{"id":3848,"date":"2022-03-15T23:18:07","date_gmt":"2022-03-15T23:18:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&#038;p=3848"},"modified":"2022-06-02T01:30:42","modified_gmt":"2022-06-02T01:30:42","slug":"corequisite-support-activity-for-6-b","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/corequisite-support-activity-for-6-b\/","title":{"raw":"Corequisite Support Activity for 6.B: Interpreting Estimated Slopes and Y-Intercepts","rendered":"Corequisite Support Activity for 6.B: Interpreting Estimated Slopes and Y-Intercepts"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>what you'll need to know<\/h3>\r\nIn this support activity you\u2019ll become familiar with the following:\r\n<ul>\r\n \t<li>Calculate the slope of a line using [latex]\\dfrac{\\text{rise}}{\\text{run}}.<\/li>\r\n \t<li>Graph a line with a given positive or negative slope.<\/li>\r\n \t<li>Sketch the graph of a line given its y-intercept and slope.<\/li>\r\n \t<li>Identify the y-intercept and slope given the equation of a line.<\/li>\r\n<\/ul>\r\n<\/div>\r\nPreviously in the course, you learned to calculate and write an equation for the line of best fit to perform a linear regression analysis for bivariate data. In the next preview assignment and in the next class, you will need to understand slope and read linear equations. You'll prepare for that in this support activity by taking a deeper look into the meaning of the slope, or constant rate of change, in a linear equation.\r\n<h2>Triangles<\/h2>\r\nWe'll use the triangles in the questions below to build up the idea of slope. Recall that a <strong>right triangle<\/strong> is a triangle that contains one right angle (90 degrees).\r\n<h3>Rise, Run, and Slope<\/h3>\r\nIn this corequisite support activity, we will think of the horizontal side of each right triangle as the<strong> base<\/strong> and refer to it as the \u201crun.\u201d In addition, we will refer to the vertical side as the \u201crise\u201d and the slanted side as the \u201cslant,\u201d as shown below:\r\n\r\n<img class=\"wp-image-1211 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184335\/Picture132-300x166.png\" alt=\"A right triangle. The vertical edge of the triangle is labeled &quot;Rise,&quot; the horizontal edge is labeled &quot;Run,&quot; and the diagonal edge is labeled &quot;Slant.&quot;\" width=\"337\" height=\"187\" \/>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 1<\/h3>\r\nConsider the right triangles labeled [latex]T_1[\/latex], [latex]T_2[\/latex], and [latex]T_3[\/latex] shown here:\r\n\r\n<img class=\"wp-image-1212 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184339\/Picture133-300x164.png\" alt=\"Two triangles, labeled T1 and T2, respectively. T1 is labeled 4 on the horizontal side, 3 on the vertical side, and 5 on the diagonal side. T2 is labeled 8 on the horizontal side, 6 on the vertical side, and 10 on the diagonal side.\" width=\"417\" height=\"228\" \/>\r\n\r\n<img class=\"wp-image-1213 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184343\/Picture134-300x221.png\" alt=\"A triangle labeled T3. Its horizontal edge is labeled 12 and its vertical edge is labeled 9.\" width=\"417\" height=\"307\" \/>\r\n\r\nPart A: What is the length of the slant in Triangle [latex]T_1[\/latex]?\r\n\r\n[reveal-answer q=\"397944\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"397944\"]See the labeled triangle.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B: What is the length of the rise in Triangle [latex]T_2[\/latex]?\r\n\r\n[reveal-answer q=\"332477\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"332477\"]See the labeled triangle.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart C: What is the length of the run in Triangle [latex]T_3[\/latex]?\r\n\r\n[reveal-answer q=\"55918\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"55918\"]See the labeled triangle[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart D: Compare the run of Triangle [latex]T_1[\/latex] with the run of Triangle [latex]T_3[\/latex]. By what factor must the run of Triangle\u00a0[latex]T_1[\/latex] be multiplied to equal the run of Triangle [latex]T_3[\/latex]?\r\n\r\n[reveal-answer q=\"400565\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"400565\"]That is, what number multiplied to latex]4[\/latex] will yield [latex]12[\/latex]?[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart E: Compare the rise of Triangle [latex]T_1[\/latex]with the rise of Triangle [latex]T_3[\/latex]. By what factor must the rise of Triangle [latex]T_1[\/latex]\u00a0 be multiplied to equal the rise of Triangle [latex]T_3[\/latex]?\r\n\r\n[reveal-answer q=\"860037\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"860037\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart F: Compute the ratio \u201crise over run\u201d for each triangle. Hint: You can think of \u201crise over run\u201d as [latex]\\frac{rise}{run}[\/latex].\r\n\r\n[latex]T_1[\/latex]:_____\u00a0 \u00a0 \u00a0 \u00a0[latex]T_2[\/latex]:____\u00a0 \u00a0 \u00a0 \u00a0[latex]T_3[\/latex]:_____\r\n\r\n[reveal-answer q=\"44168\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44168\"]For example, for [latex]T_1[\/latex] the rise is [latex]3[\/latex] and the run is [latex]4[\/latex], so [latex]\\dfrac{\\text{rise}}{\\text{run}}=\\dfrac{3}{4}[\/latex][\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart G: Consider the three ratios computed in Part F. Which ratio is the largest, which is the smallest, and which ratios are equal? It may help to write each of the ratios either as a simplified fraction or in decimal form.\r\n\r\n[reveal-answer q=\"257896\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"257896\"]Remember to put all the ratios in simplified or decimal form before answering.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart H: What is your best guess for the length of the slant of Triangle [latex]T_3[\/latex]? Explain.\r\n\r\n[reveal-answer q=\"329086\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"329086\"]Refer to your answers to Parts D and E for a hint.[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nIn Question 1, you saw that the \"slant\" of a triangle can be expressed using a ratio of it's \"rise\" over it's \"run.\" If we focus just on the slanted side, and extend it out in either direction to create a line, we call the ratio of the rise over the run for the line \"slope.\"\r\n<h3>Graphing From a Given Slope<\/h3>\r\nYou have seen that the slope of a line is computed using the ratio: [latex]\\frac{rise}{run}[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 2<\/h3>\r\nConsider a line with a slope of [latex]\\frac{2}{3}[\/latex]. By viewing this fraction in the form [latex]\\frac{rise}{run}[\/latex], we can identify that the run is 3 and the rise is 2.\r\n\r\n&nbsp;\r\n\r\nPart A: On the following xy-plane, plot any starting point. Then create a right triangle with a run of 3 and rise of 2. Make sure that you move to the right for the run and move up for the rise. The slant of this right triangle has a slope of [latex]\\frac{2}{3}[\/latex]. Extend this line segment in both directions to create a line with a slope of [latex]\\frac{2}{3}[\/latex].\r\n\r\n<img class=\"wp-image-1214 aligncenter\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184348\/Picture135-228x300.png\" alt=\"A grid\" width=\"401\" height=\"528\" \/>[reveal-answer q=\"391445\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"391445\"]It may be helpful to draw in the horizontal and vertical axes first to locate the units of distance for the run and for the rise. Use any starting point you wish from which to count out the run and the rise. [\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart B: Plot a different starting point on the same [latex]xy[\/latex]-plane and use the method introduced in Part A to create a different line with a slope of [latex]\\frac{2}{3}[\/latex].\r\n\r\n[reveal-answer q=\"756805\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"756805\"]Put Answer Here[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\nPart C: Compare and contrast the two lines from Parts A and B. What is similar and what is different?\r\n\r\n[reveal-answer q=\"718790\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"718790\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the previous question, both the run and the rise were positive; but sometimes, slopes may have negative values for the run or the rise. On an [latex]xy[\/latex]-plane, we define the positive direction on the horizontal axis to be towards the right and the positive direction on the vertical axis to be towards the top. For this reason, a positive run moves to the right and a negative run moves to the left. Similarly, a positive rise moves up and a negative rise moves down.\r\n\r\nConsider a line with a slope of [latex]-2[\/latex]. To interpret this slope using [latex]\\frac{rise}{run}[\/latex], it might be helpful to rewrite [latex]-2[\/latex] as a fraction like this: [latex]-2 = \\frac{-2}{1}[\/latex]. In this fraction form, we can determine that the run is [latex]1[\/latex] and the rise is [latex]-2[\/latex].\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 3<\/h3>\r\nOn the following [latex]xy[\/latex]-plane, plot any starting point and then create a right triangle with a run of\u00a0[latex]1[\/latex] and a rise of [latex]-2[\/latex]. Make sure to move right for the run and move down for the rise. The slant of this right triangle has a slope of [latex]-2[\/latex]. Extend this line segment in both directions to create a line with a slope of [latex]-2[\/latex].\r\n\r\n<img class=\"wp-image-1214 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184348\/Picture135-228x300.png\" alt=\"A grid\" width=\"403\" height=\"531\" \/>\r\n\r\n[reveal-answer q=\"47904\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"47904\"]Use a method like the one you used in Question 2 to create this line.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Graphing From a Given Y-Intercept and Slope<\/h3>\r\nNow that you have a better understanding of how the slope of a line is calculated, and how you can use the slope to graph a line in an\u00a0[latex]xy[\/latex]-plane from any starting point, let's use the idea of a linear equation to graph a line from a specific starting point.\r\n\r\nRecall that a linear equation gives two pieces of information for graphing a line in an\u00a0[latex]xy[\/latex]-plane: the slope and the y-intercept. Think of the y-intercept as the starting point. It's the location on the y-axis where the line intersects. Use this idea to answer Question 4.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 4<\/h3>\r\nConsider a line that has a y-intercept of [latex]4[\/latex] and a slope of [latex]-0.2[\/latex].\r\n\r\nPart A: Write the slope as a fraction and then find the run and the rise.\r\n\r\n[reveal-answer q=\"111730\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"111730\"]\u00a0There is more than one way to write [latex]-0.2[\/latex] as a fraction. One option is to simply divide by [latex]1[\/latex], like we did in Question 2. Another option is to write [latex]-0.2[\/latex] in the form [latex]- \\frac{2}{10}[\/latex] (and then simplify the fraction, if possible).[\/hidden-answer]\r\n\r\nPart B: Plot the y-intercept as your starting point on the following [latex]xy[\/latex]-plane. Then use the run and the rise from Part A to sketch this line.\r\n\r\n<img class=\"wp-image-1214 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184348\/Picture135-228x300.png\" alt=\"A grid\" width=\"406\" height=\"534\" \/>\r\n\r\n[reveal-answer q=\"844310\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"844310\"]Follow the hint in Part A to identify the run and the rise, then use a method similar to those you used in Questions 2 and 3 to sketch the line.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Linear Equations<\/h3>\r\nRecall that a linear equation can be written in the form: [latex]y=mx+b[\/latex], where b is the y-intercept and m is the slope.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Question 5<\/h3>\r\nDetermine the y-intercept and the slope of the line [latex]y=7x-3[\/latex].\r\n\r\n[reveal-answer q=\"182303\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"182303\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>question 6<\/h3>\r\nDetermine the y-intercept and the slope of the line[latex]y=4+ \\frac{2}{3}x[\/latex]\r\n\r\n[reveal-answer q=\"722601\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"722601\"]What do <em>you<\/em> think?[\/hidden-answer]\r\n\r\n<\/div>\r\nNow that you've had a closer look at what the slope of a linear equation represents in the relationship between the input and output variables it's time to move on to the course material and activity.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>what you&#8217;ll need to know<\/h3>\n<p>In this support activity you\u2019ll become familiar with the following:<\/p>\n<ul>\n<li>Calculate the slope of a line using [latex]\\dfrac{\\text{rise}}{\\text{run}}.<\/li>\n<li>Graph a line with a given positive or negative slope.<\/li>\n<li>Sketch the graph of a line given its y-intercept and slope.<\/li>\n<li>Identify the y-intercept and slope given the equation of a line.<\/li>\n<\/ul><\/div>\n<p>  Previously in the course, you learned to calculate and write an equation for the line of best fit to perform a linear regression analysis for bivariate data. In the next preview assignment and in the next class, you will need to understand slope and read linear equations. You'll prepare for that in this support activity by taking a deeper look into the meaning of the slope, or constant rate of change, in a linear equation.  <\/p>\n<h2>Triangles<\/h2>\n<p>  We'll use the triangles in the questions below to build up the idea of slope. Recall that a <strong>right triangle<\/strong> is a triangle that contains one right angle (90 degrees).  <\/p>\n<h3>Rise, Run, and Slope<\/h3>\n<p>  In this corequisite support activity, we will think of the horizontal side of each right triangle as the<strong> base<\/strong> and refer to it as the \u201crun.\u201d In addition, we will refer to the vertical side as the \u201crise\u201d and the slanted side as the \u201cslant,\u201d as shown below:    <img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1211 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184335\/Picture132-300x166.png\" alt=\"A right triangle. The vertical edge of the triangle is labeled \"Rise,\" the horizontal edge is labeled \"Run,\" and the diagonal edge is labeled \"Slant.\"\" width=\"337\" height=\"187\" \/>  <\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 1<\/h3>\n<p>  Consider the right triangles labeled [latex]T_1[\/latex], [latex]T_2[\/latex], and [latex]T_3[\/latex] shown here:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1212 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184339\/Picture133-300x164.png\" alt=\"Two triangles, labeled T1 and T2, respectively. T1 is labeled 4 on the horizontal side, 3 on the vertical side, and 5 on the diagonal side. T2 is labeled 8 on the horizontal side, 6 on the vertical side, and 10 on the diagonal side.\" width=\"417\" height=\"228\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1213 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184343\/Picture134-300x221.png\" alt=\"A triangle labeled T3. Its horizontal edge is labeled 12 and its vertical edge is labeled 9.\" width=\"417\" height=\"307\" \/><\/p>\n<p>Part A: What is the length of the slant in Triangle [latex]T_1[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q397944\">Hint<\/span><\/p>\n<div id=\"q397944\" class=\"hidden-answer\" style=\"display: none\">See the labeled triangle.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B: What is the length of the rise in Triangle [latex]T_2[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q332477\">Hint<\/span><\/p>\n<div id=\"q332477\" class=\"hidden-answer\" style=\"display: none\">See the labeled triangle.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part C: What is the length of the run in Triangle [latex]T_3[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q55918\">Hint<\/span><\/p>\n<div id=\"q55918\" class=\"hidden-answer\" style=\"display: none\">See the labeled triangle<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part D: Compare the run of Triangle [latex]T_1[\/latex] with the run of Triangle [latex]T_3[\/latex]. By what factor must the run of Triangle\u00a0[latex]T_1[\/latex] be multiplied to equal the run of Triangle [latex]T_3[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q400565\">Hint<\/span><\/p>\n<div id=\"q400565\" class=\"hidden-answer\" style=\"display: none\">That is, what number multiplied to latex]4[\/latex] will yield [latex]12[\/latex]?<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part E: Compare the rise of Triangle [latex]T_1[\/latex]with the rise of Triangle [latex]T_3[\/latex]. By what factor must the rise of Triangle [latex]T_1[\/latex]\u00a0 be multiplied to equal the rise of Triangle [latex]T_3[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q860037\">Hint<\/span><\/p>\n<div id=\"q860037\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part F: Compute the ratio \u201crise over run\u201d for each triangle. Hint: You can think of \u201crise over run\u201d as [latex]\\frac{rise}{run}[\/latex].<\/p>\n<p>[latex]T_1[\/latex]:_____\u00a0 \u00a0 \u00a0 \u00a0[latex]T_2[\/latex]:____\u00a0 \u00a0 \u00a0 \u00a0[latex]T_3[\/latex]:_____<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44168\">Hint<\/span><\/p>\n<div id=\"q44168\" class=\"hidden-answer\" style=\"display: none\">For example, for [latex]T_1[\/latex] the rise is [latex]3[\/latex] and the run is [latex]4[\/latex], so [latex]\\dfrac{\\text{rise}}{\\text{run}}=\\dfrac{3}{4}[\/latex]<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part G: Consider the three ratios computed in Part F. Which ratio is the largest, which is the smallest, and which ratios are equal? It may help to write each of the ratios either as a simplified fraction or in decimal form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q257896\">Hint<\/span><\/p>\n<div id=\"q257896\" class=\"hidden-answer\" style=\"display: none\">Remember to put all the ratios in simplified or decimal form before answering.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part H: What is your best guess for the length of the slant of Triangle [latex]T_3[\/latex]? Explain.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q329086\">Hint<\/span><\/p>\n<div id=\"q329086\" class=\"hidden-answer\" style=\"display: none\">Refer to your answers to Parts D and E for a hint.<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>In Question 1, you saw that the &#8220;slant&#8221; of a triangle can be expressed using a ratio of it&#8217;s &#8220;rise&#8221; over it&#8217;s &#8220;run.&#8221; If we focus just on the slanted side, and extend it out in either direction to create a line, we call the ratio of the rise over the run for the line &#8220;slope.&#8221;<\/p>\n<h3>Graphing From a Given Slope<\/h3>\n<p>You have seen that the slope of a line is computed using the ratio: [latex]\\frac{rise}{run}[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 2<\/h3>\n<p>Consider a line with a slope of [latex]\\frac{2}{3}[\/latex]. By viewing this fraction in the form [latex]\\frac{rise}{run}[\/latex], we can identify that the run is 3 and the rise is 2.<\/p>\n<p>&nbsp;<\/p>\n<p>Part A: On the following xy-plane, plot any starting point. Then create a right triangle with a run of 3 and rise of 2. Make sure that you move to the right for the run and move up for the rise. The slant of this right triangle has a slope of [latex]\\frac{2}{3}[\/latex]. Extend this line segment in both directions to create a line with a slope of [latex]\\frac{2}{3}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1214 aligncenter\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184348\/Picture135-228x300.png\" alt=\"A grid\" width=\"401\" height=\"528\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q391445\">Hint<\/span><\/p>\n<div id=\"q391445\" class=\"hidden-answer\" style=\"display: none\">It may be helpful to draw in the horizontal and vertical axes first to locate the units of distance for the run and for the rise. Use any starting point you wish from which to count out the run and the rise. <\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part B: Plot a different starting point on the same [latex]xy[\/latex]-plane and use the method introduced in Part A to create a different line with a slope of [latex]\\frac{2}{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q756805\">Hint<\/span><\/p>\n<div id=\"q756805\" class=\"hidden-answer\" style=\"display: none\">Put Answer Here<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Part C: Compare and contrast the two lines from Parts A and B. What is similar and what is different?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q718790\">Hint<\/span><\/p>\n<div id=\"q718790\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<\/div>\n<p>In the previous question, both the run and the rise were positive; but sometimes, slopes may have negative values for the run or the rise. On an [latex]xy[\/latex]-plane, we define the positive direction on the horizontal axis to be towards the right and the positive direction on the vertical axis to be towards the top. For this reason, a positive run moves to the right and a negative run moves to the left. Similarly, a positive rise moves up and a negative rise moves down.<\/p>\n<p>Consider a line with a slope of [latex]-2[\/latex]. To interpret this slope using [latex]\\frac{rise}{run}[\/latex], it might be helpful to rewrite [latex]-2[\/latex] as a fraction like this: [latex]-2 = \\frac{-2}{1}[\/latex]. In this fraction form, we can determine that the run is [latex]1[\/latex] and the rise is [latex]-2[\/latex].<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 3<\/h3>\n<p>On the following [latex]xy[\/latex]-plane, plot any starting point and then create a right triangle with a run of\u00a0[latex]1[\/latex] and a rise of [latex]-2[\/latex]. Make sure to move right for the run and move down for the rise. The slant of this right triangle has a slope of [latex]-2[\/latex]. Extend this line segment in both directions to create a line with a slope of [latex]-2[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1214 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184348\/Picture135-228x300.png\" alt=\"A grid\" width=\"403\" height=\"531\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q47904\">Hint<\/span><\/p>\n<div id=\"q47904\" class=\"hidden-answer\" style=\"display: none\">Use a method like the one you used in Question 2 to create this line.<\/div>\n<\/div>\n<\/div>\n<h3>Graphing From a Given Y-Intercept and Slope<\/h3>\n<p>Now that you have a better understanding of how the slope of a line is calculated, and how you can use the slope to graph a line in an\u00a0[latex]xy[\/latex]-plane from any starting point, let&#8217;s use the idea of a linear equation to graph a line from a specific starting point.<\/p>\n<p>Recall that a linear equation gives two pieces of information for graphing a line in an\u00a0[latex]xy[\/latex]-plane: the slope and the y-intercept. Think of the y-intercept as the starting point. It&#8217;s the location on the y-axis where the line intersects. Use this idea to answer Question 4.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>question 4<\/h3>\n<p>Consider a line that has a y-intercept of [latex]4[\/latex] and a slope of [latex]-0.2[\/latex].<\/p>\n<p>Part A: Write the slope as a fraction and then find the run and the rise.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q111730\">Hint<\/span><\/p>\n<div id=\"q111730\" class=\"hidden-answer\" style=\"display: none\">\u00a0There is more than one way to write [latex]-0.2[\/latex] as a fraction. One option is to simply divide by [latex]1[\/latex], like we did in Question 2. Another option is to write [latex]-0.2[\/latex] in the form [latex]- \\frac{2}{10}[\/latex] (and then simplify the fraction, if possible).<\/div>\n<\/div>\n<p>Part B: Plot the y-intercept as your starting point on the following [latex]xy[\/latex]-plane. Then use the run and the rise from Part A to sketch this line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1214 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5738\/2022\/01\/12184348\/Picture135-228x300.png\" alt=\"A grid\" width=\"406\" height=\"534\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q844310\">Hint<\/span><\/p>\n<div id=\"q844310\" class=\"hidden-answer\" style=\"display: none\">Follow the hint in Part A to identify the run and the rise, then use a method similar to those you used in Questions 2 and 3 to sketch the line.<\/div>\n<\/div>\n<\/div>\n<h3>Linear Equations<\/h3>\n<p>Recall that a linear equation can be written in the form: [latex]y=mx+b[\/latex], where b is the y-intercept and m is the slope.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Question 5<\/h3>\n<p>Determine the y-intercept and the slope of the line [latex]y=7x-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q182303\">Hint<\/span><\/p>\n<div id=\"q182303\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>question 6<\/h3>\n<p>Determine the y-intercept and the slope of the line[latex]y=4+ \\frac{2}{3}x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q722601\">Hint<\/span><\/p>\n<div id=\"q722601\" class=\"hidden-answer\" style=\"display: none\">What do <em>you<\/em> think?<\/div>\n<\/div>\n<\/div>\n<p>Now that you&#8217;ve had a closer look at what the slope of a linear equation represents in the relationship between the input and output variables it&#8217;s time to move on to the course material and activity.<\/p>\n","protected":false},"author":428269,"menu_order":7,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3848","chapter","type-chapter","status-publish","hentry"],"part":4241,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3848","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/users\/428269"}],"version-history":[{"count":11,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3848\/revisions"}],"predecessor-version":[{"id":4814,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3848\/revisions\/4814"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/parts\/4241"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapters\/3848\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/media?parent=3848"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/pressbooks\/v2\/chapter-type?post=3848"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/contributor?post=3848"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/wp-json\/wp\/v2\/license?post=3848"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}