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Straight Line Graphs (Linear Graphs)
𝒚 = 𝒎𝒙 + 𝒄 The graph is a LINE !!! 𝑯𝒊𝒈𝒉𝒆𝒔𝒕 𝑷𝒐𝒘𝒆𝒓 𝒐𝒇 𝒙 𝒊𝒔 𝟏 → 𝒙 𝟏 =𝒙 𝒎 𝒊𝒔 𝒕𝒉𝒆 𝒈𝒓𝒂𝒅𝒊𝒆𝒏𝒕 →𝒔𝒕𝒆𝒆𝒑𝒏𝒆𝒔𝒔 𝒐𝒇 𝒍𝒊𝒏𝒆 𝒄 𝒊𝒔 𝒕𝒉𝒆 𝒚−𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕: 𝒘𝒉𝒆𝒓𝒆 𝒕𝒉𝒆 𝒍𝒊𝒏𝒆 𝒉𝒊𝒕𝒔 𝒕𝒉𝒆 𝒚−𝒂𝒙𝒊𝒔
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The curve is called a Parabola!! 𝒂 𝒊𝒔 𝑵𝑬𝑽𝑬𝑹 𝟎 , 𝒃𝒖𝒕 𝒃 𝒂𝒏𝒅 𝒄 𝒄𝒂𝒏 𝒃𝒆 𝟎!
Quadratic Graphs The curve is called a Parabola!! 𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 The graph is a CURVE !!! 𝑯𝒊𝒈𝒉𝒆𝒔𝒕 𝑷𝒐𝒘𝒆𝒓 𝒐𝒇 𝒙 𝒊𝒔 𝟐 → 𝒙 𝟐 𝒂,𝒃 𝒂𝒏𝒅 𝒄 𝒂𝒓𝒆 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 𝒂 𝒊𝒔 𝑵𝑬𝑽𝑬𝑹 𝟎 , 𝒃𝒖𝒕 𝒃 𝒂𝒏𝒅 𝒄 𝒄𝒂𝒏 𝒃𝒆 𝟎!
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𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 𝑾𝒉𝒂𝒕 𝒅𝒐𝒆𝒔 𝒂 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕?
Quadratic Graphs The graph is a CURVE !!! The curve is called a Parabola!! 𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 𝑾𝒉𝒂𝒕 𝒅𝒐𝒆𝒔 𝒂 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕? 𝑻𝒉𝒆 𝑽𝑨𝑳𝑼𝑬 𝒐𝒇 𝒂 𝒕𝒆𝒍𝒍𝒔 𝒖𝒔 𝒉𝒐𝒘 𝑪𝑳𝑶𝑺𝑬𝑫 𝒕𝒉𝒆 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔 𝑻𝒉𝒆 𝒃𝒊𝒈𝒈𝒆𝒓 𝒂 𝒊𝒔 𝒕𝒉𝒆 𝒎𝒐𝒓𝒆 𝑪𝑳𝑶𝑺𝑬𝑫 𝒕𝒉𝒆 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔!
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𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 𝑾𝒉𝒂𝒕 𝒅𝒐𝒆𝒔 𝒂 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕?
Quadratic Graphs The graph is a CURVE !!! The curve is called a Parabola!! 𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 𝑾𝒉𝒂𝒕 𝒅𝒐𝒆𝒔 𝒂 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕? 𝑻𝒉𝒆 𝑺𝑰𝑮𝑵 𝒐𝒇 𝒂 𝒕𝒆𝒍𝒍𝒔 𝒖𝒔 𝒘𝒉𝒆𝒕𝒉𝒆𝒓 𝒕𝒉𝒆 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔 𝑯𝑨𝑷𝑷𝒀 𝒐𝒓 𝑺𝑨𝑫 𝑰𝒇 𝒂 𝒊𝒔 𝑷𝑶𝑺𝑰𝑻𝑰𝑽𝑬→𝑯𝑨𝑷𝑷𝒀 𝑮𝑹𝑨𝑷𝑯 𝑰𝒇 𝒂 𝒊𝒔 𝑵𝑬𝑮𝑨𝑻𝑰𝑽𝑬→𝑺𝑨𝑫 𝑮𝑹𝑨𝑷𝑯
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𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 𝑾𝒉𝒂𝒕 𝒅𝒐𝒆𝒔 𝒃 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕?
Quadratic Graphs The graph is a CURVE !!! The curve is called a Parabola!! 𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 𝑾𝒉𝒂𝒕 𝒅𝒐𝒆𝒔 𝒃 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕? 𝑻𝒉𝒆 𝑺𝑰𝑮𝑵 𝒐𝒇 𝒃 𝒕𝒆𝒍𝒍𝒔 𝒖𝒔 𝒕𝒉𝒆 𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝑺𝑯𝑰𝑭𝑻 𝑰𝒇 𝒃 𝒊𝒔 𝑷𝑶𝑺𝑰𝑻𝑰𝑽𝑬→𝑺𝑯𝑰𝑭𝑻𝑺 𝑻𝑶 𝑻𝑯𝑬 𝑳𝑬𝑭𝑻 For a positive 𝑰𝒇 𝒃 𝒊𝒔 𝑵𝑬𝑮𝑨𝑻𝑰𝑽𝑬→𝑺𝑯𝑰𝑭𝑻𝑺 𝑻𝑶 𝑻𝑯𝑬 𝑹𝑰𝑮𝑯𝑻
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𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 𝑾𝒉𝒂𝒕 𝒅𝒐𝒆𝒔 𝒄 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕?
Quadratic Graphs The graph is a CURVE !!! The curve is called a Parabola!! 𝒚 =𝒂 𝒙 𝟐 +𝒃𝒙+ 𝒄 𝑾𝒉𝒂𝒕 𝒅𝒐𝒆𝒔 𝒄 𝒓𝒆𝒑𝒓𝒆𝒔𝒆𝒏𝒕? 𝑨𝒈𝒂𝒊𝒏 𝒄 𝒊𝒔 𝒕𝒉𝒆 𝒚−𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝒄 𝒊𝒔 𝒘𝒉𝒆𝒓𝒆 𝒕𝒉𝒆 𝒄𝒖𝒓𝒗𝒆 𝒄𝒖𝒕𝒔 𝒕𝒉𝒆 𝒚−𝒂𝒙𝒊𝒔
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Plotting Quadratic Graphs
Well, we need the exact same things as we did for straight line graphs We must set up a table of values We take some values of 𝒙 We find the corresponding values of 𝒚 We write the co-ordinates We plot the co-ordinates BUT WE GET A CURVE NOT A LINE 𝒚= 𝒙 𝟐
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𝒚= 𝒙 𝟐 (-2,4) (-1,1) (0,0) (1,1) (2,4) 𝒙 -2 -1 1 2 𝒙𝟐 𝒚 𝒘𝒉𝒆𝒏 𝒙=−𝟐
1 2 𝒙𝟐 𝒚 𝒘𝒉𝒆𝒏 𝒙=−𝟐 𝒚= 𝒙 𝟐 𝒚= (−𝟐) 𝟐 =−𝟐×−𝟐=𝟒 𝒚=𝟒 𝒘𝒉𝒆𝒏 𝒙=𝟏 𝒚= 𝒙 𝟐 𝒚= (𝟏) 𝟐 =𝟏×𝟏=𝟏 𝒚=𝟏 4 1 1 4 𝒘𝒉𝒆𝒏 𝒙=−𝟏 𝒚= 𝒙 𝟐 𝒚= (−𝟏) 𝟐 =−𝟏×−𝟏=𝟏 𝒚=𝟏 𝒘𝒉𝒆𝒏 𝒙=𝟐 𝒚= 𝒙 𝟐 𝒚= (𝟐) 𝟐 =𝟐×𝟐=𝟒 𝒚=𝟒 Co-ordinates (-2,4) (-1,1) (0,0) (1,1) (2,4) 𝒘𝒉𝒆𝒏 𝒙=𝟎 𝒚= 𝒙 𝟐 𝒚= (𝟎) 𝟐 =𝟎×𝟎=𝟎 𝒚=𝟎
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𝒚= −𝒙 𝟐 (-2,-4) (-1,-1) (0,0) (1,-1) (2,-4) 𝒙 -2 -1 1 2 − 𝒙 𝟐 𝒚 -4 -1
1 2 − 𝒙 𝟐 𝒚 -4 -1 -1 -4 Co-ordinates (-2,-4) (-1,-1) (0,0) (1,-1) (2,-4)
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𝒚= 𝒙 𝟐 Compare to 𝒚= 𝒂𝒙 𝟐 +𝒃𝒙+𝒄 𝒂=𝟏 , 𝒃=𝟎, 𝒄=𝟎
𝒂=𝟏 , 𝒃=𝟎, 𝒄=𝟎 𝒂=𝟏: 𝒕𝒉𝒆 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔 𝒉𝒂𝒑𝒑𝒚 𝒑𝒐𝒔𝒊𝒕𝒊𝒗𝒆 𝒂𝒏𝒅 𝒏𝒐𝒕 𝒗𝒆𝒓𝒚 𝒄𝒍𝒐𝒔𝒆𝒅 (𝟏) 𝒃=𝟎: 𝒕𝒉𝒆 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔 𝒏𝒐𝒕 𝒔𝒉𝒊𝒇𝒕𝒆𝒅 𝒍𝒆𝒇𝒕 𝒐𝒓 𝒓𝒊𝒈𝒉𝒕, 𝒊 𝒕 ′ 𝒔 𝒊𝒏 𝒕𝒉𝒆 𝒎𝒊𝒅𝒅𝒍𝒆 𝒄=𝟎: 𝒕𝒉𝒆 𝒈𝒓𝒂𝒑𝒉 𝒉𝒊𝒕𝒔 𝒕𝒉𝒆 𝒚−𝒂𝒙𝒊𝒔 𝒂𝒕 𝒚=𝟎 𝑾𝒉𝒆𝒏 𝒂 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔 𝒉𝒂𝒑𝒑𝒚, 𝒊𝒕 𝒉𝒂𝒔 𝒂 𝒎𝒊𝒏𝒊𝒎𝒖𝒎 𝒑𝒐𝒊𝒏𝒕 𝒕𝒉𝒆 𝒃𝒐𝒕𝒕𝒐𝒎 The minimum point for 𝒚= 𝒙 𝟐 is (𝟎,𝟎)
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𝒚= −𝒙 𝟐 𝑾𝒉𝒆𝒏 𝒂 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔 𝒔𝒂𝒅, 𝒊𝒕 𝒉𝒂𝒔 𝒂 𝒎𝒂𝒙𝒊𝒎𝒖𝒎 𝒑𝒐𝒊𝒏𝒕 𝒕𝒉𝒆 𝒕𝒐𝒑
Compare to 𝒚= 𝒂𝒙 𝟐 +𝒃𝒙+𝒄 𝒂=−𝟏 , 𝒃=𝟎, 𝒄=𝟎 𝑻𝒉𝒆 𝒄𝒐𝒎𝒎𝒆𝒏𝒕𝒔 𝒇𝒐𝒓 ′𝒃′ 𝒂𝒏𝒅 ′𝒄′ 𝒂𝒓𝒆 𝒕𝒉𝒆 𝒔𝒂𝒎𝒆 𝒂𝒔 𝒇𝒐𝒓 𝒚= 𝒙 𝟐 𝑶𝒏𝒍𝒚 𝒕𝒉𝒆 𝒄𝒐𝒎𝒎𝒆𝒏𝒕𝒔 𝒇𝒐𝒓 ′𝒂′ 𝒄𝒉𝒂𝒏𝒈𝒆 𝒂=−𝟏: 𝒕𝒉𝒆 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔 𝒔𝒂𝒅 (𝒏𝒆𝒈𝒂𝒕𝒊𝒗𝒆) 𝑾𝒉𝒆𝒏 𝒂 𝒈𝒓𝒂𝒑𝒉 𝒊𝒔 𝒔𝒂𝒅, 𝒊𝒕 𝒉𝒂𝒔 𝒂 𝒎𝒂𝒙𝒊𝒎𝒖𝒎 𝒑𝒐𝒊𝒏𝒕 𝒕𝒉𝒆 𝒕𝒐𝒑 The maximum point for 𝒚= −𝒙 𝟐 is (𝟎,𝟎)
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𝒚= 𝒂𝒙 𝟐 +𝒃𝒙+𝒄 Who remembers what: 𝒂 does to the graph?
Let’s see ! 𝒄 does to the graph?
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