{"id":5099,"date":"2017-12-31T21:32:32","date_gmt":"2017-12-31T21:32:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/?post_type=chapter&p=5099"},"modified":"2023-11-08T13:20:48","modified_gmt":"2023-11-08T13:20:48","slug":"summary","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/summary\/","title":{"raw":"Summary","rendered":"Summary"},"content":{"raw":"You have been tasked with creating a dazzling new logo for the school newspaper, The Parabola. \u00a0So of course you start thinking about how you might incorporate parabolas into the design. \u00a0The standard parabola, defined by the function \u00a0[latex]f\\left(x\\right)=x^2[\/latex], is simply too boring by itself. \u00a0However by scaling, shifting, and reflecting, perhaps you might come up something more interesting.\r\n\r\nFirst of all, the basic parabola shape seems too steep. \u00a0You can use a vertical compression to scale it down. \u00a0Remember, a vertical stretch or compression takes the form \u00a0[latex]y=af\\left(x\\right)[\/latex], for a constant \u00a0[latex]a[\/latex].\r\n

Vertical compression by a factor of 2: \u00a0\u00a0[latex]y={\\Large\\frac{1}{2}}x^2[\/latex]<\/p>\r\n

Vertical compression by a factor of 3: \u00a0\u00a0[latex]y={\\Large\\frac{1}{3}}x^2[\/latex]<\/p>\r\n

Vertical compression by a factor of 4: \u00a0\u00a0[latex]y={\\Large\\frac{1}{4}}x^2[\/latex]<\/p>\r\nLet\u2019s see what all of these functions look like when graphed together. \u00a0We\u2019ll graph \u00a0[latex]y=x^2[\/latex] in black, and then overlay the compressions by 2, 3, and 4, in purple, blue, and green, respectively.\r\n\r\n\"Four\r\n\r\nNot bad - in fact it\u2019s looking like a pretty decent logo already, but let\u2019s add a few more features. \u00a0The school colors are red and gray, so let\u2019s put two upside-down parabolas in the foreground. \u00a0The transformation that reflects a graph about the [latex]x[\/latex]-axis is: \u00a0[latex]y=f-\\left(x^{ }\\right)[\/latex]. \u00a0But we also want to shift the graphs upwards after reflecting. \u00a0Therefore we should combine the reflection with a vertical shift. \u00a0Altogether, the two transformations are accomplished by \u00a0[latex]y=-f\\left(x\\right)+k[\/latex] for a constant [latex]k[\/latex].\r\n

Reflected and shifted up by 2: \u00a0 [latex]y=-x^2+2[\/latex]<\/p>\r\n

Reflected and shifted up by 4: \u00a0\u00a0[latex]y=-x^2+4[\/latex]<\/p>\r\n\"Two\r\n

The red curve is [latex]y=-x^2+4[\/latex], and the gray curve is [latex]y=-x^2+2[\/latex].<\/p>\r\nFinally, let\u2019s combine the parabolas all together and create our logo! \u00a0We\u2019ll get rid of the axes, colorize the background, and include the name of the paper for our finished product. \u00a0Who knew that mathematics could help to create art?\r\n\r\n\"Logo","rendered":"

You have been tasked with creating a dazzling new logo for the school newspaper, The Parabola. \u00a0So of course you start thinking about how you might incorporate parabolas into the design. \u00a0The standard parabola, defined by the function \u00a0[latex]f\\left(x\\right)=x^2[\/latex], is simply too boring by itself. \u00a0However by scaling, shifting, and reflecting, perhaps you might come up something more interesting.<\/p>\n

First of all, the basic parabola shape seems too steep. \u00a0You can use a vertical compression to scale it down. \u00a0Remember, a vertical stretch or compression takes the form \u00a0[latex]y=af\\left(x\\right)[\/latex], for a constant \u00a0[latex]a[\/latex].<\/p>\n

Vertical compression by a factor of 2: \u00a0\u00a0[latex]y={\\Large\\frac{1}{2}}x^2[\/latex]<\/p>\n

Vertical compression by a factor of 3: \u00a0\u00a0[latex]y={\\Large\\frac{1}{3}}x^2[\/latex]<\/p>\n

Vertical compression by a factor of 4: \u00a0\u00a0[latex]y={\\Large\\frac{1}{4}}x^2[\/latex]<\/p>\n

Let\u2019s see what all of these functions look like when graphed together. \u00a0We\u2019ll graph \u00a0[latex]y=x^2[\/latex] in black, and then overlay the compressions by 2, 3, and 4, in purple, blue, and green, respectively.<\/p>\n

\"Four<\/p>\n

Not bad – in fact it\u2019s looking like a pretty decent logo already, but let\u2019s add a few more features. \u00a0The school colors are red and gray, so let\u2019s put two upside-down parabolas in the foreground. \u00a0The transformation that reflects a graph about the [latex]x[\/latex]-axis is: \u00a0[latex]y=f-\\left(x^{ }\\right)[\/latex]. \u00a0But we also want to shift the graphs upwards after reflecting. \u00a0Therefore we should combine the reflection with a vertical shift. \u00a0Altogether, the two transformations are accomplished by \u00a0[latex]y=-f\\left(x\\right)+k[\/latex] for a constant [latex]k[\/latex].<\/p>\n

Reflected and shifted up by 2: \u00a0 [latex]y=-x^2+2[\/latex]<\/p>\n

Reflected and shifted up by 4: \u00a0\u00a0[latex]y=-x^2+4[\/latex]<\/p>\n

\"Two<\/p>\n

The red curve is [latex]y=-x^2+4[\/latex], and the gray curve is [latex]y=-x^2+2[\/latex].<\/p>\n

Finally, let\u2019s combine the parabolas all together and create our logo! \u00a0We\u2019ll get rid of the axes, colorize the background, and include the name of the paper for our finished product. \u00a0Who knew that mathematics could help to create art?<\/p>\n

\"Logo<\/p>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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