hi guys! in this video we're gonna learn
the basics of quadratic function. before
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okay, let's get started. first I'll show
you a sample so you have a more
in-depth impression of the quadratic
function. there is a 20-meter fence which
can be used to enclose a rectangular
area. what is the area of the rectangle
with different length values? we assume
the length of the rectangles area is X
20 meters is the perimeter of the fence's
area because the area is a rectangle, and
the perimeter is the sum of twice the
length and twice the width. the width can
be described as 20 minus 2x divided by 2
which can be simplified into 10-X
we know the area of the rectangle is the
product of the length and the width that
is x times parenthesis 10 minus X which
equals 10x minus x squared or negative x
squared plus 10x so the area can be
described as negative x squared plus 10x
now let's make a table to see the values
better. in this table the area of the
rectangle is y changes when the length x
changes. this relationship between the
two variables can be formed in a
function y equals negative x squared
plus 10x. generally speaking the
function is a form of y equals ax
squared plus BX plus C, where a B and C
are numbers and a cannot be equal to 0
the function describes the relationship
between the two variables. X is the input
and Y is the output. the value of y
changes when the value of x changes. in
other words, its value depends on the
value of x. you may say this is familiar
yeah, if y equals zero, the function turns
into a quadratic equation x squared plus
BX plus C equals zero. actually the
quadratic equation we learned about in
the previous videos is a special form of
the quadratic function. please check out
the link in the description if you want
to learn more about the quadratic
equation. just like the linear function,
the quadratic function can also be
graphed. let's try to graph the function
of the previous example. in the table
there are groups of numbers. 2 and 16, 3
and 21, 4 and 24, 5 and 25, 6 and 24, 7 and 21
and 8 and 16. all of them should be on the
graph. we can see the graph of the
quadratic function y equals ax squared
plus BX plus C is always a u-shaped
curve which is called the parabola. now
let's talk about some important
characteristics of the parabola. first
the coefficient a is very important and
can determine how wide or narrow the
graphs are whether the graph opens
upward or downward. the parabola opens
upward if a is greater than 0. the parabola
opens downward if a is less than zero
the smaller the value of a is, the wider
the parabola opens. and the bigger the
value of a is, the narrower the parabola
opens. second when we look at the
parabola we can see that it has the
lowest point if it opens upward. that
means the function Y has a minimum value
in this case. on the contrary it has a
highest point if it opens downward and Y
has a maximum value. the lowest or the
highest point is called the vertex but
how can you determine the value of the
vertex? the result is when x equals
negative B divided by 2a, the function
will have a vertex, which value is C
minus B Square divided by 4a. keep this
in mind and it will help you solve
problems with the quadratic function
now let's see how to prove it. let's use
the complete the square formula, just add
and subtract B Square divided by 4a at
the right side of the function and get y
equals ax squared plus bx plus b squared
divided by 4 a minus b squared divided
by 4 a + c. factoring out a from the
first three terms, the function turns
into y equals a times parentheses x
squared plus b / a x plus b squared
divided by 4 a squared plus C minus b
squared divided by 4a. x squared plus B
divided by a X plus b squared divided by
4 a squared is the square of parentheses
X plus B divided by 2a so we get y equals
a times parentheses X plus B divided by
2 a squared plus C minus B squared
divided by 4 a. when x equals negative B
divided by 2a, the first part equals 0 so
when a is greater than 0 and x equals
negative B divided by 2a, the function
has a minimum value of C minus B square divided by 4a. when a is less than 0 and
x equals negative B divided by 2a, the
function has a maximum value of C minus
B squared divided by 4a, so we get the
value of the vertex negative B divided
by 2a and C minus B squared divided by 4a. it's a little complicated but it can
help you understand the vertex better
and make you more comfortable with the
completing the square formula. now let's
look back at the graph of the quadratic
function. just by looking at it you can
see that it is symmetrical and it's line
of symmetry passes through the vertex
that means the vertex line x equals
negative B divided by 2a is the axis of
symmetry of the parabola. okay now that
we've finished learning the basic
concepts of the quadratic function. let's
look at an example together to better
understand the
concept. y equals x squared plus 3x plus
2. what is the vertex and the axis of
symmetry of this function? here a equals
one so another parabola opens upward and
has the minimum value. to get the value
of the vertex, we just plug the value of
a, B and C in. negative B divided by 2a
and C minus B squared divided by 4a
negative B divided by 2a equals negative
three over two C minus B squared divided
by four a equals two minus 9 over
4 which equals negative one over 4
so the vertex is negative 3 over 2 and
negative 1 over 4 but don't forget
the axis of symmetry is the vertex line
of x equals negative B divided by 2a
which is x equals negative 3 over
2. I think we've already learned enough
for today. in the next video we will
learn how to draw a graph of the
quadratic function so please stay tuned
if you have any questions leave a
comment below or email us at question @beat-math.com we will try to answer
your questions in a future video. and the
most common question will be given
priority in our next video. see you next
time
