<p class="paragraph-header">Rational function of the form \(y=\dfrac{ax^2+bx+c}{px+q}\)</p>
<p>Long division converts \(y=\dfrac{ax^2+bx+c}{px+q}\), where&nbsp;\(x\ne-\dfrac{q}{p}\) and \(p\ne0\), into \(y=mx+n+\dfrac{k}{px+q}\). The line \(y=mx+n\) is the oblique asymptote.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="300" data-height="220" style="width:300px;height:220px;max-width:100%;display:block;margin:0 auto;"></div><p>\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=0\) has real solutions, so the curve has two turning points and some \(y\)-values are not attained.</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="300" data-height="220" style="width:300px;height:220px;max-width:100%;display:block;margin:0 auto;"></div><p>\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=0\) has no real solutions, so the curve has no turning point.</p></div></div></div>

Rational function of the form \(y=\dfrac{ax^2+bx+c}{px+q}\)

Long division converts \(y=\dfrac{ax^2+bx+c}{px+q}\), where \(x\ne-\dfrac{q}{p}\) and \(p\ne0\), into \(y=mx+n+\dfrac{k}{px+q}\). The line \(y=mx+n\) is the oblique asymptote.

Rational Functions with Oblique Asymptotes

Rational Functions with Oblique Asymptotes

Video Solution:

Solution:

Solution locked