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# Graphs

## Local extrema

Given a function $f$, a point $\xo$ is

• a local maximum if, for all $x$ around $\xo$, $$f(x)\le f(\xo);$$
• a local minimum if, for all $x$ around $\xo$, $$f(x)\ge f(\xo).$$

An extremum is either a minimum or a maximum. A local extremum is sometimes called a turning point or a relative extremum.

$0$ is a minimum of $x^2$ and a maximum of $-x^2$.

At a local extremum, the tangent must be horizontal $$f'(\xo)=0.$$

At a local maximum $\xo$, $f$ "increases" to its left, so $f'(\xo)\ge 0$, and "decreases" to its right, so $f'(\xo)\le 0$. Thus, $f'(\xo)=0$.

Local minimum (A), inflection point (B), local maximum (C) of a function

## Stationary points

A stationary point $\xo$ of a function $f$ has horizontal tangent $$f'(\xo) = 0$$

A point of inflection is a stationary point that is not an extremum.

In practice, a stationary point $\xo$ is

• a local maximum if $f''(\xo)\lt 0$,
• a local minimum if $f''(\xo) \gt 0$,
• a point of inflection if $f''$ changes sign around $\xo$.

$0$ is a stationary point of $$f_1(x) = x^2,\quad f_2(x)= x^3,\quad f_3(x) = -x^2.$$ It is minimum of $f_1$ ($f''_1(0)= 2$), a maximum of $f_3$ ($f''_3(0)= -2$) and an inflection point of $f_2$ ($f''_2(x) = 6x$).

We show that, when $f''(\xo)\lt 0$, $\xo$ is a local maximum. $f'$ is decreasing and $0$ at $\xo$. On the left of $\xo$, $f'$ is positive, so $f$ is increasing, so $f(x)\le f(\xo)$. Similarly, on its right, $f(x)\le f(\xo)$.

Local minimum (A), inflection point (B), local maximum (C) of a function

## Asymptotes

An asymptote is a straight line tangent to a function at infinity.

• A vertical asymptote is a line of the form $x=a$ when $$\lim_{x\to \pm a}f(x) = \pm\infty.$$ The function has an infinite limit at $a$ when $x$ approaches $a$ either from below ($x\to a-$) or from above ($x\to a+$).
• A horizontal asymptote is a line of the form $y=b$ when $$\lim_{x\to \pm\infty}f(x) = b.$$ The function has a limit when $x$ goes to infinity.
• An oblique asymptote is a line $y = ax + b$ ($a\ne 0$) when $$\lim_{x\to\pm\infty}\big(f(x)-ax\big) = b.$$

$f(x) = 2(x-1)/x$ has two asymptotes: $x = 0$ and $y=2$.

Asymptotes: Vertical $(x=0)$ horizontal $(y=1)$ and oblique $(y=x-1)$

## Graph Transformations 1

The graph of modified functions can often be deduced from simple geometrical transformations of the graph of the original function. To deduce the transformation, we often consider how a few points on the graph are modified from the original graph and then extrapolate.

We explain which modified function corresponds to the appropriate transformation on the graph of $f$. $a\ne 0$ is a fixed number.

• Horizontal translation of $a$: $f(x-a)$
• Vertical translation of $a$: $f(x)+a$
• Horizontal scaling of ratio $1/a$: $f(ax)$. It is a stretch or dilation if $\vert a\vert \lt 1$ and a compression if $\vert a\vert \gt 1$.
• Vertical scaling of ratio $1/a$: $af(x)$
• Horizontal reflection (along vertical axis): $f(-x)$
• Vertical reflection (along horizontal axis): $-f(x)$
• Rotation of angle $\pi$ about the origin:$-f(-x)$
Transformation of graph of $f(x)$ Translation $f(x-1)$ horizontal scaling $f(2x)$ vertical scaling $2f(x)$ horizontal reflection $f(-x)$ vertical reflection $-f(x)$ symmetry $-f(-x)$

## Graph Transformations 2

It is often useful to be able to apply more elaborate graph transformations. The methodology is unchanged: draw the function, see how it changes a few points and extrapolate to get the full graph.

• $f(\vert x\vert)$: Horizontal reflection of the graph of $f$ for $x\ge0$.
• $\vert f(x)\vert$: Vertical reflection of the negative part of the graph of $f$.
• $1/f(x)$: roots of $f$ (solutions of $f(x)=0$) are transformed into vertical asymptotes; monotonicity is transformed into inverse monotonicity (if $f$ is increasing, $1/f$ is decreasing); when $\vert f(x)\vert = 1$, graphs $f$ and $1/f$ intersect.
• $y^2=f(x)$: when $f(x)\lt 0$, $y$ is undefined; if $f(x)\ge 0$, $y$ is the graph superposition of the graphs of the functions $\sqrt{f(x)}$ and $-\sqrt{f(x)}$; monotonicity of $\sqrt{f(x)}$ and $f$ is same; when $f(x) = 1$, graphs $f$ and $\sqrt{f}$ intersect.
Transformation of graph of $\sin x$. A: $f(\vert x\vert)$; B: $\vert f(x)\vert$; C: $1/f(x)$; D: $y^2=f(x)$