Introduction to Quadratic Function | Examples of Quadratic Function
Summary
TLDRIn 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
📚 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.
🔢 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.
🌀 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.
Mindmap
Keywords
💡Quadratic Function
💡Polynomial
💡Standard Form
💡Vertex Form
💡Parabola
💡Axis of Symmetry
💡Vertex
💡Degree of Polynomial
💡Linear Function
💡Second Differences
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
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