<p>Changing the sign of \(a\) reflects the graph in its midline. The period is unchanged; assume \(b&gt;0\).</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\text{Sine/cosine amplitude}=|a|\qquad\text{Extremes}=c\pm|a|\)</p></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 600px;"><colgroup><col style="width: 25.0000%;"><col style="width: 17.5000%;"><col style="width: 17.5000%;"><col style="width: 40.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Curve</strong></th><th style="text-align: center; --cell-justify: center"><strong>Reflection line</strong></th><th style="text-align: center; --cell-justify: center"><strong>Amplitude</strong></th><th style="text-align: center; --cell-justify: center"><strong>What changes</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(y=a\sin(bx),\ a&lt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(y=a\sin(bx)+c,\ a&lt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(y=a\cos(bx),\ a&lt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(y=a\cos(bx)+c,\ a&lt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(y=a\tan(bx),\ a&lt;0\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
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<p>For&nbsp;\(y=2-3\cos2x\), the midline is \(y=2\) and the amplitude is \(3\).</p>

Changing the sign of \(a\) reflects the graph in its midline. The period is unchanged; assume \(b>0\).

\(\text{Sine/cosine amplitude}=|a|\qquad\text{Extremes}=c\pm|a|\)

Peaks and troughs swap; period unchanged.

Reflected in the shifted midline; extrema remain \(c\pm\lvert a\rvert\).

Starts at a minimum instead of a maximum; period unchanged.

Branches decrease instead of increase; asymptotes and period are unchanged.

For \(y=2-3\cos2x\), the midline is \(y=2\) and the amplitude is \(3\).

Curve	Reflection line	Amplitude	What changes
\(y=a\sin(bx),\ a<0\)
\(y=a\sin(bx)+c,\ a<0\)
\(y=a\cos(bx),\ a<0\)
\(y=a\cos(bx)+c,\ a<0\)
\(y=a\tan(bx),\ a<0\)

Trigonometry - Trigonometric Graphs and Parameters

Trigonometry - Trigonometric Graphs and Parameters

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