15. Curve Sketching · Concavity · Symmetry

Adapted from: Classical single-variable calculus (concavity, inflection, asymptotes, symmetry, even/odd, sketching checklist). Rewritten for teaching — not a verbatim textbook dump.

After max/min tests, we assemble a full picture of $y=f(x)$: cup vs cap, inflection, asymptotes, and symmetry.

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Concavity

Intuitively: an arc is concave upward (a cup) or concave downward (a cap).

More precisely: near each $x_0$, if the graph lies above its tangent line, the arc is concave upward; if it lies below its tangent, concave downward. Real graphs often switch — down from $A$ to $B$, up from $B$ to $C$, down again from $C$ to $D$.

Fig. 15-1 — (a) cup (b) cap (c) changing concavity

The second derivative reads the shape:

Theorem 15.1 (concavity)

(a) If $f''(x)>0$ on $(a,b)$, the graph is concave upward for $a<x<b$.

(b) If $f''(x)<0$ on $(a,b)$, the graph is concave downward for $a<x<b$.

Examples

(a) $f(x)=x^{2}$: $f''=2>0$ everywhere → always concave up (upward parabola).

(b) $y=\sqrt{1-x^{2}}$ (upper unit semicircle). Implicitly $x+yy'=0$, then $1+yy''+(y')^{2}=0$, so $y''=-[1+(y')^{2}]/y$. For $y>0$, $y''<0$ → always concave down.

Check: concavity

Quiz

Concavity

1 / 1

f″>0 on an interval means the graph is…

Pick an answer to see if you are right.

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Points of inflection

An inflection point is where concavity changes (up on one side, down on the other). If $f''$ exists nearby and is continuous at $x_0$, a sign change forces $f''(x_0)=0$.

Theorem 15.2

If the graph has an inflection at $x_0$, $f''$ exists in a neighborhood of $x_0$, and $f''$ is continuous at $x_0$, then $f''(x_0)=0$.

Warning. $f''(x_0)=0$ does not guarantee an inflection.

Examples

$f$	$f''$	Inflection?
$x^{3}$	$6x$	Yes at $0$ ($f''$ changes $-$ to $+$)
$x^{4}$	$12x^{2}$	No at $0$ ($f''\ge 0$ always)
$\tfrac13 x^{3}+\tfrac12 x^{2}-6x+8$	$2x+1$	Yes at $x=-\tfrac12$

Hands On: Sign of f″

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Vertical asymptotes

The line $x=x_0$ is a vertical asymptote if $f(x)\to+\infty$ or $-\infty$ as $x\to x_0$ from the left and/or the right.

For $f=g/h$ with $g,h$ continuous: if $h(x_0)=0$ and $g(x_0)\neq 0$, then $x=x_0$ is typically a vertical asymptote.

Horizontal asymptotes

$y=y_0$ is a horizontal asymptote if $\lim_{x\to+\infty}f(x)=y_0$ or $\lim_{x\to-\infty}f(x)=y_0$.

Examples

$f$	Vertical	Horizontal
$1/x$	$x=0$	$y=0$
$1/(x-2)$	$x=2$	$y=0$
$(x-2)/((x-1)(x+3))$	$x=1$, $x=-3$	$y=0$
$(x+4)/(x-3)$	$x=3$	$y=1$

Check: asymptotes

Quiz

Asymptotes

1 / 1

For (x+4)/(x−3), the horizontal asymptote is…

Pick an answer to see if you are right.

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Symmetry

Two points $P,Q$ are symmetric with respect to a line $\ell$ if $\ell$ is the perpendicular bisector of $PQ$. They are symmetric with respect to a point $B$ if $B$ is the midpoint of $PQ$.

A curve is symmetric wrt $\ell$ (or $B$) if every point on the curve has its partner on the curve. Then $\ell$ is an axis of symmetry.

Fig. 15-2 — reflection in a line; axis of symmetry

Coordinate facts:

• $(x,y)$ ↔ $(-x,y)$: symmetry in the $y$-axis
• $(x,y)$ ↔ $(x,-y)$: symmetry in the $x$-axis
• $(x,y)$ ↔ $(-x,-y)$: symmetry in the origin

Fig. 15-3 — y-axis, x-axis, and origin reflections

For an equation $F(x,y)=0$:

1. $y$-axis symmetry iff $F(x,y)=0\Rightarrow F(-x,y)=0$
2. $x$-axis symmetry iff $F(x,y)=0\Rightarrow F(x,-y)=0$
3. Origin symmetry iff $F(x,y)=0\Rightarrow F(-x,-y)=0$

Examples. $y=x^{2}$ (even powers) → $y$-axis. $x=y^{2}$ → $x$-axis. Circles, ellipses, and standard hyperbolas are symmetric in both axes and the origin.

Swap tip. $(a,b)$ and $(b,a)$ are symmetric wrt the line $y=x$ (segment slope $-1$, midpoint on $y=x$).

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Inverse functions and symmetry

Theorem 15.3

If $f$ is one-to-one with inverse $f^{-1}$, the graphs of $f$ and $f^{-1}$ are symmetric to each other with respect to $y=x$.

Reason: $(a,b)$ on $y=f(x)$ means $f(a)=b$, so $f^{-1}(b)=a$ — i.e. $(b,a)$ on the inverse graph.

Examples. $y=2x$ and $y=\tfrac12 x$; or $y=x^{2}$ (right branch as a function) reflected to $x=y^{2}$.

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Even and odd functions

• Even: $f(-x)=f(x)$ (domain symmetric) ⇔ graph symmetric in the $y$-axis
• Odd: $f(-x)=-f(x)$ ⇔ graph symmetric in the origin

Polynomials with only even powers are even; only odd powers are odd.

Check: even / odd

Quiz

Even and odd

1 / 1

An odd function’s graph is symmetric with respect to…

Pick an answer to see if you are right.

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Hints for sketching $y=f(x)$

1. Compute $y'$ (and $y''$ when convenient).
2. Critical numbers ($y'=0$ or undefined); classify extrema (1st/2nd derivative tests).
3. Mono intervals from the sign of $y'$.
4. Concavity from $y''$; check $y''=0$ / undefined for inflection (need a sign change).
5. Vertical asymptotes (e.g. poles of $g/h$).
6. Horizontal asymptotes ($\lim_{x\to\pm\infty}$).
7. End behavior if limits are $\pm\infty$.
8. $x$- and $y$-intercepts.
9. Corners: one-sided derivatives disagree finitely (e.g. $|x|$ at $0$).
10. Cusps: $y'\to+\infty$ or $-\infty$ from both sides (e.g. $\sqrt{|x|}$ at $0$).
11. Oblique asymptotes $y=mx+b$ with $f(x)-(mx+b)\to 0$ as $x\to\pm\infty$.

Key takeaways

• $f''>0$ ⇒ concave up; $f''<0$ ⇒ concave down.
• Inflection = concavity change; $f''=0$ is necessary (if continuous) but not sufficient.
• Asymptotes: vertical (blow-up), horizontal (limits at $\infty$), oblique (linear end behavior).
• Even / odd ↔ $y$-axis / origin symmetry; inverses reflect across $y=x$.
• Sketch with a fixed checklist: derivatives → extrema → mono → concavity → asymptotes → intercepts → special points.

Further Reading

• 16. Review of Trigonometry — next: radians and unit-circle trig
• 15b. Curve Sketching — Practice — solved-problem drills (concavity, asymptotes, sketches)
• 14. Maximum and Minimum Values — critical numbers and derivative tests
• 14b. Max/Min — Practice — optimization drills
• 13. Law of the Mean — monotone from $f'$