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Mathematics I · Week 3 — Quadratic functions

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Mathematics I · Week 3 — Quadratic functions

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6 min read

2026-08-16

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Now · Week map

Week 3 — quadratic functions

Quiz 2 scope: Weeks 1–8 per IITM May 2026 foundation courses. Source baseline: IITM BS admissions important-dates calendar · May 2026 cycle. Times on assessments are operational conventions — verify hall ticket.

Part of the Quiz 2 prep system · Weeks 1–8 index · Pattern atlas · Formula chains.

Week map

Standard form → factoring/roots → vertex → discriminant → graph features

Classify → Represent → Execute → Trap-check

Recognize: Ask: What does $D=0$ imply about the graph?
Procedure: Look for two numbers that multiply to $ac$ and add to $b$ when $a=1$ . Set each factor to zero. Verify by substitution. If monic and prime over integers, state that and switch to formula.
Variations / traps: Watch for: Sign error in $h = -b/(2a)$ when $b$ is negative.

Formula chain (compressed)

f(x)=ax²+bx+c → roots via factor/formula → Δ=b²−4ac → vertex xᵥ=−b/(2a).

Standard form — f(x) = ax² + bx + c, a≠0 — identify quadratic
Discriminant — Δ = b² − 4ac — count real roots
Quadratic formula — x = (−b ± √Δ)/(2a) — factoring hard
Vertex x — xᵥ = −b/(2a) — turning point
Opening — a>0 cup (min), a<0 cap (max) — graph sketch

Open interactive formula desk · Week 3 tab.

Deep study

Mathematics I · Week 3 — Quadratic functions

Parabolas link algebra (polynomial degree 2), graphs (vertex and symmetry), and root-finding.

Week map

Standard form

ax^2+bx+c

→ graph shape → factoring and roots → quadratic formula → vertex and axis → discriminant → parameter problems.

Quadratic notation

$f(x) = ax^2 + bx + c$ → standard quadratic → $a \neq 0$ → $f(x) = 2x^2 - 4x + 1$ .
$a$ → leading coefficient → sign sets cup up ( $a>0$ ) or down ( $a<0$ ).
Root / zero → input where $f(x)=0$ → solve $ax^2+bx+c=0$ .
$h = -\frac{b}{2a}$ → $x$ -coordinate of vertex → axis of symmetry vertical line $x=h$ .
$D = b^2 - 4ac$ → discriminant → classifies real roots.

Vertex form

f(x) = a(x-h)^2 + k

→ vertex

(h,k)

→ graph shifts

h

horizontal,

k

vertical.

Mini-example:

f(x) = 3(x-2)^2 - 5

has vertex

(2,-5)

, opens upward (

a=3>0

Roots and the discriminant

$D$	Real roots
$D > 0$	two distinct
$D = 0$	one repeated (touch axis)
$D < 0$	none (no horizontal crossing)

Quadratic formula:

x = \frac{-b \pm \sqrt{D}}{2a}

when solving

ax^2+bx+c=0

Mini-example:

x^2 - 5x + 6 = 0

. Factor

(x-2)(x-3)=0

→ roots

2,3

. Check

D = 25-24 = 1 > 0

Factoring strategy (monic $a=1$ )

Find integers

p,q

with

pq = c

and

p+q = b

for

x^2+bx+c

x^2 + x - 12

: need product

-12

, sum

1

→

4

and

-3

→

(x+4)(x-3)

If no integer pair works, use formula or completing the square.

Completing the square sketch

x^2 + 6x + 1 = (x^2+6x+9) - 9 + 1 = (x+3)^2 - 8

. Vertex

(-3,-8)

Graph features

Axis of symmetry: $x = h$ .
$y$ -intercept: $f(0) = c$ .
Direction: min value at vertex if $a>0$ ; max if $a<0$ .
Width: larger $|a|$ → narrower parabola.

Pattern families

Easy — Factor and solve

Monic quadratics with integer roots.
Read vertex from $(x-h)^2+k$ when given.
State $y$ -intercept from $c$ .

Medium — Vertex and transformation

Convert standard to vertex form by completing square or formula for $h,k$ .
Given roots, write quadratic in factored form $a(x-r_1)(x-r_2)$ .
Use symmetry: roots equidistant from axis at $h$ .

Hard — Discriminant parameters

Find values of $k$ so $x^2 + kx + 9 = 0$ has exactly one real root → $D=0$ .
Inequalities on $D$ for “no real roots” or “two real roots”.
Word constraints linking vertex height to domain (e.g. minimum cost).

Worked mini-examples

Example 1 — Factoring.

2x^2 - 8x + 6 = 0

. Divide 2:

x^2 - 4x + 3 = 0

→

(x-1)(x-3)=0

→

x=1,3

Example 2 — Vertex.

f(x) = -x^2 + 4x + 1

h = -4/(2\cdot -1) = 2

f(2) = -4+8+1 = 5

. Vertex

(2,5)

, max (since

a=-1

Example 3 — Discriminant.

3x^2 - 2x + k = 0

has no real roots when

D = 4 - 12k < 0

→

k > \frac{1}{3}

Example 4 — From roots. Roots

-2

and

5

f(x) = a(x+2)(x-5)

. If

a=1

f(x) = x^2 - 3x - 10

c = -10

y

-intercept.

Traps

Sign error in $h = -b/(2a)$ when $b$ negative.
Vertex $(h,k)$ confused with roots (zeros on $x$ -axis).
Forgetting $a$ when using formula: divide by $2a$ , not 2.
$D=0$ means one repeated root, not “no roots”.
Opening direction from sign of $a$ , not $c$ .

Diagnostic (try yourself)

Solve $x^2 + 7x + 10 = 0$ by factoring.
Find the vertex and axis of symmetry of $f(x) = 2x^2 - 12x + 5$ .
For what values of $m$ does $x^2 + mx + 16 = 0$ have exactly one real solution?
A parabola has roots at $x = 1$ and $x = 9$ and passes through $(0, 18)$ . Write $f(x)$ in standard form.
Without fully solving, how many real roots does $2x^2 + 3x - 7 = 0$ have? Use the discriminant.

ChatGPT prep archive

Chatty reference archive

Archived import for extra depth — complements the notes above, not official IITM material.

Core concepts

Standard form $f(x) = ax^2 + bx + c$ ; parabola opens up if $a>0$ , down if $a<0$ .
Roots from factoring, completing the square, or quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ .
Vertex $(h,k)$ from $h = -b/(2a)$ , $k = f(h)$ ; vertex form $a(x-h)^2 + k$ .
Discriminant $D = b^2 - 4ac$ : positive two real roots, zero one repeated, negative no real roots.

Notation & vocabulary

Form	Use
$ax^2+bx+c$	coefficients, discriminant
$a(x-h)^2+k$	vertex, transformations
$D=b^2-4ac$	root count (real)

Pattern families

Easy — Factor simple quadratics

Look for two numbers that multiply to

ac

and add to

b

when

a=1

. Set each factor to zero. Verify by substitution. If monic and prime over integers, state that and switch to formula.

Medium — Vertex and axis

Compute

h = -b/(2a)

and evaluate

f(h)

for vertex. Axis of symmetry is vertical line

x=h

. Compare vertex

y

-value with roots to sketch direction and width from

a

Hard — Parameter in discriminant

Set discriminant condition for desired root type:

D>0

D=0

, or

D<0

. Solve inequality for parameter

k

. Boundary cases

D=0

often mark transition points—test them separately.

Drill these on the pattern atlas — filter to week 3.

Traps

Sign error in $h = -b/(2a)$ when $b$ is negative.
Confusing vertex $(h,k)$ with roots.
Using quadratic formula without checking for factorization first on easy integers.
Assuming two roots always means positive discriminant without checking reality.

Retrieval prompts

What does $D=0$ imply about the graph?
How do you convert $f(x)=x^2-6x+5$ to vertex form?
Which direction opens for $f(x)=-2x^2+3x$ ?

Practice loop

Read Deep study (if present) or core concepts once.
Recite the formula chain without looking.
Open one easy pattern on the interactive atlas for week 3.
Attempt without solutions; mark studied after an honest try.
Say one trap aloud before closing the tab.