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\title{Graphing}
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$f(X)$ is a function on a domain $I$, which is usually an interval.
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{\bf Step 1:} Write down where $f$ is defined clearly, int arms of sub-intervals and note down the points where $f$ is not defined.
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{\bf Step 2:} Compute $f'(x)$ and $f''(x)$ correctly; note down points where they are not defined.
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{\bf Step 3:} Write down the intervals for $f$ where it is defined, and analyze the behaviour of $f'$ in these intervals. This will give you the interval where $f$ is increasing or decreasing; increasing if $f'>0,$ and decreasing if $f'<0.$
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{\bf Step 4:} Find the critical points of $f$ (points $c$ where $f'(c)=0$ of $f'(c)$DNE. Note that points where $f$ or $f'$ are not defined are points of interest but if $F(c)$ is not defined, $c$ is not a critical point.
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{\bf Step 5:} At these critical points where $f'(c)=0$ analyse whether they are local minimum or local maximum. You may use the first derivative test or the second derivative test to do this ($f''(c)>0$ means local minimum and $f''(c)<0$ means local maximum).
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{\bf Step 6:} Analyse the concavity of $f$ in the intervals: If $f''(x)>0$ on $I$, then $f$ is concave up. If $f''(x)<0$ on $I$, then $f$ is concave down.
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{\bf Step 7:} Identify the inflection points: A point $c$ is an inflection point if $f$ is continuous at $c$ and changes concavity at $c$.
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{\bf Step 8: } Study the existence of asymptotes and note them down (see notes on asymptotes). Analyse the end behaviour of the function.
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{\bf Step 9:} Write down the $x$ and $y$-intercepts: $x$-intercepts are obtained as the roots of $f(x)$ i.e find the $x$ such that $y=f(x)=0.$
For $y$-intercepts, put $x=0$ and write down the value of $y$.
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{\bf Step 10:} Put all the information obtained together and draw the graph.
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