{"id":42814,"date":"2025-06-29T10:38:12","date_gmt":"2025-06-29T10:38:12","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=42814"},"modified":"2025-06-29T10:38:14","modified_gmt":"2025-06-29T10:38:14","slug":"determine-the-midline-amplitude-and-period-then-write-the-equation-of-the-graph","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/determine-the-midline-amplitude-and-period-then-write-the-equation-of-the-graph\/","title":{"rendered":"Determine the midline; amplitude and period then write the equation of the graph"},"content":{"rendered":"\n<p>Determine the midline; amplitude and period then write the equation of the graph:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"346\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-509.png\" alt=\"\" class=\"wp-image-42815\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-509.png 700w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/image-509-300x148.png 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>=<\/p>\n\n\n\n<p>Based on the provided graph, here are the determined characteristics and the corresponding equation.<\/p>\n\n\n\n<p><strong>Midline:<\/strong>&nbsp;The midline is the horizontal line that passes exactly halfway between the graph&#8217;s maximum and minimum points. First, we identify the maximum y-value, which is y = 2 (seen at x = -2 and x = 1). Next, we identify the minimum y-value, which is y = -6 (seen at x = -0.5 and x = 2.5). The midline is the average of these two values.<br>Midline = (Maximum Value + Minimum Value) \/ 2<br>Midline = (2 + (-6)) \/ 2 = -4 \/ 2 = -2<br>Therefore, the midline is the line&nbsp;<strong>y = -2<\/strong>.<\/p>\n\n\n\n<p><strong>Amplitude:<\/strong>&nbsp;The amplitude is the vertical distance from the midline to either a maximum or minimum point. It represents the height of the waves. It can be calculated as half the difference between the maximum and minimum values.<br>Amplitude = (Maximum Value &#8211; Minimum Value) \/ 2<br>Amplitude = (2 &#8211; (-6)) \/ 2 = 8 \/ 2 = 4<br>Thus, the amplitude is&nbsp;<strong>4<\/strong>.<\/p>\n\n\n\n<p><strong>Period:<\/strong>&nbsp;The period is the horizontal length of one complete cycle of the wave. We can measure this distance from one peak to the next consecutive peak. The first visible peak is at x = -2, and the next peak is at x = 1.<br>Period = x\u2082 &#8211; x\u2081 = 1 &#8211; (-2) = 3<br>So, the period of the function is&nbsp;<strong>3<\/strong>.<\/p>\n\n\n\n<p><strong>Equation of the Graph:<\/strong><br>We can model the graph using a sine or cosine function in the form y = A cos(B(x &#8211; C)) + D or y = A sin(B(x &#8211; C)) + D.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>A<\/strong>\u00a0is the amplitude, which is 4.<\/li>\n\n\n\n<li><strong>D<\/strong>\u00a0is the vertical shift, which is the midline, y = -2.<\/li>\n\n\n\n<li><strong>B<\/strong>\u00a0is related to the period (P) by the formula P = 2\u03c0 \/ B. We found P = 3, so 3 = 2\u03c0 \/ B, which gives\u00a0<strong>B = 2\u03c0 \/ 3<\/strong>.<\/li>\n\n\n\n<li><strong>C<\/strong>\u00a0is the phase shift, or horizontal shift. Let&#8217;s use a cosine function. A standard cosine function starts at its maximum value. This graph has a maximum at x = -2. This means the graph is shifted 2 units to the left, so\u00a0<strong>C = -2<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>Putting it all together into the cosine form:<br>y = A cos(B(x &#8211; C)) + D<br>y = 4 cos( (2\u03c0\/3)(x &#8211; (-2)) ) &#8211; 2<\/p>\n\n\n\n<p>The final equation is:<br><strong>y = 4 cos( (2\u03c0\/3)(x + 2) ) &#8211; 2<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-382.jpeg\" alt=\"\" class=\"wp-image-42817\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-382.jpeg 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-382-300x300.jpeg 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-382-150x150.jpeg 150w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-382-768x768.jpeg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Determine the midline; amplitude and period then write the equation of the graph: The Correct Answer and Explanation is: = Based on the provided graph, here are the determined characteristics and the corresponding equation. Midline:&nbsp;The midline is the horizontal line that passes exactly halfway between the graph&#8217;s maximum and minimum points. First, we identify the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-42814","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42814","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=42814"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42814\/revisions"}],"predecessor-version":[{"id":42818,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/42814\/revisions\/42818"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=42814"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=42814"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=42814"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}