Week 3: Quadratic Functions, Graphs & Equations

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2026-09-12T00:00:00.000Z

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Parabolas, vertex, axis of symmetry, discriminant, nature of roots, Vieta's relations, and quadratic optimization. # Mathematics I — Week 3: Quadratic Functions, Graphs & Equations IITM’s Week 3 covers **quadratic functions, minima/maxima, vertex and slope-related interpretation, and quadratic equations**.

Mathematics I — Week 3: Quadratic Functions, Graphs & Equations

IITM’s Week 3 covers quadratic functions, minima/maxima, vertex and slope-related interpretation, and quadratic equations.

TL;DR

A quadratic function has the form

f(x)=ax^2+bx+c,\qquad a\neq 0.

Its graph is a parabola.

The most important Week 3 ideas are:

standard form, factored form, vertex form
roots / $x$ -intercepts
$y$ -intercept
axis of symmetry
vertex
minimum or maximum
discriminant
quadratic formula
relation between roots and coefficients
how changing $a,b,c$ changes the graph

1. What is a quadratic function?

A quadratic function is:

\boxed{ f(x)=ax^2+bx+c }

where:

a,b,c\in\mathbb R

and:

a\neq0.

Why must

a\neq0

Because if:

a=0,

then:

f(x)=bx+c,

which is linear, not quadratic.

2. Why "quadratic"?

The word comes from the Latin quadratus, meaning "square."

That is because the highest power of the variable is:

x^2.

So:

3x^2+2x+1

is quadratic.

But:

x^3+2x

is cubic.

3. Degree

For:

f(x)=ax^2+bx+c,

the degree is:

2.

The degree is the highest exponent of

x

with a nonzero coefficient.

4. Graph of a quadratic

The graph of a quadratic function is a parabola.

Example:

y=x^2

looks roughly like:


text
      y
      ↑
      |
   *     *
    *   *
     * *
      *
------+------→ x

It is symmetric around the

y

-axis.

5. Direction of opening

The coefficient

a

determines whether the parabola opens upward or downward.

If $a>0$

The parabola opens upward:

\cup

It has a minimum.

If $a<0$

The parabola opens downward:

\cap

It has a maximum.

This is one of the fastest graph facts to identify.

6. Effect of magnitude of $a$

Compare:

y=x^2

y=3x^2

y=\frac12x^2.

Larger:

|a|

makes the parabola appear narrower/steeper.

Smaller:

|a|

makes it appear wider/flatter.

7. Standard form

The standard form is:

\boxed{ f(x)=ax^2+bx+c }

This form is especially useful for:

identifying coefficients
using the quadratic formula
finding the discriminant
getting the $y$ -intercept immediately

At:

x=0,

we get:

f(0)=c.

Therefore the

y

-intercept is:

\boxed{(0,c)}.

8. Roots / zeros / $x$ -intercepts

A root or zero is a value of

x

satisfying:

f(x)=0.

For:

f(x)=ax^2+bx+c,

roots satisfy:

ax^2+bx+c=0.

Graphically, roots are where the parabola intersects the

x

-axis.

9. Example

Consider:

f(x)=x^2-5x+6.

Set:

f(x)=0.

Then:

x^2-5x+6=0.

Factor:

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

Therefore:

x=2

or:

x=3.

So the

x

-intercepts are:

(2,0),\qquad(3,0).

10. Factored form

If roots are:

r_1,\qquad r_2,

then a quadratic can be written as:

\boxed{ f(x)=a(x-r_1)(x-r_2) }

This form makes the roots immediately visible.

Example:

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

has roots:

1

and:

-4.

11. Axis of symmetry

Every parabola has a vertical line through its center called the axis of symmetry.

For:

f(x)=ax^2+bx+c,

the axis is:

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

This is one of the most important Week 3 formulas.

12. Vertex

The vertex is the turning point of the parabola.

Its

x

-coordinate is:

x_v=-\frac{b}{2a}.

Then:

y_v=f\left(-\frac{b}{2a}\right).

Therefore:

\boxed{ V= \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) }

13. Minimum or maximum

If:

a>0,

the vertex is the minimum point.

If:

a<0,

the vertex is the maximum point.

So once you know the vertex and the sign of

a

, you know the extremum.

14. Example: vertex

Consider:

f(x)=x^2-6x+5.

Here:

a=1,\qquad b=-6.

Thus:

x_v= -\frac{-6}{2(1)} = 3.

Now:

f(3)=9-18+5=-4.

Therefore:

\boxed{V=(3,-4)}.

Since:

a>0,

the parabola opens upward.

So:

\boxed{\text{minimum value}=-4}.

15. Vertex form

A quadratic can also be written as:

\boxed{ f(x)=a(x-h)^2+k }

This is called vertex form.

The vertex is immediately:

\boxed{(h,k)}.

The axis of symmetry is:

\boxed{x=h}.

16. Example

Consider:

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

Then:

h=3,\qquad k=5.

Therefore vertex:

(3,5).

Since:

a=2>0,

it opens upward.

Hence minimum value:

5.

17. Completing the square

Completing the square converts standard form into vertex form.

Start with:

f(x)=ax^2+bx+c.

Factor

a

from the first two terms:

f(x) = a\left( x^2+\frac ba x \right)+c.

Now add and subtract:

\left(\frac{b}{2a}\right)^2.

Then:

f(x) = a\left[ \left( x+\frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 \right]+c.

So:

f(x)= a\left( x+\frac{b}{2a} \right)^2 + c-\frac{b^2}{4a}.

Thus the vertex is:

\boxed{ \left( -\frac{b}{2a}, c-\frac{b^2}{4a} \right) }

18. Alternative expression for vertex $y$ -coordinate

Since:

\Delta=b^2-4ac,

we can write:

c-\frac{b^2}{4a} = -\frac{b^2-4ac}{4a} = -\frac{\Delta}{4a}.

Therefore:

\boxed{ V= \left( -\frac{b}{2a}, -\frac{\Delta}{4a} \right) }

This is a useful connection between the graph and discriminant.

19. Quadratic equation

A quadratic equation is:

\boxed{ ax^2+bx+c=0 }

with:

a\neq0.

The goal is to find values of

x

satisfying the equation.

20. Solving by factorization

Example:

x^2-7x+12=0.

We want two numbers whose:

product is $12$
sum is $-7$

Those numbers are:

-3,\quad-4.

So:

(x-3)(x-4)=0.

Hence:

x=3

or:

x=4.

21. Zero-product principle

The reason factorization works is:

AB=0

implies:

A=0

or:

B=0.

Thus:

(x-3)(x-4)=0

means:

x-3=0

or:

x-4=0.

22. Quadratic formula

For:

ax^2+bx+c=0,

the roots are:

\boxed{ x= \frac{-b\pm\sqrt{b^2-4ac}}{2a} }

The expression:

b^2-4ac

is called the discriminant.

23. Discriminant

Define:

\boxed{ \Delta=b^2-4ac }

It tells us the nature of the roots.

24. Case 1: $\Delta>0$

If:

\Delta>0,

there are two distinct real roots.

Graphically, the parabola crosses the

x

-axis at two distinct points.


text
\        /
 \      /
--\----/------ x
   \  /
    \/

25. Case 2: $\Delta=0$

If:

\Delta=0,

there is one repeated real root.

The two roots are equal:

x= -\frac b{2a}.

Graphically, the parabola just touches the

x

-axis at its vertex.

26. Case 3: $\Delta<0$

If:

\Delta<0,

there are no real roots.

Graphically, the parabola never intersects the

x

-axis.

Over complex numbers, there are still two complex roots.

But for real-coordinate graph interpretation:

\boxed{\text{no real }x\text{-intercepts}}

27. Root/discriminant cheat sheet

Discriminant	Roots	Graph
$\Delta>0$	two distinct real	crosses $x$ -axis twice
$\Delta=0$	one repeated real	touches $x$ -axis
$\Delta<0$	no real roots	no $x$ -axis intersection

28. Deriving the quadratic formula

This is worth understanding once instead of memorizing blindly.

Start:

ax^2+bx+c=0.

Divide by

a

x^2+\frac ba x+\frac ca=0.

Move the constant:

x^2+\frac ba x = -\frac ca.

Complete the square:

x^2+\frac ba x + \frac{b^2}{4a^2} = -\frac ca + \frac{b^2}{4a^2}.

Then:

\left( x+\frac b{2a} \right)^2 = \frac{b^2-4ac}{4a^2}.

Take square roots:

x+\frac b{2a} = \pm \frac{\sqrt{b^2-4ac}}{2a}.

Thus:

x= \frac{-b\pm\sqrt{b^2-4ac}}{2a}.

29. Sum and product of roots

Suppose the roots are:

\alpha,\beta.

Then:

a(x-\alpha)(x-\beta) = ax^2-a(\alpha+\beta)x+a\alpha\beta.

Compare this with:

ax^2+bx+c.

Therefore:

\boxed{ \alpha+\beta=-\frac ba }

and:

\boxed{ \alpha\beta=\frac ca }

These are extremely useful.

30. Example

For:

2x^2-7x+3=0,

we have:

a=2,\quad b=-7,\quad c=3.

Therefore sum of roots:

\alpha+\beta = -\frac{-7}{2} = \frac72.

Product:

\alpha\beta = \frac32.

Without even solving the equation.

31. Constructing a quadratic from roots

If roots are:

\alpha,\beta,

then:

(x-\alpha)(x-\beta)=0

gives:

x^2-(\alpha+\beta)x+\alpha\beta=0.

Example:

Roots:

2,\quad5.

Then:

(x-2)(x-5)=0

so:

x^2-7x+10=0.

32. Repeated root

If both roots equal:

r,

then:

f(x)=a(x-r)^2.

This is exactly why the parabola touches the

x

-axis at one point.

That point is also the vertex.

33. Relationship between roots and axis of symmetry

If roots are:

r_1,\qquad r_2,

then the axis of symmetry lies exactly halfway between them:

\boxed{ x= \frac{r_1+r_2}{2} }

But:

r_1+r_2=-\frac ba.

Therefore:

x= -\frac b{2a}.

So the familiar vertex formula is also just:

the midpoint of the roots.

Very useful intuition.

34. Example

Suppose roots are:

2,\qquad8.

Then axis:

x= \frac{2+8}{2} = 5.

The parabola is symmetric around:

x=5.

35. Domain of a quadratic

For the ordinary quadratic:

f(x)=ax^2+bx+c,

there are no restrictions on

x

Therefore:

\boxed{ \text{Domain}=\mathbb R }

36. Range of a quadratic

The range depends on the vertex.

If:

a>0,

the vertex gives the minimum:

y_{\min}=k.

Then:

\boxed{ \text{Range}=[k,\infty) }

for vertex:

(h,k).

If:

a<0,

the vertex gives the maximum:

\boxed{ \text{Range}=(-\infty,k] }

37. Example

For:

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

vertex:

(2,-5).

Since:

a=1>0,

minimum:

-5.

Thus:

\boxed{ \text{Range}=[-5,\infty) }

38. Transformations from $y=x^2$

The base parabola is:

y=x^2.

Now consider:

y=(x-h)^2+k.

This shifts the graph:

right by $h$ if $h>0$
left by $|h|$ if $h<0$
up by $k$ if $k>0$
down by $|k|$ if $k<0$

39. The sign trap

For:

y=(x-3)^2,

the graph shifts right by 3, not left.

Why?

The vertex occurs when:

x-3=0,

so:

x=3.

Similarly:

y=(x+4)^2

has vertex at:

x=-4.

The sign inside the bracket appears reversed.

40. Reflection

If:

y=f(x),

then:

y=-f(x)

reflects the graph across the

x

-axis.

Thus:

y=x^2

becomes:

y=-x^2.

Upward-opening becomes downward-opening.

41. Vertical stretching/compression

For:

y=ax^2:

$|a|>1$ gives vertical stretching
$0<|a|<1$ gives vertical compression

Visually:

stretched → narrower
compressed → wider

42. Finding a quadratic from three points

A quadratic has three unknown coefficients:

a,b,c.

So three suitable points usually determine it uniquely.

Suppose it passes through:

(0,1),\quad(1,4),\quad(2,9).

Let:

f(x)=ax^2+bx+c.

From

(0,1)

c=1.

From

(1,4)

a+b+1=4

so:

a+b=3.

From

(2,9)

4a+2b+1=9

so:

2a+b=4.

Subtract:

(2a+b)-(a+b)=4-3

thus:

a=1.

Then:

b=2.

Therefore:

\boxed{ f(x)=x^2+2x+1 }

which is:

(x+1)^2.

43. Slope of a quadratic is not constant

For a straight line:

y=mx+c,

slope is constant.

For a quadratic:

y=ax^2+bx+c,

the steepness changes with

x

This is why the graph curves.

Later, calculus formalizes this using the derivative:

f'(x)=2ax+b.

You do not need derivatives yet to understand the idea:

the slope changes continuously as you move along the parabola.

44. What happens at the vertex?

At the vertex:

decreasing switches to increasing, or
increasing switches to decreasing.

So the tangent there becomes horizontal.

In later calculus language:

\text{slope}=0.

Indeed:

2ax+b=0

gives:

x=-\frac b{2a}.

Exactly the vertex formula.

This is a nice preview of Week 8 derivatives.

45. Increasing and decreasing behavior

For:

a>0,

the parabola decreases until:

x=-\frac b{2a},

then increases afterward.

So:

f(x)\text{ decreases on } \left(-\infty,-\frac b{2a}\right)

and increases on:

\left(-\frac b{2a},\infty\right).

For:

a<0,

the behavior reverses.

46. Sign of a quadratic from its graph

Suppose:

a>0

and there are two roots:

r_1<r_2.

Then:

f(x)>0

outside the roots:

x<r_1

or:

x>r_2.

And:

f(x)<0

between them:

r_1<x<r_2.

For a downward-opening parabola, signs reverse.

This becomes useful for quadratic inequalities.

47. Example

Consider:

f(x)=(x-2)(x-5).

Roots:

2,\quad5.

Since:

a=1>0,

the graph opens upward.

Therefore:

f(x)>0

for:

x<2

or:

x>5.

And:

f(x)<0

for:

2<x<5.

48. Quadratic inequalities

Suppose:

x^2-5x+6>0.

Factor:

(x-2)(x-3)>0.

Critical points:

2,\quad3.

Since the parabola opens upward:

f(x)>0

outside the roots.

Thus:

\boxed{ x<2\quad\text{or}\quad x>3 }

Common trap: root vs vertex

Roots are points where:

y=0.

The vertex is the turning point.

They are different concepts.

Example:

f(x)=x^2-4.

Roots:

x=\pm2.

Vertex:

(0,-4).

Common trap: vertex

x

-coordinate

For:

f(x)=ax^2+bx+c,

the vertex is not:

-\frac ba.

It is:

\boxed{ -\frac b{2a} }

The factor

2

matters.

Common trap: discriminant

For:

ax^2+bx+c=0,

the discriminant is:

\boxed{ b^2-4ac }

not:

b^2+4ac.

Common trap:

\pm

Quadratic formula:

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

The

\pm

produces two possible roots:

x_1= \frac{-b+\sqrt\Delta}{2a}

and:

x_2= \frac{-b-\sqrt\Delta}{2a}.

Unless:

\Delta=0,

in which case both coincide.

Common trap:

(-b)^2

-b^2

These are different:

(-b)^2=b^2

but:

-b^2=-(b^2).

For example:

(-3)^2=9

while:

-3^2=-9.

Parentheses matter.

Common trap: interpreting

c

In:

f(x)=ax^2+bx+c,

c

gives the $y$ -intercept, not necessarily an

x

-intercept.

Because:

f(0)=c.

55. Three useful forms of the same quadratic

Suppose:

f(x)=x^2-6x+5.

Standard form

x^2-6x+5

Useful for:

coefficients
discriminant
quadratic formula

Factored form

(x-1)(x-5)

Useful for:

roots

Vertex form

(x-3)^2-4

Useful for:

vertex
minimum
axis of symmetry

Each representation exposes different information.

This is an important general mathematical pattern:

Same object, different representation, different information becomes obvious.

56. Full graph anatomy example

Take:

f(x)=x^2-4x+3.

Step 1: opening

a=1>0

so it opens upward.

Step 2: $y$ -intercept

c=3

so:

(0,3).

Step 3: roots

x^2-4x+3 = (x-1)(x-3).

So:

x=1,3.

Step 4: axis

x= -\frac{-4}{2} = 2.

Step 5: vertex

f(2)=4-8+3=-1.

Therefore:

V=(2,-1).

Step 6: range

Since it opens upward:

y\geq -1.

So:

\boxed{ [-1,\infty) }

That is essentially a complete quadratic analysis.

57. Fast exam workflow

Given:

f(x)=ax^2+bx+c,

a good order is:


text
1. Read a,b,c
2. Check sign of a
3. Find y-intercept = c
4. Calculate Δ = b² - 4ac
5. Find roots if needed
6. Find axis x = -b/(2a)
7. Find vertex
8. Infer min/max and range
9. Sketch

This minimizes duplicated work.

58. Connections to previous weeks

Week 1 → functions

A quadratic is a particular function:

f:\mathbb R\to\mathbb R

usually written:

f(x)=ax^2+bx+c.

Week 2 → coordinate geometry

The graph consists of points:

(x,y)

satisfying:

y=ax^2+bx+c.

Roots are simply intersections with the

x

-axis.

The

y

-intercept is the intersection with the

y

-axis.

So Week 3 builds directly on Week 2.

59. One deeper mental model

Think of:

y=ax^2+bx+c

as producing an output

y

for every input

x

The graph is simply a picture of all input-output pairs:

\{(x,f(x)):x\in\mathbb R\}.

That idea applies to every function you'll study later.

60. Formula sheet

Standard form

\boxed{ f(x)=ax^2+bx+c }

Axis of symmetry

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

Vertex

\boxed{ \left( -\frac b{2a}, f\left(-\frac b{2a}\right) \right) }

or:

\boxed{ \left( -\frac b{2a}, -\frac{\Delta}{4a} \right) }

Vertex form

\boxed{ f(x)=a(x-h)^2+k }

Quadratic formula

\boxed{ x= \frac{-b\pm\sqrt{b^2-4ac}}{2a} }

Discriminant

\boxed{ \Delta=b^2-4ac }

Sum of roots

\boxed{ \alpha+\beta=-\frac ba }

Product of roots

\boxed{ \alpha\beta=\frac ca }

61. What you should actually master

By the end of Week 3, you should be able to:

Recognize a quadratic function.
Identify $a,b,c$ correctly.
Determine whether its parabola opens up or down.
Find the $y$ -intercept.
Find roots by factorization.
Use the quadratic formula.
Calculate and interpret the discriminant.
Find the axis of symmetry.
Find the vertex.
Determine minimum/maximum value.
Find the range.
Convert between standard, factored and vertex forms.
Use root sum/product relations.
Sketch a parabola from algebraic information.
Solve simple quadratic inequalities.

If those are comfortable, Week 3 is under control.

62. Concept map


text
Quadratic
f(x)=ax²+bx+c
│
├── coefficient a
│   ├── a>0 → minimum
│   └── a<0 → maximum
│
├── intercepts
│   ├── y-intercept → c
│   └── x-intercepts → roots
│
├── discriminant
│   ├── Δ>0 → 2 real roots
│   ├── Δ=0 → repeated root
│   └── Δ<0 → no real roots
│
├── symmetry
│   └── x=-b/(2a)
│
├── vertex
│
└── representations
    ├── standard
    ├── factored
    └── vertex

Quick Recall & Flashcards

Test your active recall before proceeding to practice problems. Format:

Question :: Answer.

Vertex $x$ -coordinate for $y = ax^2 + bx + c$ :: $x = -\frac{b}{2a}$
Discriminant formula for $ax^2 + bx + c = 0$ :: $D = b^2 - 4ac$
Roots nature when discriminant $D > 0$ :: Two distinct real roots
Roots nature when discriminant $D = 0$ :: One real repeated root (tangent to axis)
Roots nature when discriminant $D < 0$ :: Two complex conjugate roots (no x-intercepts)
Vieta's formula for sum of roots :: $\alpha + \beta = -\frac{b}{a}$
Vieta's formula for product of roots :: $\alpha \beta = \frac{c}{a}$
Parabola opening direction if $a > 0$ :: Upward (has a global minimum at vertex)
Parabola opening direction if $a < 0$ :: Downward (has a global maximum at vertex)

Practice & Self-Check

Attempt these diagnostic questions to verify your operational understanding.

Question 1

What is the

x

-coordinate of the vertex of the parabola

f(x) = 2x^2 - 8x + 11

NAT

3 Unit Assessment

Question 2

\alpha

and

\beta

are the roots of

2x^2 - 6x + 5 = 0

, what is the value of

\alpha + \beta

MCQ

3 Unit Assessment

A. 3

B. -3

C. 5/2

D. -5/2

What this note is really saying

Mathematics I — Week 3: Quadratic Functions, Graphs & Equations

TL;DR

1. What is a quadratic function?

2. Why "quadratic"?

3. Degree

4. Graph of a quadratic

5. Direction of opening

If a>0a>0a>0

If a<0a<0a<0

6. Effect of magnitude of aaa

7. Standard form

8. Roots / zeros / xxx-intercepts

9. Example

10. Factored form

11. Axis of symmetry

12. Vertex

13. Minimum or maximum

14. Example: vertex

15. Vertex form

16. Example

17. Completing the square

18. Alternative expression for vertex yyy-coordinate

19. Quadratic equation

20. Solving by factorization

21. Zero-product principle

22. Quadratic formula

23. Discriminant

24. Case 1: Δ>0\Delta>0Δ>0

25. Case 2: Δ=0\Delta=0Δ=0

26. Case 3: Δ<0\Delta<0Δ<0

27. Root/discriminant cheat sheet

28. Deriving the quadratic formula

29. Sum and product of roots

30. Example

31. Constructing a quadratic from roots

32. Repeated root

33. Relationship between roots and axis of symmetry

34. Example

35. Domain of a quadratic

36. Range of a quadratic

37. Example

38. Transformations from y=x2y=x^2y=x2

39. The sign trap

40. Reflection

41. Vertical stretching/compression

42. Finding a quadratic from three points

43. Slope of a quadratic is not constant

44. What happens at the vertex?

45. Increasing and decreasing behavior

46. Sign of a quadratic from its graph

47. Example

48. Quadratic inequalities

55. Three useful forms of the same quadratic

Standard form

Factored form

Vertex form

56. Full graph anatomy example

Step 1: opening

Step 2: yyy-intercept

Step 3: roots

Step 4: axis

Step 5: vertex

Step 6: range

57. Fast exam workflow

58. Connections to previous weeks

Week 1 → functions

Week 2 → coordinate geometry

59. One deeper mental model

60. Formula sheet

Standard form

Axis of symmetry

Vertex

Vertex form

Quadratic formula

Discriminant

Sum of roots

Product of roots

61. What you should actually master

62. Concept map

If $a>0$

If $a<0$

6. Effect of magnitude of $a$

8. Roots / zeros / $x$ -intercepts

18. Alternative expression for vertex $y$ -coordinate

24. Case 1: $\Delta>0$

25. Case 2: $\Delta=0$

26. Case 3: $\Delta<0$

38. Transformations from $y=x^2$

Step 2: $y$ -intercept