Important graphs that appear in CBSE boards โ drawn visually so Ishu (My Wife) can
sketch them accurately in the exam. Visual learning = permanent memory! ๐ฏ
๐ Trigonometric Graphs
AOI ยท AO Diff
Sine Function
$y = \sin x$
Domain: $(-\infty, \infty)$, Range: $[-1, 1]$
Period: $2\pi$, Odd function
Passes through $(0,0)$, max at $\pi/2$
$\int_0^\pi \sin x\,dx = 2$ โ common AOI!
Cosine Function
$y = \cos x$
Domain: $(-\infty, \infty)$, Range: $[-1, 1]$
Period: $2\pi$, Even function ($\cos(-x)=\cos x$)
Max at $x=0$, zero at $\pi/2$
Tangent Function
$y = \tan x$
Vertical asymptotes at $x = \pm\pi/2, \pm3\pi/2$
Period: $\pi$. Range: $(-\infty, \infty)$
Strictly increasing on each interval
๐ต Inverse Trig Graphs
Board Fav
Inverse Sine
$y = \sin^{-1}x$
Domain: $[-1, 1]$, Range: $[-\pi/2, \pi/2]$
Strictly increasing, odd function
Key points: $(-1,-\pi/2),(0,0),(1,\pi/2)$
Inverse Cosine
$y = \cos^{-1}x$
Domain: $[-1, 1]$, Range: $[0, \pi]$
Strictly decreasing
Key points: $(-1,\pi),(0,\pi/2),(1,0)$
Inverse Tangent
$y = \tan^{-1}x$
Domain: $\mathbb{R}$, Range: $(-\pi/2, \pi/2)$
Horizontal asymptotes at $y=\pm\pi/2$
$\tan^{-1}(0)=0$, strictly increasing
๐ Curve & Conic Section Graphs
AOI Must Know
Parabola
$y^2 = 4ax$ and $y = x^2$
$y^2=4ax$: opens right, vertex at origin
$y=x^2$: opens up, vertex at origin
Area under $y=x^2$ from $0$ to $a$ = $\frac{a^3}{3}$
Area = $\pi r^2$
Circle
$x^2 + y^2 = r^2$
Centre at origin, radius $r$
Area = $\pi r^2$ (via integration: $4\int_0^r\sqrt{r^2-x^2}\,dx$)
Upper half: $y=\sqrt{r^2-x^2}$
Diff + Integral
Exponential & Log
$y = e^x$ and $y = \ln x$
$e^x$: always positive, passes $(0,1)$
$\ln x$: domain $(0,\infty)$, passes $(1,0)$
They are mirror images in $y=x$
๐ท Special Curves
Continuity Ch
Modulus Function
$y = |x|$
V-shaped, vertex at $(0,0)$
Continuous everywhere but NOT differentiable at $x=0$!