Graphs of functions — plotting, reading key features

What is a graph of a function?

Why this definition? A function assigns exactly one output to each input. On a graph, this means: vertical line through any $x$ in the domain hits the curve at exactly one point. This is the vertical line test: if a vertical line crosses the curve twice, it's not a function.

How to plot a function from scratch

Step-by-step derivation of the plotting process:

Identify the domain: What $x$ -values are allowed?
- For $f(x) = \sqrt{x}$ , domain is $x \geq 0$ (can't square-root negatives in reals).
- For $f(x) = \frac{1}{x}$ , domain is $x \neq 0$ (division by zero undefined).
Sample strategic points: Choose $x$ -values that reveal structure.
- Include $x = 0$ (if in domain) — often special.
- Include points where $f(x) = 0$ (roots/zeros).
- Include points where $f(x)$ might blow up (asymptotes).
- Spread points across the domain (negative, zero, positive).
Compute $y = f(x)$ for each sample: Just plug in.
Plot $(x, y)$ pairs: Mark each point on graph paper or axes.
Connect smoothly: Draw a curve through the points. For polynomials and smooth functions, use a flowing curve. For piecewise or discontinuous functions, respect jumps/breaks.
Extend the pattern: If the domain is infinite, sketch behavior as $x \to \infty$ or $x \to -\infty$ .

Why these steps? Sampling gives you anchor points. Connecting reveals the trend. Strategic choices (like zeros, undefined points) highlight the function's critical features.

Key features to read from a graph

Formula Reading features

From a graph, extract:

Domain: All $x$ where the curve exists.
Range: All $y$ -values the curve reaches.
Intercepts:
- $x$ -intercept (root/zero): where curve crosses $x$ -axis, i.e., $f(x) = 0$ .
- $y$ -intercept: where curve crosses $y$ -axis, i.e., $x = 0$ , so point is $(0, f(0))$ .
Increasing/Decreasing intervals:
- Increasing on $(a, b)$ if $x_1 < x_2 \implies f(x_1) < f(x_2)$ (curve goes uphill left-to-right).
- Decreasing if curve goes downhill.
Local maxima/minima: Peaks and valleys. Points where the function changes from increasing to decreasing (max) or vice versa (min).
Asymptotes: Lines the curve approaches but never touches.
- Vertical asymptote: $x = a$ if $f(x) \to \pm\infty$ as $x \to a$ .
- Horizontal asymptote: $y = L$ if $f(x) \to L$ as $x \to \pm\infty$ .
Symmetry:
- Even function: $f(-x) = f(x)$ → symmetric about $y$ -axis.
- Odd function: $f(-x) = -f(x)$ → symmetric about origin (rotate 180°).
Continuity: Any breaks, jumps, holes?

Why these features? They summarize the function's entire behavior. Instead of a formula, you get: "This function is positive on $(−\infty, 2)$ , zero at $x=2$ , then negative, with a vertical asymptote at $x=5$ ." That's a complete story.

Worked examples

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

Setup: Plot and identify features.

Step 1: Domain? All real $x$ (no restrictions). Domain = $\mathbb{R}$ .

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: $(2, 1)$ .

Why these points? $x=0$ gives the $y$ -intercept. Spread over positive integers to see the pattern.

Step 3: Plot and connect. Straight line through $(0, -3)$ , $(1, -1)$ , $(2, 1)$ .

Step 4: Features.

$y$ -intercept: $(0, -3)$ .
$x$ -intercept: Set $f(x) = 0 \implies 2x - 3 = 0 \implies x = \frac{3}{2}$ . Point: $(\frac{3}{2}, 0)$ .
Increasing on $\mathbb{R}$ (slope = 2 > 0).
No max/min (line extends forever).
Range: $\mathbb{R}$ (all $y$ values covered).

Figure — Graphs of functions — plotting, reading key features

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

Step 1: Domain = $\mathbb{R}$ .

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: $(4, 3)$ .

Why these? Spread across integers. Notice $x=1, 3$ give zeros.

Step 3: Plot Parabola opens upward (coefficient of $x^2$ is positive).

Step 4: Features.

$y$ -intercept: $(0, 3)$ .
$x$ -intercepts: $(1, 0)$ and $(3, 0)$ .
Vertex (minimum point): Parabola's turning point. For $ax^2 + bx + c$ , vertex at $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$ . Then $g(2) = -1$ . Vertex: $(2, -1)$ .
Decreasing on $(-\infty, 2)$ , increasing on $(2, \infty)$ .
Range: $[-1, \infty)$ (lowest $y$ is $-1$ , goes up forever).
Symmetry: Parabola is symmetric about $x = 2$ (the vertex $x$ -coordinate).

Why this step-by-step? Each point confirms the parabola's shape. The vertex formula comes from completing the square: $x^2 - 4x + 3 = (x-2)^2 - 1$ , so minimum at $(2, -1)$ .

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

Step 1: Domain? Denominator $\neq 0 \implies x \neq 2$ . Domain: $\mathbb{R} \setminus \{2\}$ .

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

$x = 0$ : $h(0) = \frac{1}{0-2} = -\frac{1}{2}$ . 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: Behavior near $x=2$ .

As $x \to 2^-$ (from left), $x - 2 \to 0^-$ (small negative), so $\frac{1}{x-2} \to -\infty$ .
As $x \to 2^+$ (from right), $x - 2 \to 0^+$ (small positive), so $\frac{1}{x-2} \to +\infty$ .
Vertical asymptote at $x = 2$ .

Step 4: Behavior as $x \to \pm\infty$ .

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

Step 5: Features.

$y$ -intercept: $(0, -\frac{1}{2})$ .
No $x$ -intercept (numerator = 1, never zero).
Vertical asymptote: $x = 2$ .
Horizontal asymptote: $y = 0$ .
Two branches: the left branch (for $x < 2$ ) lies below the $x$ -axis, curving down to $-\infty$ near $x=2$ . Note it passes through quadrant III (for $x<0$ ) and quadrant IV (for $0<x<2$ ), since $y<0$ throughout. The right branch (for $x > 2$ ) lies in quadrant I, coming down from $+\infty$ toward $y=0$ .
Range: $\mathbb{R} \setminus \{0\}$ (never equals zero).

Why these steps? Near the asymptote, the function "blows up." Far away, it flattens toward zero. Sampling on both sides reveals the two-branch structure.

Common mistakes

Why it feels right: You want a smooth curve, and your brain interpolates.

Fix: Check the domain first. If $x=2$ is excluded (like $\frac{1}{x-2}$ ), leave a gap or asymptote, don't connect through it.

Why it feels right: Drawing to infinity is hard on paper.

Fix: Draw a dashed vertical line at $x=2$ (the asymptote), and show the curve approaching it on both sides. Label it "asymptote." This communicates that the function explodes there.

Why it feels right: Visually, "up" = increasing.

Fix: Increasing means as you move right, $y$ increases. A curve can go up-then-down (like a parabola). Check each interval separately. For $x^2 - 4x + 3$ , it's decreasing on $(-\infty, 2)$ and increasing on $(2, \infty)$ .

Fix: Always check $f(0)$ if $0$ is in the domain. It's the easiest point to compute and anchors the vertical position of the curve.

Active recall practice

Recall Feynman explanation (explain to a 12-year-old)

Imagine you have a magic box (a function). You drop a number in (input $x$ ), and it spits out another number (output $f(x)$ ). A graph is like taking a photo of the box's behavior for every possible input. You draw a dot for each (input, output) pair. When you connect all the dots, you get a curve that shows the box's "mood." Does it always spit out bigger numbers as you give it bigger inputs? (Increasing.) Does it suddenly freak out and spit out infinity at some input? (Asymptote.) Does it give the same output for $x$ and $-x$ ? (Even function, mirror image.) Reading the graph is like reading the box's diary: you know everything it does without having to test every single input yourself.

For increasing/decreasing: "If it climbs right, it's increasing in sight."

Connections

Domain and range — First step before plotting.
Types of functions — Shape depends on type (linear = line, quadratic = parabola, etc.).
Vertical line test — Tool to verify it's a function from the graph.
Transformations of functions — Shifting/stretching graphs without recomputing points.
Asymptotes — Behavior at boundaries (limits).
Symmetry (even and odd functions) — Reduces work: plot half, reflect.
Solving equations graphically — Find $f(x) = k$ by reading $y = k$ intersection.
Calculus: derivatives and graphs — Slope of the tangent = increasing/decreasing intervals.

#flashcards/maths

What is the graph of a function? :: The set of all points $(x, f(x))$ in the plane, where $x$ is in the domain. Visually: a curve where each $x$ maps to exactly one $y$ .

What is the vertical line test?

If a vertical line crosses the curve more than once, it's not a function (one input would give multiple outputs).

How do you find the

y

-intercept from a graph?

Look where the curve crosses the

y

-axis. This is the point

(0, f(0))

(plug in

x=0

How do you find

x

-intercepts (zeros) from a graph?

Look where the curve crosses the

x

-axis. These are points where

f(x) = 0

What does it mean for a function to be increasing on an interval?

x

increases,

f(x)

also increases. Visually: the curve goes uphill (left to right).

What is a vertical asymptote?

A vertical line

x = a

where

f(x) \to \pm\infty

x \to a

. The curve shoots up/down near

x = a

but never touches the line.

What is a horizontal asymptote?

A horizontal line

y = L

where

f(x) \to L

x \to \pm\infty

. The curve flattens out toward this height at the far left/right.

How do you identify the domain from a graph?

The domain is all

x

-values where the curve exists. Look for gaps, asymptotes, or endpoints.

How do you identify the range from a graph?

The range is all

y

-values the curve reaches. Look at the lowest and highest

y

on the curve, including behavior at infinity.

What is the vertex of a parabola

ax^2 + bx + c

The turning point (max or min), located at

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

. Plug this

x

into the function to get the

y

-coordinate.

How do you spot an even function from its graph?

The graph is symmetric about the

y

-axis. If you fold the graph along the

y

-axis, both halves match.

How do you spot an odd function from its graph?

The graph has rotational symmetry about the origin (rotate 180° and it looks the same). Equivalently:

f(-x) = -f(x)

Why sample strategic points when plotting?

To capture the function's key behavior (zeros, asymptotes, turning points) with fewer calculations. Random points might miss important features.

What is a local maximum on a graph?

A point where the function reaches a peak (higher than nearby points). The curve changes from increasing to decreasing.

What is a local minimum on a graph?

A point where the function reaches a valley (lower than nearby points). The curve changes from decreasing to increasing.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek function ko machine samjho — input daalo, output milta hai. Lekin ek-ek karke f(3), f(4), f(5) nikaalte rahoge toh bore ho jaoge aur pura picture nahi dikhega. Yahi pe graph ka jaadu aata hai: ye function ki puri kahani ek saath dikha deta hai. Curve dekhte hi tumhe pata chal jaata hai ki function kahan badh raha hai, kahan gir raha hai, kahan explode ho raha hai — bina algebra kiye. Basically graph function ki personality hai, ek visual story jo tumhara brain instantly pehchaan leta hai.

Ab plotting ka process simple hai — pehle domain nikaalo (kaunse x-values allowed hain), phir strategic points choose karo jaise x=0, roots (jahan f(x)=0 hota hai), ya jahan function blow up karta hai. In points ke liye y=f(x) compute karo, plot karo, aur smoothly connect kar do. Ek important cheez yaad rakhna — vertical line test: agar koi vertical line curve ko do baar cut kare, toh wo function nahi hai, kyunki ek input ka sirf ek hi output ho sakta hai.

Graph se tum bahut saari cheezein padh sakte ho — domain, range, x aur y intercepts, increasing/decreasing intervals, local maxima-minima (peaks aur valleys), asymptotes (jin lines ko curve chhoo nahi paata bas paas jaata hai), aur symmetry (even function y-axis ke around symmetric, odd function origin ke around). Ye sab features milke function ki poori behavior summarize kar dete hain. Isliye graph reading itni powerful skill hai — formula yaad rakhne se zyada, tum bas dekh kar function ko samajh jaate ho. Exam mein aur real understanding mein dono jagah ye tumhe aage le jaayega.

Graphs of functions — plotting, reading key features

What is a graph of a function?

How to plot a function from scratch

Key features to read from a graph

Worked examples

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

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

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

Common mistakes

Active recall practice

Connections

Concept Map

Hinglish (regional understanding)

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Functions

Connections