{"id":16490,"date":"2019-10-03T16:54:03","date_gmt":"2019-10-03T16:54:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/putting-it-together-graph-linear-equations-and-inequalities\/"},"modified":"2024-04-30T23:14:26","modified_gmt":"2024-04-30T23:14:26","slug":"putting-it-together-graph-linear-equations-and-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/putting-it-together-graph-linear-equations-and-inequalities\/","title":{"raw":"Putting It Together: Graphs","rendered":"Putting It Together: Graphs"},"content":{"raw":"\"Watercolor\r\n\r\nJoan's uncle Devon is\u00a0a semiprofessional blogger who writes about media (he would really like to be a paid\u00a0film critic, but no one has offered yet). Recently, Devon posted a short blog piece complaining\u00a0about the number of ads on TV these days compared to\u00a0when he was younger. He was not very scientific about it, and a couple\u00a0of his\u00a0readers disputed a\u00a0claim he made and\u00a0tried to start an argument.\r\n\r\nDevon\u00a0asks\u00a0Joan, who is taking an algebra class at night, to\u00a0help him gather and cite some data to support his claim\u00a0that there are\u00a0\"way more\" TV ads now than when he was a kid. Together, they\u00a0get\u00a0online and do a little research and find a website that reports some interesting data on the number of minutes of TV commercials per hour since 2009. The data is 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
[latex]0[\/latex]<\/td>\r\n[latex]8.5[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]1[\/latex]<\/td>\r\n[latex]9.25[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]2[\/latex]<\/td>\r\n[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]3[\/latex]<\/td>\r\n[latex]10.75[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]4[\/latex]<\/td>\r\n[latex]11.5[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]5[\/latex]<\/td>\r\n[latex]12.25[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]6[\/latex]<\/td>\r\n[latex]13[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe data show that, yes, there are more commercials now than in [latex]2009[\/latex]. Joan decides the table is not very exciting because of the success she had with her manager when she reported her increase in sales using a graph.\u00a0Also, Devon\u00a0doubts that his readers will actually read a table. He agrees that a graph is\u00a0the perfect thing to post on his blog to convince his\u00a0skeptical readers!\r\n\r\nThey\u00a0get to work.\r\n\r\nFirst, they figure out what information they 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\u00a0[latex]2009[\/latex]?\r\n\r\nJoan\u00a0remembers that the coordinate plane\u00a0gives graphs structure and meaning. A straight line on a page will not tell Devon's\u00a0readers much. Joan\u00a0draws the axes and labels the horizontal one \"Years Since [latex]2009[\/latex],\" because that is the first data point they have. \u00a0Then Devon labels the vertical axis from\u00a0[latex]1[\/latex] to\u00a0[latex]18[\/latex], because the minute data ranges from\u00a0[latex]8.5[\/latex] to\u00a0[latex]13[\/latex] minutes.\r\n\r\n\"A\r\n\r\nThey plot the ordered pairs from the table of values on the\u00a0coordinate plane as shown below.\r\n\r\n\"The\r\n\r\nThe points\u00a0give them a guide for drawing the line which is shown below.\r\n\r\n\"An\r\n\r\nDevon is excited to\u00a0post the graph on his\u00a0blog to\u00a0show people how much more time they are being exposed to commercials in one hour of watching TV since [latex]2009[\/latex].\r\n\r\nThen, it happens . . .\r\n\r\nOne of Devon's readers asks if he 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, he realizes that he don't have to guess!\u00a0 He has all the information he needs to write the equation of the line he drew, and he recalls that, with an equation, he can put in any value for the years since [latex]2009[\/latex].\r\n

Finding the Equation<\/h3>\r\nDevon remembers that knowing the slope and y-intercept of a line can help him write the equation of the line.\u00a0 He realizes he has the [latex]y[\/latex]-intercept: [latex](0,8.5)[\/latex]. \u00a0Now, he just needs the slope.\r\n\r\nDevon checks his math notes to\u00a0find the definition of slope, and uses two of his 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 Devon has the two pieces of information he needs 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 [latex]2009[\/latex].\r\n\r\nLet's review how this is done.\u00a0 First, choose your variables: [latex]x[\/latex] = the year and [latex]y[\/latex] = the number of minutes. You can substitute the values for m and b 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 Devon's reader's question, next we can figure out what [latex]10[\/latex] years from now would be in relation to [latex]0[\/latex] representing [latex]2009[\/latex] on your graph. \u00a0If [latex]10[\/latex] years from now is [latex]2026[\/latex], then it's [latex]17[\/latex] years from [latex]2009[\/latex]. \u00a0The x<\/em> value you need in order to answer Devon's question is [latex]17[\/latex].\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\nNow Devon has a new data point to share with his bloggers: [latex](17, 21.25)[\/latex]. \u00a0This means that in [latex]2026[\/latex] there will be more than\u00a0[latex]20[\/latex] minutes of commercials in one hour of TV. Yuck!","rendered":"

\"Watercolor<\/p>\n

Joan’s uncle Devon is\u00a0a semiprofessional blogger who writes about media (he would really like to be a paid\u00a0film critic, but no one has offered yet). Recently, Devon posted a short blog piece complaining\u00a0about the number of ads on TV these days compared to\u00a0when he was younger. He was not very scientific about it, and a couple\u00a0of his\u00a0readers disputed a\u00a0claim he made and\u00a0tried to start an argument.<\/p>\n

Devon\u00a0asks\u00a0Joan, who is taking an algebra class at night, to\u00a0help him gather and cite some data to support his claim\u00a0that there are\u00a0“way more” TV ads now than when he was a kid. Together, they\u00a0get\u00a0online and do a little research and find a website that reports some interesting data on the number of minutes of TV commercials per hour since 2009. The data is 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
[latex]0[\/latex]<\/td>\n[latex]8.5[\/latex]<\/td>\n<\/tr>\n
[latex]1[\/latex]<\/td>\n[latex]9.25[\/latex]<\/td>\n<\/tr>\n
[latex]2[\/latex]<\/td>\n[latex]10[\/latex]<\/td>\n<\/tr>\n
[latex]3[\/latex]<\/td>\n[latex]10.75[\/latex]<\/td>\n<\/tr>\n
[latex]4[\/latex]<\/td>\n[latex]11.5[\/latex]<\/td>\n<\/tr>\n
[latex]5[\/latex]<\/td>\n[latex]12.25[\/latex]<\/td>\n<\/tr>\n
[latex]6[\/latex]<\/td>\n[latex]13[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The data show that, yes, there are more commercials now than in [latex]2009[\/latex]. Joan decides the table is not very exciting because of the success she had with her manager when she reported her increase in sales using a graph.\u00a0Also, Devon\u00a0doubts that his readers will actually read a table. He agrees that a graph is\u00a0the perfect thing to post on his blog to convince his\u00a0skeptical readers!<\/p>\n

They\u00a0get to work.<\/p>\n

First, they figure out what information they 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\u00a0[latex]2009[\/latex]?<\/p>\n

Joan\u00a0remembers that the coordinate plane\u00a0gives graphs structure and meaning. A straight line on a page will not tell Devon’s\u00a0readers much. Joan\u00a0draws the axes and labels the horizontal one “Years Since [latex]2009[\/latex],” because that is the first data point they have. \u00a0Then Devon labels the vertical axis from\u00a0[latex]1[\/latex] to\u00a0[latex]18[\/latex], because the minute data ranges from\u00a0[latex]8.5[\/latex] to\u00a0[latex]13[\/latex] minutes.<\/p>\n

\"A<\/p>\n

They plot the ordered pairs from the table of values on the\u00a0coordinate plane as shown below.<\/p>\n

\"The<\/p>\n

The points\u00a0give them a guide for drawing the line which is shown below.<\/p>\n

\"An<\/p>\n

Devon is excited to\u00a0post the graph on his\u00a0blog to\u00a0show people how much more time they are being exposed to commercials in one hour of watching TV since [latex]2009[\/latex].<\/p>\n

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

One of Devon’s readers asks if he 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, he realizes that he don’t have to guess!\u00a0 He has all the information he needs to write the equation of the line he drew, and he recalls that, with an equation, he can put in any value for the years since [latex]2009[\/latex].<\/p>\n

Finding the Equation<\/h3>\n

Devon remembers that knowing the slope and y-intercept of a line can help him write the equation of the line.\u00a0 He realizes he has the [latex]y[\/latex]-intercept: [latex](0,8.5)[\/latex]. \u00a0Now, he just needs the slope.<\/p>\n

Devon checks his math notes to\u00a0find the definition of slope, and uses two of his 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 Devon has the two pieces of information he needs 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 [latex]2009[\/latex].<\/p>\n

Let’s review how this is done.\u00a0 First, choose your variables: [latex]x[\/latex] = the year and [latex]y[\/latex] = the number of minutes. You can substitute the values for m and b 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 Devon’s reader’s question, next we can figure out what [latex]10[\/latex] years from now would be in relation to [latex]0[\/latex] representing [latex]2009[\/latex] on your graph. \u00a0If [latex]10[\/latex] years from now is [latex]2026[\/latex], then it’s [latex]17[\/latex] years from [latex]2009[\/latex]. \u00a0The x<\/em> value you need in order to answer Devon’s question is [latex]17[\/latex].<\/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

Now Devon has a new data point to share with his bloggers: [latex](17, 21.25)[\/latex]. \u00a0This means that in [latex]2026[\/latex] there will be more than\u00a0[latex]20[\/latex] minutes of commercials in one hour of TV. Yuck!<\/p>\n","protected":false},"author":169554,"menu_order":30,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"1862cc0d72da4d8b8b3b0a908f80aa96","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16490","chapter","type-chapter","status-publish","hentry"],"part":8524,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16490","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/users\/169554"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16490\/revisions"}],"predecessor-version":[{"id":20468,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16490\/revisions\/20468"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8524"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/16490\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/media?parent=16490"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=16490"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=16490"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/wp-json\/wp\/v2\/license?post=16490"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}