Reading equations as graphs

2026-09-04

What a formula looks like, and what each number in it does. Every plot draws the same base function several times with one constant changed, so the effect of that constant is the difference between the curves. Read left to right: as $x$ grows, what happens to $y$ ?

Building blocks

Equation	Shape	As $x$ grows
$y = c$	flat line at height $c$	nothing changes
$y = x$	straight line through the origin, slope 1	$y$ grows at the same rate
$y = x^2$	U shape, bottom at the origin	grows faster and faster
$y = x^3$	S shape through the origin	grows even faster, odd symmetry
$y = \sqrt{x}$	half of a sideways U, starts at 0	grows, but slower and slower
$y = 1/x$	two hyperbola branches, never touches the axes	shrinks toward 0
$y = e^x$	hugs the axis on the left, shoots up on the right	grows, multiplying itself
$y = \ln x$	shoots down near 0, climbs slowly	grows, slower than any root
$y = \sin x$	wave between $-1$ and $1$	repeats every $2\pi$

constants y = c c = 2 c = 0.5 c = −1

the line y = xy = x

Adding and subtracting a number

Adding to the whole expression moves the graph up; subtracting moves it down. Adding to $x$ inside the function moves the graph left; subtracting moves it right. The shape never changes.

y = x + c x + 2 x x − 2

y = (x − c)² (x + 1)² x² (x − 2)²

Change	Effect
$f(x) + c$	up by $c$ (down if $c < 0$ )
$f(x - c)$	right by $c$ (left if $c < 0$ )
$f(x + c)$	left by $c$

Multiplying by a number

Multiplying the whole expression by $c$ makes it steeper ( $c > 1$ ), flatter ( $0 < c < 1$ ), or flips it upside down ( $c < 0$ ). Multiplying $x$ inside squeezes the graph horizontally.

y = c·x 2x x 0.5x −x

y = c·x² 2x² x² 0.5x² −x²

y = sin(c·x) sin 2x sin x sin(x/2)

y = c·sin x 2 sin x sin x 0.5 sin x

Change	Effect
$c \cdot f(x)$ , $c > 1$	taller: stretched away from the $x$ -axis
$c \cdot f(x)$ , $0 < c < 1$	shorter: squashed toward the $x$ -axis
$-f(x)$	flipped over the $x$ -axis
$f(c \cdot x)$ , $c > 1$	narrower: everything happens $c$ times sooner
$f(c \cdot x)$ , $0 < c < 1$	wider: everything happens later
$f(-x)$	mirrored over the $y$ -axis

Dividing by a number

Dividing by $c$ is multiplying by $1/c$ : the graph gets flatter. Dividing by $x$ is a different animal: $1/x$ blows up near $0$ and fades toward $0$ far away.

y = x / c x x/2 x/4

y = c / x 1/x 2/x −1/x

Equation	Near $x = 0$	Far from $0$
$c/x$	shoots to $\pm\infty$ (vertical asymptote)	fades to $0$ (horizontal asymptote)
$c/x^2$	shoots to $+\infty$ on both sides	fades to $0$ faster
$1/(x - a)$	asymptote moves to $x = a$

Powers of x

For $x > 1$ a bigger exponent grows faster; for $0 < x < 1$ a bigger exponent is smaller. Every $x^c$ passes through $(1, 1)$ .

y = xᶜ near 1 x^0.5 x^1 x^2 x^3

even vs odd x² x³

Exponent	Shape
even ( $x^2, x^4$ )	U shape, symmetric, always $\ge 0$
odd ( $x^3, x^5$ )	S shape, negative on the left, positive on the right
between 0 and 1 ( $\sqrt{x}$ )	rises quickly then flattens; only for $x \ge 0$
negative ( $x^{-1}$ )	$1/x$ : asymptotes at both axes

Exponentials: a number to the power x

Base bigger than 1 grows, base between 0 and 1 decays, and the further the base is from 1 the faster it happens. Every $c^x$ passes through $(0, 1)$ . Multiplying $x$ by $k$ in $e^{kx}$ sets the rate: $k > 0$ grows, $k < 0$ decays.

y = cˣ eˣ 2ˣ 1.5ˣ 0.5ˣ

y = e^(k·x) e^2x eˣ e^0.5x e^−x

Equation	Behavior
$c^x$ , $c > 1$	growth; doubles every $\ln 2 / \ln c$ units of $x$
$c^x$ , $0 < c < 1$	decay toward $0$ , never reaching it
$e^{kx}$	growth for $k > 0$ , decay for $k < 0$ ; doubling time $\ln 2 / k$
$A e^{kx}$	starts at $A$ when $x = 0$
$e^{x} + c$	horizontal asymptote moves from $0$ to $c$

Logarithms

The inverse of exponentials: every log passes through $(1, 0)$ , climbs forever but ever more slowly, and plunges to $-\infty$ as $x \to 0$ . A bigger base is a flatter curve. Multiplying $x$ inside a log only shifts the curve up, because $\ln(cx) = \ln x + \ln c$ .

y = log_c x log₂ x ln x log₁₀ x

y = ln(c·x) ln 4x ln 2x ln x

Who wins over time

On the way up, exponentials eventually beat every power, and every power beats every log. On the way down, $e^{-x}$ fades fastest, then $1/x^2$ , then $1/x$ .

growth 2ˣ x² x ln x

decay e^−x 1/x² 1/x

Rank (large $x$ )	Family	Example
fastest	exponential	$2^x$ , $e^x$
	polynomial, higher power first	$x^3$ , $x^2$ , $x$
	roots	$\sqrt{x}$
slowest	logarithmic	$\ln x$

Waves: A sin(ωx + φ) + C

Four numbers, four separate effects. Amplitude $A$ sets the height, angular frequency $\omega$ sets how many waves fit in a stretch, phase $\phi$ slides the wave sideways by $-\phi/\omega$ , and $C$ lifts the center line.

amplitude A 2 sin x sin x 0.5 sin x

frequency ω sin 2x sin x sin x/2

phase φ sin(x + π/2) sin x sin(x − π/2)

offset C sin x + 1 sin x sin x − 1

Reading an equation

You see	Expect
only $x$ to the first power	a straight line; the coefficient is the slope
highest power even	both ends go the same way (up if the coefficient is positive)
highest power odd	ends go opposite ways
$x$ in a denominator	a vertical asymptote where the denominator is $0$
$x$ in an exponent	growth or decay that outruns any power
$x$ inside a log	only defined for positive arguments, slow growth
$x$ inside sin or cos	a wave; what multiplies $x$ sets the frequency
a number added outside	the whole graph moves up or down
a number added inside	the whole graph moves left or right
a number multiplied outside	taller or shorter; negative flips it
a number multiplied inside	narrower or wider; negative mirrors it
$\sqrt{\;}$ or even root	starts at the point where the inside becomes $0$
$\lvert \cdots \rvert$	the negative part folds up above the axis

References

Desmos graphing calculator (opens in a new tab): add sliders for $a$ , $b$ , $h$ , $k$ and watch each constant move the curve
Paul's Online Math Notes: Transformations (opens in a new tab): shifts, stretches and reflections with worked examples
OpenStax, Precalculus 2e (opens in a new tab): function transformations, exponential, logarithmic and periodic graphs
Ivan Savov, No bullshit guide to math & physics (opens in a new tab): sections 1.11 to 1.13 on functions and their graphs
Khan Academy: Algebra 2 (opens in a new tab): transformations of functions, with practice problems