝒕 𝒗𝒆𝒄𝒕𝒐𝒓 𝒊𝒏 𝒅𝒊𝒓𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒗

𝒗 𝟏 𝒗 , 𝒗 𝟐 𝒗