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# 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.

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.