{"id":6012,"date":"2016-10-03T20:19:35","date_gmt":"2016-10-03T20:19:35","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/beginalgebra\/?post_type=chapter&p=6012"},"modified":"2017-07-27T16:47:30","modified_gmt":"2017-07-27T16:47:30","slug":"conclusion-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-beginalgebra\/chapter\/conclusion-2\/","title":{"raw":"Putting It Together: Graphing","rendered":"Putting It Together: Graphing"},"content":{"raw":"\"Watercolor\r\n\r\nSuppose you are a semiprofessional blogger who writes about media (you'd really like to be a paid\u00a0film critic, but no one has offered yet). Recently you\u00a0posted a short blog piece complaining\u00a0about the number of ads on TV these days, compared to\u00a0when you were younger. You weren't very scientific about it, and a couple\u00a0of your readers disputed your claim and\u00a0tried to start an argument with you.\r\n\r\nWell, that got your attention. It made you wonder whether your perception of there being \"way more\" TV ads\u00a0was accurate. You get\u00a0online and do a little research, and you find a website that reports some interesting data on the number of minutes of TV commercials per hour since 2009, shown below:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Years since 2009<\/th>\r\nMinutes<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
0<\/td>\r\n8.5<\/td>\r\n<\/tr>\r\n
1<\/td>\r\n9.25<\/td>\r\n<\/tr>\r\n
2<\/td>\r\n10<\/td>\r\n<\/tr>\r\n
3<\/td>\r\n10.75<\/td>\r\n<\/tr>\r\n
4<\/td>\r\n11.5<\/td>\r\n<\/tr>\r\n
5<\/td>\r\n12.25<\/td>\r\n<\/tr>\r\n
6<\/td>\r\n13<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe data show that, yes, there are more commercials now than in 2009, but the table isn't very exciting, and you doubt that your readers will care. You may not be a famous film critic yet, but you realize you can use what you've learned in math class to present this information as a graph\u2014the perfect thing to post on your blog and convince\u00a0your skeptical readers!\r\n\r\nYou get to work.\r\n\r\nWhat information do you need to draw a graph of the line that represents the change in the number of minutes of commercials in one hour of TV since 2009?\r\n

The Cartesian Coordinate Plane<\/h3>\r\nYou\u00a0remember that the coordinate plane\u00a0gives your graph structure and meaning. A straight line on a page won't tell your readers much. You draw the axes and label the horizontal one \"Years Since 2009,\" because that's the first data point you have. \u00a0You label the vertical axis from 1 to 18 because your minute data range from 8.5 to 13 minutes, and that will give you room on either side.\r\n\r\n\"A\r\n\r\nThen you plot the ordered pairs from your table of values on your coordinate plane, as below.\r\n\r\n\"The\r\n\r\nThe points\u00a0give you a guide for drawing your\u00a0line, which you do, as below.\r\n\r\n\"An\r\n\r\nThings are looking pretty good, so you\u00a0post your graph to your blog to\u00a0show people how much more time they are being exposed to commercials in one hour of TV watching since 2009.\r\n\r\nThen, it happens . . .\r\n\r\nOne of your readers asks if you can guess how many minutes of commercials will be in one hour of television ten years from now (assuming the current trend continues). After thinking about the question for a while, you realize you don't have to guess! \u00a0You have all the information you need to write the equation of the line you drew, and you recall\u00a0that, with an equation, you can put in any value for the years since 2009.\r\n

Finding the Equation<\/h3>\r\nYou remember that knowing the slope and y-intercept of a line can help you write the equation of the line. \u00a0You realize you have the y-intercept: [latex](0,8.5)[\/latex]. \u00a0Now, you just need the slope.\r\n\r\nYou check your math notes to\u00a0find the definition of slope, and use two of your data points to calculate it:\r\n

[latex]\\begin{array}{l} \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}\\\\\\\\m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\\\\\\\\m=\\frac{{11.5}-{10.75}}{{4}-{3}}\\\\\\\\m=\\frac{{0.75}}{{1}} = 0.75\\end{array}[\/latex]<\/p>\r\nNow you have the two pieces of information you need to write the equation of the line that represents how many minutes of commercials will be in one hour of TV in any<\/em> year before or after 2009.\r\n\r\nFirst choose your variables: x<\/em> = the year and y<\/em> = the number of minutes. You can substitute the values for m<\/em> and b<\/em> into the slope-intercept form of a line:\r\n

[latex]\\begin{array}{l}{ y }= {m x} + {b}\\\\{ y }= {0.75 x} + {8.5}\\end{array}[\/latex]<\/p>\r\nRemembering that the whole point of this exercise was to answer your reader's question, next you figure out what 10 years from now would be in relation to 0 representing 2009 on your graph. \u00a0If 10 years from now is 2026, then it's 17 years from 2009. \u00a0The x<\/em> value you need in order to answer the question is 17.\r\n

[latex]\\begin{array}{l}{ y }= {0.75 (17)} + {8.5}\\\\\\\\{ y }= {12.75} + {8.5}\\\\\\\\{ y }= {21.25}\\end{array}[\/latex]<\/p>\r\nYou have a new data point: [latex](17,21.25)[\/latex]. \u00a0This means that in 2026 there will be more than\u00a020 minutes of commercials in one hour of TV. Yuck!\r\n\r\n \r\n\r\n ","rendered":"

\"Watercolor<\/p>\n

Suppose you are a semiprofessional blogger who writes about media (you’d really like to be a paid\u00a0film critic, but no one has offered yet). Recently you\u00a0posted a short blog piece complaining\u00a0about the number of ads on TV these days, compared to\u00a0when you were younger. You weren’t very scientific about it, and a couple\u00a0of your readers disputed your claim and\u00a0tried to start an argument with you.<\/p>\n

Well, that got your attention. It made you wonder whether your perception of there being “way more” TV ads\u00a0was accurate. You get\u00a0online and do a little research, and you find a website that reports some interesting data on the number of minutes of TV commercials per hour since 2009, shown below:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Years since 2009<\/th>\nMinutes<\/th>\n<\/tr>\n<\/thead>\n
0<\/td>\n8.5<\/td>\n<\/tr>\n
1<\/td>\n9.25<\/td>\n<\/tr>\n
2<\/td>\n10<\/td>\n<\/tr>\n
3<\/td>\n10.75<\/td>\n<\/tr>\n
4<\/td>\n11.5<\/td>\n<\/tr>\n
5<\/td>\n12.25<\/td>\n<\/tr>\n
6<\/td>\n13<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The data show that, yes, there are more commercials now than in 2009, but the table isn’t very exciting, and you doubt that your readers will care. You may not be a famous film critic yet, but you realize you can use what you’ve learned in math class to present this information as a graph\u2014the perfect thing to post on your blog and convince\u00a0your skeptical readers!<\/p>\n

You get to work.<\/p>\n

What information do you need to draw a graph of the line that represents the change in the number of minutes of commercials in one hour of TV since 2009?<\/p>\n

The Cartesian Coordinate Plane<\/h3>\n

You\u00a0remember that the coordinate plane\u00a0gives your graph structure and meaning. A straight line on a page won’t tell your readers much. You draw the axes and label the horizontal one “Years Since 2009,” because that’s the first data point you have. \u00a0You label the vertical axis from 1 to 18 because your minute data range from 8.5 to 13 minutes, and that will give you room on either side.<\/p>\n

\"A<\/p>\n

Then you plot the ordered pairs from your table of values on your coordinate plane, as below.<\/p>\n

\"The<\/p>\n

The points\u00a0give you a guide for drawing your\u00a0line, which you do, as below.<\/p>\n

\"An<\/p>\n

Things are looking pretty good, so you\u00a0post your graph to your blog to\u00a0show people how much more time they are being exposed to commercials in one hour of TV watching since 2009.<\/p>\n

Then, it happens . . .<\/p>\n

One of your readers asks if you can guess how many minutes of commercials will be in one hour of television ten years from now (assuming the current trend continues). After thinking about the question for a while, you realize you don’t have to guess! \u00a0You have all the information you need to write the equation of the line you drew, and you recall\u00a0that, with an equation, you can put in any value for the years since 2009.<\/p>\n

Finding the Equation<\/h3>\n

You remember that knowing the slope and y-intercept of a line can help you write the equation of the line. \u00a0You realize you have the y-intercept: [latex](0,8.5)[\/latex]. \u00a0Now, you just need the slope.<\/p>\n

You check your math notes to\u00a0find the definition of slope, and use two of your data points to calculate it:<\/p>\n

[latex]\\begin{array}{l} \\text{Slope}=\\frac{\\text{rise}}{\\text{run}}\\\\\\\\m=\\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\\\\\\\\m=\\frac{{11.5}-{10.75}}{{4}-{3}}\\\\\\\\m=\\frac{{0.75}}{{1}} = 0.75\\end{array}[\/latex]<\/p>\n

Now you have the two pieces of information you need to write the equation of the line that represents how many minutes of commercials will be in one hour of TV in any<\/em> year before or after 2009.<\/p>\n

First choose your variables: x<\/em> = the year and y<\/em> = the number of minutes. You can substitute the values for m<\/em> and b<\/em> into the slope-intercept form of a line:<\/p>\n

[latex]\\begin{array}{l}{ y }= {m x} + {b}\\\\{ y }= {0.75 x} + {8.5}\\end{array}[\/latex]<\/p>\n

Remembering that the whole point of this exercise was to answer your reader’s question, next you figure out what 10 years from now would be in relation to 0 representing 2009 on your graph. \u00a0If 10 years from now is 2026, then it’s 17 years from 2009. \u00a0The x<\/em> value you need in order to answer the question is 17.<\/p>\n

[latex]\\begin{array}{l}{ y }= {0.75 (17)} + {8.5}\\\\\\\\{ y }= {12.75} + {8.5}\\\\\\\\{ y }= {21.25}\\end{array}[\/latex]<\/p>\n

You have a new data point: [latex](17,21.25)[\/latex]. \u00a0This means that in 2026 there will be more than\u00a020 minutes of commercials in one hour of TV. Yuck!<\/p>\n

 <\/p>\n

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