Introduction to Quadratic Function | Examples of Quadratic Function

Summary

In this video, Teacher Gone discusses quadratic functions, starting with their definition as second-degree polynomials and the requirement that the coefficient of x² (a) must not be zero. The video also covers different forms of quadratic functions, such as the standard form and vertex form, and explains the graph of a quadratic function, which is a U-shaped parabola. Teacher Gone illustrates how to identify quadratic functions from equations, tables, and graphs while comparing them to linear functions. The video is aimed at providing a clear understanding of quadratic functions and their properties.

Takeaways

• 📚 The video focuses on explaining the quadratic function, which is a second-degree polynomial.
• 🔢 A quadratic function is represented by f(x) = ax² + bx + c, or equivalently y = ax² + bx + c, where a ≠ 0.
• ❗ If a = 0, the function becomes a linear function instead of a quadratic function.
• 🔍 The graph of a quadratic function is a U-shaped curve called a parabola.
• 🧮 Quadratic functions have two main forms: standard form (f(x) = ax² + bx + c) and vertex form (f(x) = a(x-h)² + k).
• 📝 The vertex of a parabola is the highest or lowest point depending on whether the parabola opens upward or downward.
• 📏 The axis of symmetry of a parabola is a vertical line passing through the vertex, often the y-axis.
• 🔬 To determine if an equation is quadratic, the highest exponent of the variable x must be 2.
• 📊 The second difference in a table of values is constant for quadratic functions, which helps identify them.
• 📈 Quadratic functions can be recognized from their graphs (parabolas), tables of values, and equations.

Q & A

What is a quadratic function?

A quadratic function is a second-degree polynomial represented as f(x) = ax² + bx + c or y = ax² + bx + c, where a, b, and c are real numbers, and a ≠ 0.

Why should 'a' in a quadratic function not be equal to zero?

If 'a' is equal to zero, the function becomes a linear function instead of a quadratic one because it eliminates the x² term, reducing the degree of the polynomial to one.

What is the standard form of a quadratic function?

The standard form of a quadratic function is f(x) = ax² + bx + c, or equivalently, y = ax² + bx + c.

What is the vertex form of a quadratic function?

The vertex form of a quadratic function is f(x) = a(x - h)² + k, or y = a(x - h)² + k, where (h, k) represents the vertex of the parabola.

What shape does the graph of a quadratic function take?

The graph of a quadratic function is a parabola, which is U-shaped.

What is the axis of symmetry in a quadratic function?

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. In many cases, it corresponds to the y-axis.

How do you identify a quadratic function based on its equation?

To identify a quadratic function, check the degree of the polynomial. If the highest exponent of the variable x is 2, then the function is quadratic.

How can you determine if a function represented by a table of values is quadratic?

You can determine if a function is quadratic by calculating the first and second differences between consecutive values. If the second differences are constant, the function is quadratic.

What does the graph of a linear function look like?

The graph of a linear function is a straight line.

How can you identify a quadratic function based on its graph?

A quadratic function can be identified by its graph, which is a U-shaped curve called a parabola.

Outlines

00:00

📚 Introduction to Quadratic Functions

In this opening, the teacher introduces the topic of quadratic functions. The function is defined as a second-degree polynomial, represented by either f(x) = ax^2 + bx + c or y = ax^2 + bx + c. It is emphasized that the coefficient a must not equal zero, as this would make the function linear instead of quadratic. Key characteristics of quadratic functions are discussed, including their classification as second-degree polynomials. The distinction between quadratic and linear functions is highlighted, with linear functions being first-degree polynomials.

05:02

🔢 Forms of Quadratic Functions and Their Graphs

This section delves into two key forms of quadratic functions: standard form f(x) = ax^2 + bx + c and vertex form f(x) = a(x - h)^2 + k. The teacher explains that both forms can be represented as y = ax^2 + bx + c and y = a(x - h)^2 + k, respectively. The graph of a quadratic function is described as a parabola, which is U-shaped. The comparison between linear and quadratic graphs is made clear, with linear graphs forming straight lines and quadratic graphs forming parabolas.

10:03

🌀 Understanding the Parts of a Parabola

Here, the teacher discusses the key parts of a parabola, which is the graph of a quadratic function. The concept of the vertex, which can be the lowest or highest point of the parabola, depending on whether it opens upward or downward, is introduced. Additionally, the axis of symmetry, typically represented by the y-axis, is explained as the line that divides the parabola into two mirror-image halves.

🤔 Identifying Quadratic Functions in Various Representations

This part focuses on identifying quadratic functions using different methods of representation: equations, tables of values, and graphs. The teacher provides examples, starting with an equation f(x) = 6x - 11, which is identified as linear due to its first-degree nature. The second example, f(x) = 2x^2 + x - 7, is correctly classified as a quadratic function because its degree is two.

📈 Using Tables of Values to Confirm Quadratic Functions

The teacher explains how to determine if a function is quadratic by analyzing the first and second differences in a table of values. By calculating these differences, it is shown that if the second differences are constant, the function is quadratic. Examples are provided, demonstrating the process of finding the first and second differences and confirming whether a function is quadratic.

📝 Recognizing Quadratic Functions by Their Graphs

This section explains how to identify quadratic functions by examining their graphs. Linear functions are represented by straight lines, while quadratic functions are represented by parabolas. The teacher illustrates how to distinguish between these types of functions visually and emphasizes the unique characteristics of the parabola as the graph of a quadratic function.

Keywords

💡Quadratic Function

A quadratic function is a second-degree polynomial, commonly written as f(x) = ax² + bx + c or y = ax² + bx + c. It plays a central role in the video as the main topic of discussion. The speaker emphasizes that a quadratic function has a degree of 2, meaning the highest exponent of the variable x is 2. Additionally, the video explains that the coefficient 'a' must not be zero; otherwise, it becomes a linear function.

💡Polynomial

A polynomial is an algebraic expression that involves variables and coefficients, where the variables have non-negative integer exponents. In the video, the term is specifically used to describe quadratic functions, which are second-degree polynomials. The speaker contrasts quadratic polynomials with linear polynomials, which have a degree of 1.

💡Standard Form

The standard form of a quadratic function is expressed as f(x) = ax² + bx + c, where a, b, and c are real numbers. The video explains this form as the most common way to represent quadratic functions, making it easier to identify the coefficients and terms that shape the parabola.

💡Vertex Form

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The video describes this form as useful for identifying the vertex, or the turning point, of the quadratic graph. The form is emphasized for its role in determining the shape and position of the parabola.

💡Parabola

A parabola is the U-shaped graph of a quadratic function. In the video, the speaker highlights that every quadratic function produces a parabola, either opening upwards or downwards. The graph's shape is influenced by the sign of the coefficient 'a' in the quadratic function.

💡Axis of Symmetry

The axis of symmetry is the vertical line that divides the parabola into two mirror-image halves. In the video, the speaker explains that this axis often corresponds to the y-axis or a vertical line passing through the vertex of the parabola. The axis of symmetry helps in graphing and analyzing the properties of quadratic functions.

💡Vertex

The vertex is the highest or lowest point on the graph of a quadratic function, depending on whether the parabola opens upwards or downwards. The video explains that the vertex serves as the critical point in the graph, where the direction of the curve changes. In the vertex form, the vertex is given by the coordinates (h, k).

💡Degree of Polynomial

The degree of a polynomial is the highest exponent of the variable in the function. In the case of quadratic functions, the degree is 2, as emphasized in the video. The speaker contrasts this with linear functions, which have a degree of 1, thus showing how the degree determines the type of function.

💡Linear Function

A linear function is a first-degree polynomial, represented by an equation like f(x) = mx + b. In the video, the speaker contrasts linear functions with quadratic functions, explaining that linear functions result in straight-line graphs, whereas quadratic functions create parabolas. The key difference lies in the degree of the polynomial.

💡Second Differences

Second differences refer to the differences between consecutive first differences in a sequence of function values, and they are constant for quadratic functions. The video includes an example where the speaker uses a table of values to calculate second differences, confirming that a quadratic function is represented when these differences are constant.

Highlights

Introduction to quadratic functions and their general form: f(x) = ax² + bx + c.

Key point: 'a' must not be zero; otherwise, the function becomes linear.

Explanation of the vertex form: f(x) = a(x - h)² + k.

Graph of a quadratic function is a parabola, which can either open upward or downward.

Definition of a parabola's vertex: lowest point when the parabola opens upward and highest point when it opens downward.

Introduction of the axis of symmetry, typically along the y-axis.

Identifying whether a function is quadratic by looking at the highest degree of the polynomial (degree 2 for quadratic).

Example 1: f(x) = 6x - 11 is not quadratic because its highest exponent is 1 (linear).

Example 2: f(x) = 2x² + x - 7 is quadratic due to the highest exponent being 2.

Using table of values and first and second differences to identify if a function is quadratic.

First differences calculation example: differences between y-values of consecutive points to see if a pattern exists.

Second differences being constant is a hallmark of a quadratic function in a table of values.

Example 3: Table of values where the second differences are consistent indicates a quadratic function.

Distinction between quadratic and linear functions by observing their graphs: linear functions form straight lines, while quadratic functions form parabolas.

Reminder to subscribe to the channel for more educational content.

Transcripts

hi guys it's me teacher going in

today's video we will talk about

quadratic function

so without further ado let's do this

topic

so we have here a definition of

quadratic function

let me read this one a quadratic

function

is a second degree polynomial

represented as

f of x is equal to ax squared plus b

x plus c or we can also represent it

as y is equal to a x squared

plus b x plus c because we all know that

f of x in function notation is equal to

y and then another thing the opportunity

on the iron about quadratic function

is that your a should be not equal to

zero

because i put the giving zero ca that

function

is a linear function so again

where a is not equal to zero where a and

b

a b and c are real numbers so that's the

definition or basic definition of

quadratic function

first degree polynomial alumni

degree polynomial is a linear function

so bear in mind that

when you saw a function that is a second

degree polynomial

you need to think of it as a quadratic

function

so another thing about quadratic

function is that

yuma forms america the first one is

standard form our standard form

is represented by f of x is equal to ax

squared

plus bx plus c or we can also represent

it

as y is equal to

ax squared plus bx

plus c because that f of x is equal to

y another thing done about quadratic

function

is unity targeting vertex form we have

here

f of x is equal to a times the quantity

of x minus

h squared plus k where in

we can also write it using y is equal to

a times x minus h squared

plus k so in the later part

the other part of our video

standard form at a vertex format n so

another note about quadratic function

is that you need to bear in mind that

the graph

the graph of a quadratic function is a

parabola

so in your graded mathematics or in your

previous function in a linear

we can produce a straight line but here

in quadratic function and graphene is a

parabola

it is u-shaped okay

okay so another thing about the

quadratic function i

am graphing a u-shaped line called

the parabola so right now we will talk

about the different parts of a parabola

so anua you need to think

two different types of parabola

so what are the parts of the parabola

basically

you need to know the first part in a

predicament

opening so as you can see you arrows

so this is the

opening

of the

[Music]

[Music]

or you say you can consider the highest

point of the parabola

since you adding parabola opens upward

automatic

your vertex is considered as the lowest

point of the parabola

so what if

axis of symmetry so ditos a graph net

and since

you're adding y-axis at the nothing

y-axis

parabola that is considered as the axis

of symmetry

so

again in this parabola you add the axis

of symmetry a y-axis

axis of

symmetry okay

[Music]

now for this part of our video an ecb

number netting

part anatomy identify whether the given

representation of a function

is a quadratic function let's say we

have three different ways on how to

represent the function one is by

equation

at human legend representation here

one is utilizing table values

type no representation is a function a

by graph

identify whether the given

representation is a quadratic function

or not

una let's have here number one

f of x is equal to 6x minus 11. so

basically

a new definition is a quadratic function

they must have been on second degree

polynomial

but here in our function the degree of

this polynomial is

one okay or in other words the highest

exponent of

the variable x is one meaning this one

is not

a quadratic function i'm not in

quadratic function is a second degree

polynomial

answer if

one as the degree of polynomial is a

function

function the quadratic function this one

is merely

a linear function

okay linear function n now

let's move on to item number two we have

f of x is equal to two x squared

plus x minus seven so an analysis look

at the exponent or the high exponent of

the variable x this one is two

tuan canyon highest exponent meaning the

degree of this polynomial

function is two so it is nothing

quadratic function

okay beyond so you know nothing

quadratic functions

degree or your highest exponent and

variable is two

now let's move on with the third one

um function table of values

first the fair and

sastaya so the surprise first difference

says

zero minus one subtract means zero

at one that's equal to negative one then

after that

number two negative one and then coding

difference number three zero so you have

three

minus zero boxing of me and that is

minus three that is seven the numerator

seven next

21 minus 10 so we have 21

minus 10 so that is

11 followed by 36

minus 21. 36

minus 21 it will give you 15.

so 15 dio atom a number these are your

first differences

quadratic

differences as you can see

you adding first differences

so let's subtract three and negative one

so that is

three minus negative one so hypothesis

objective that is

four positive four so forty total again

followed by seven

minus three seven minus three it will be

four so

okay function

okay followed by eleven minus seven

so you have four so sir

so you have 15 minus eleven so that is

four

this table values represent a quadratic

conscious battle again represents a

quadratic function so let's move on with

the item number four number five

ethereum uh combined

whether chinese and quadratic function

so

in number four as you can see that is a

straight line

so basically my knowledge here about

a linear function alumni is a graph

non-linear function is a straight line

so basically

this one is a linear

function so

in other words that is not

quadratic

okay so another thing number five let's

move on to

the last item this one is a quadratic

function

quadratic function in different forms

the quadratic function

imparts num graph and quadratic function

which is parabola

actually different ways on how to

identify whether the

representation of a function is

quadratic or not

so again come back holland's youtube

channel could don't forget to

like and subscribe bell button for

updated statement of future uploads

again i am teacher gone my name is