Graphs of functions — plotting, reading key features

2.2.6 · Maths › Functions

Ek function ka graph kya hota hai?

Yeh definition kyun? Ek function har input ko bilkul ek output assign karta hai. Graph par iska matlab hai: domain mein kisi bhi $x$ se guzarne wali vertical line curve ko exactly ek point par hit karti hai. Yahi vertical line test hai: agar koi vertical line curve ko do baar cross kare, toh woh function nahi hai.

Scratch se function ko kaise plot karein

Plotting process ki step-by-step derivation:

Domain identify karo: Kaun se $x$ -values allowed hain?
- $f (x) = x$ ke liye, domain $x \geq 0$ hai (reals mein negatives ka square-root nahi le sakte).
- $f (x) = \frac{1}{x}$ ke liye, domain $x \neq = 0$ hai (zero se division undefined hai).
Strategic points sample karo: Aise $x$ -values chuno jo structure reveal karein.
- $x = 0$ include karo (agar domain mein hai) — aksar special hota hai.
- Woh points include karo jahan $f (x) = 0$ ho (roots/zeros).
- Woh points include karo jahan $f (x)$ blow up ho sakta hai (asymptotes).
- Domain mein points spread karo (negative, zero, positive).
Har sample ke liye $y = f (x)$ compute karo: Bas plug in karo.
$(x, y)$ pairs plot karo: Har point ko graph paper ya axes par mark karo.
Smoothly connect karo: Points ke through curve draw karo. Polynomials aur smooth functions ke liye, flowing curve use karo. Piecewise ya discontinuous functions ke liye, jumps/breaks ka respect karo.
Pattern extend karo: Agar domain infinite hai, toh $x \to \infty$ ya $x \to - \infty$ par behavior sketch karo.

Yeh steps kyun? Sampling tumhe anchor points deta hai. Connect karna trend reveal karta hai. Strategic choices (jaise zeros, undefined points) function ke critical features highlight karte hain.

Graph se padhne wale key features

Formula Features padhna

Ek graph se, yeh extract karo:

Domain: Saare $x$ jahan curve exist karta hai.
Range: Saare $y$ -values jo curve reach karta hai.
Intercepts:
- $x$ -intercept (root/zero): jahan curve $x$ -axis cross karta hai, yaani $f (x) = 0$ .
- $y$ -intercept: jahan curve $y$ -axis cross karta hai, yaani $x = 0$ , toh point hai $(0, f (0))$ .
Increasing/Decreasing intervals:
- Increasing on $(a, b)$ agar $x_{1} < x_{2} ⟹ f (x_{1}) < f (x_{2})$ (curve left-to-right upar jaata hai).
- Decreasing agar curve neeche jaata hai.
Local maxima/minima: Peaks aur valleys. Woh points jahan function increasing se decreasing (max) ya vice versa (min) mein badalta hai.
Asymptotes: Woh lines jinke paas curve jaata hai lekin kabhi touch nahi karta.
- Vertical asymptote: $x = a$ agar $f (x) \to \pm \infty$ as $x \to a$ .
- Horizontal asymptote: $y = L$ agar $f (x) \to L$ as $x \to \pm \infty$ .
Symmetry:
- Even function: $f (- x) = f (x)$ → $y$ -axis ke baare mein symmetric.
- Odd function: $f (- x) = - f (x)$ → origin ke baare mein symmetric (180° rotate karo).
Continuity: Koi breaks, jumps, holes hain?

Yeh features kyun? Yeh function ke poore behavior ko summarize karte hain. Formula ki jagah, tumhe milta hai: "Yeh function $(- \infty, 2)$ par positive hai, $x = 2$ par zero hai, phir negative hai, $x = 5$ par vertical asymptote hai." Yeh ek complete kahani hai.

Worked examples

Example 1: Linear function $f (x) = 2 x - 3$

Setup: Plot karo aur features identify karo.

Step 1: Domain? Saare real $x$ (koi restriction nahi). Domain = $R$ .

Step 2: Sample points.

$x = 0$ : $f (0) = 2 (0) - 3 = - 3$ . Point: $(0, - 3)$ .
$x = 1$ : $f (1) = 2 (1) - 3 = - 1$ . Point: $(1, - 1)$ .
$x = 2$ : $f (2) = 2 (2) - 3 = 1$ . Point: $(2, 1)$ .

Yeh points kyun? $x = 0$ $y$ -intercept deta hai. Positive integers mein spread karo pattern dekhne ke liye.

Step 3: Plot karo aur connect karo. $(0, - 3)$ , $(1, - 1)$ , $(2, 1)$ se guzarne wali straight line.

Step 4: Features.

$y$ -intercept: $(0, - 3)$ .
$x$ -intercept: Set $f (x) = 0 ⟹ 2 x - 3 = 0 ⟹ x = \frac{3}{2}$ . Point: $(\frac{3}{2}, 0)$ .
$R$ par increasing (slope = 2 > 0).
Koi max/min nahi (line hamesha extend karti hai).
Range: $R$ (saare $y$ values covered).

Figure — Graphs of functions — plotting, reading key features

Example 2: Quadratic $g (x) = x^{2} - 4 x + 3$

Step 1: Domain = $R$ .

Step 2: Sample points.

$x = 0$ : $g (0) = 0 - 0 + 3 = 3$ . Point: $(0, 3)$ .
$x = 1$ : $g (1) = 1 - 4 + 3 = 0$ . Point: $(1, 0)$ .
$x = 2$ : $g (2) = 4 - 8 + 3 = - 1$ . Point: $(2, - 1)$ .
$x = 3$ : $g (3) = 9 - 12 + 3 = 0$ . Point: $(3, 0)$ .
$x = 4$ : $g (4) = 16 - 16 + 3 = 3$ . Point: $(4, 3)$ .

Yeh kyun? Integers mein spread karo. Notice karo $x = 1, 3$ zeros dete hain.

Step 3: Parabola plot karo jo upar ki taraf khulja hai ( $x^{2}$ ka coefficient positive hai).

Step 4: Features.

$y$ -intercept: $(0, 3)$ .
$x$ -intercepts: $(1, 0)$ aur $(3, 0)$ .
Vertex (minimum point): Parabola ka turning point. $a x^{2} + b x + c$ ke liye, vertex $x = - \frac{b}{2 a} = - \frac{- 4}{2 ( 1 )} = 2$ par. Phir $g (2) = - 1$ . Vertex: $(2, - 1)$ .
$(- \infty, 2)$ par decreasing, $(2, \infty)$ par increasing.
Range: $[- 1, \infty)$ (sabse chhota $y$ hai $- 1$ , upar hamesha jaata hai).
Symmetry: Parabola $x = 2$ ke baare mein symmetric hai (vertex ka $x$ -coordinate).

Yeh step-by-step kyun? Har point parabola ki shape confirm karta hai. Vertex formula completing the square se aata hai: $x^{2} - 4 x + 3 = (x - 2)^{2} - 1$ , toh minimum $(2, - 1)$ par.

Example 3: Rational function $h (x) = \frac{1}{x - 2}$

Step 1: Domain? Denominator $\neq = 0 ⟹ x \neq = 2$ . Domain: $R ∖ {2}$ .

Step 2: Sample points ( $x = 2$ avoid karo).

$x = 0$ : $h (0) = \frac{1}{0 - 2} = - \frac{1}{2}$ . Point: $(0, - \frac{1}{2})$ .
$x = 1$ : $h (1) = \frac{1}{1 - 2} = - 1$ . Point: $(1, - 1)$ .
$x = 1.5$ : $h (1.5) = \frac{1}{1.5 - 2} = \frac{1}{- 0.5} = - 2$ . Point: $(1.5, - 2)$ .
$x = 2.5$ : $h (2.5) = \frac{1}{2.5 - 2} = \frac{1}{0.5} = 2$ . Point: $(2.5, 2)$ .
$x = 3$ : $h (3) = \frac{1}{3 - 2} = 1$ . Point: $(3, 1)$ .
$x = 4$ : $h (4) = \frac{1}{4 - 2} = \frac{1}{2}$ . Point: $(4, \frac{1}{2})$ .

Step 3: $x = 2$ ke paas behavior.

Jab $x \to 2^{-}$ (left se), $x - 2 \to 0^{-}$ (chhota negative), toh $\frac{1}{x - 2} \to - \infty$ .
Jab $x \to 2^{+}$ (right se), $x - 2 \to 0^{+}$ (chhota positive), toh $\frac{1}{x - 2} \to + \infty$ .
Vertical asymptote $x = 2$ par.

Step 4: $x \to \pm \infty$ par behavior.

$x \to \infty$ : $\frac{1}{x - 2} \to 0$ .
$x \to - \infty$ : $\frac{1}{x - 2} \to 0$ .
Horizontal asymptote $y = 0$ par.

Step 5: Features.

$y$ -intercept: $(0, - \frac{1}{2})$ .
Koi $x$ -intercept nahi (numerator = 1, kabhi zero nahi).
Vertical asymptote: $x = 2$ .
Horizontal asymptote: $y = 0$ .
Do branches: left branch ( $x < 2$ ke liye) $x$ -axis ke neeche rehta hai, $x = 2$ ke paas $- \infty$ ki taraf curve karta hai. Note karo yeh quadrant III (jab $x < 0$ ) aur quadrant IV (jab $0 < x < 2$ ) se guzarta hai, kyunki $y < 0$ throughout rehta hai. Right branch ( $x > 2$ ke liye) quadrant I mein hai, $+ \infty$ se neeche $y = 0$ ki taraf aata hai.
Range: $R ∖ {0}$ (kabhi zero nahi hota).

Yeh steps kyun? Asymptote ke paas, function "blow up" karta hai. Door jaane par, yeh zero ki taraf flatten hota hai. Dono taraf sampling se two-branch structure reveal hoti hai.

Common mistakes

Yeh sahi kyun lagta hai: Tum ek smooth curve chahte ho, aur tumhara brain interpolate karta hai.

Fix: Pehle domain check karo. Agar $x = 2$ excluded hai (jaise $\frac{1}{x - 2}$ mein), toh wahan gap ya asymptote chhoddo, us se connect mat karo.

Yeh sahi kyun lagta hai: Paper par infinity tak draw karna mushkil hai.

Fix: $x = 2$ par ek dashed vertical line draw karo (asymptote), aur dikhao ki curve dono taraf se us ki taraf approach kar raha hai. Use "asymptote" label karo. Yeh communicate karta hai ki function wahan explode karta hai.

Yeh sahi kyun lagta hai: Visually, "upar" = increasing.

Fix: Increasing ka matlab hai jab tum right move karo, $y$ badhta hai. Ek curve upar-phir-neeche ja sakta hai (jaise parabola). Har interval alag check karo. $x^{2} - 4 x + 3$ ke liye, yeh $(- \infty, 2)$ par decreasing hai aur $(2, \infty)$ par increasing hai.

Fix: Hamesha $f (0)$ check karo agar $0$ domain mein hai. Yeh compute karne ka sabse aasaan point hai aur curve ki vertical position anchor karta hai.

Active recall practice

Recall Feynman explanation (12 saal ke bachche ko samjhao)

Socho tumhare paas ek magic box hai (function). Tum usme ek number daalo (input $x$ ), aur woh doosra number nikalta hai (output $f (x)$ ). Graph aise hai jaise har possible input ke liye box ke behavior ki photo lena. Tum har (input, output) pair ke liye ek dot draw karte ho. Jab tum saare dots connect karte ho, tumhe ek curve milta hai jo box ki "mood" dikhata hai. Kya yeh hamesha bade numbers spew out karta hai jab tum use bade inputs dete ho? (Increasing.) Kya yeh achanak freak out karta hai aur kisi input par infinity spew karta hai? (Asymptote.) Kya yeh $x$ aur $- x$ ke liye same output deta hai? (Even function, mirror image.) Graph padhna box ki diary padhne jaisa hai: bina har single input test kiye tum jaante ho woh kya karta hai.

Increasing/decreasing ke liye: "Agar right ki taraf chadhta hai, toh increasing dikhaai deta hai."

Connections

Domain and range — Plot karne se pehle pehla step.
Types of functions — Shape type par depend karta hai (linear = line, quadratic = parabola, etc.).
Vertical line test — Graph se verify karne ka tool ki yeh function hai.
Transformations of functions — Points recompute kiye bina graphs shift/stretch karna.
Asymptotes — Boundaries par behavior (limits).
Symmetry (even and odd functions) — Kaam kam karo: aadha plot karo, reflect karo.
Solving equations graphically — $f (x) = k$ find karo $y = k$ intersection padh ke.
Calculus: derivatives and graphs — Tangent ka slope = increasing/decreasing intervals.

#flashcards/maths

Ek function ka graph kya hota hai? :: Plane mein saare points $(x, f (x))$ ka set, jahan $x$ domain mein hai. Visually: ek curve jahan har $x$ exactly ek $y$ par map karta hai.

Vertical line test kya hai?

Agar koi vertical line curve ko ek se zyada baar cross kare, toh yeh function nahi hai (ek input multiple outputs deta).

Graph se

y

-intercept kaise find karte ho?

Dekho kahan curve

y

-axis cross karta hai. Yeh point

(0, f (0))

hai (

x = 0

plug in karo).

Graph se

x

-intercepts (zeros) kaise find karte ho?

Dekho kahan curve

x

-axis cross karta hai. Yeh woh points hain jahan

f (x) = 0

Ek function ka kisi interval par increasing hone ka matlab kya hai?

Jab

x

badhta hai,

f (x)

bhi badhta hai. Visually: curve upar jaata hai (left se right).

Vertical asymptote kya hota hai?

Ek vertical line

x = a

jahan

f (x) \to \pm \infty

x \to a

. Curve

x = a

ke paas upar/neeche shoot karta hai lekin line kabhi touch nahi karta.

Horizontal asymptote kya hota hai?

Ek horizontal line

y = L

jahan

f (x) \to L

x \to \pm \infty

. Curve bahut door left/right par is height ki taraf flatten hota hai.

Graph se domain kaise identify karte ho?

Domain woh saare

x

-values hain jahan curve exist karta hai. Gaps, asymptotes, ya endpoints dekho.

Graph se range kaise identify karte ho?

Range woh saare

y

-values hain jo curve reach karta hai. Curve par sabse neeche aur upar

y

dekho, infinity par behavior bhi.

Parabola

a x^{2} + b x + c

ka vertex kya hota hai?

Turning point (max ya min),

x = - \frac{b}{2 a}

par located.

y

-coordinate get karne ke liye yeh

x

function mein plug karo.

Even function ko graph se kaise pehchante ho?

Graph

y

-axis ke baare mein symmetric hota hai. Agar graph ko

y

-axis ke along fold karo, dono halves match karte hain.

Odd function ko graph se kaise pehchante ho?

Graph origin ke baare mein rotational symmetry rakhta hai (180° rotate karo aur wahi dikhta hai). Equivalently:

f (- x) = - f (x)

Plot karte waqt strategic points sample kyun karte hain?

Function ke key behavior (zeros, asymptotes, turning points) ko kam calculations mein capture karne ke liye. Random points important features miss kar sakte hain.

Graph par local maximum kya hota hai?

Woh point jahan function peak reach karta hai (nearby points se upar). Curve increasing se decreasing mein badalta hai.

Graph par local minimum kya hota hai?

Woh point jahan function valley reach karta hai (nearby points se neeche). Curve decreasing se increasing mein badalta hai.