Ways to represent a quadratic function

🏆Practice ways of representing the quadratic function

Ways to represent a quadratic function

Standard representation

The standard representation of the quadratic function looks like this:

y=ax2+bx+cy=ax^2+bx+c

When:

aa –

  • The coefficient of X2X^2
  • determines whether the parabola will be a maximum or minimum parabola (sad or smiling) and how open or closed it will be.
  • aa must be different from 00.
  • If aa is positive – the parabola is a minimum parabola – smiling
  • If aa is negative – the parabola is a maximum parabola – sad
  • The larger aa is – the narrower the function will be, and vice versa.

bb -

  • The coefficient of XX
  • can be any negative or positive number

   c    c –

  • The free term
  • can be any positive or negative number and determines the intersection point with the Y Y  axis


Let's see an example:

We are given the function:
y=5x2+2x+7y=5x^2+2x+7
What can we conclude?
A=5A=5 therefore the parabola is smiling
C=7C=7 therefore the function intersects the YY-axis at the point (0,7)(0,7)

Click here to learn more about the standard form of a quadratic function

Vertex presentation

The vertex form of a quadratic function allows us to identify its vertex directly from the function!

The vertex form of a quadratic function is:
Y=a(X−p)2+cY=a(X-p)^2+c

When:
PP - represents the XX value of the vertex.
CC - represents the YY value of the vertex.

For example:
In the function
Y=6(X−5)2+2Y=6(X-5)^2+2

The vertex of the parabola is: 
(2,5)(2,5)

Note-
The vertex formula is structured such that there is always a – before the PP, meaning XX vertex, but this does not necessarily mean that the XX vertex is negative.
If the parabola has a negative XX vertex, a ++ will appear before the PP in the formula because –– times –– equals ++.

For example:
For example:
In the function
Y=6(X+3)2+8Y=6(X+3)^2+8

The vertex of the parabola is:
(−3,8)(-3,8)
There is a ++ before the 33 in the formula, so it is −3-3.

Click here to learn more about the vertex form of a quadratic function

Representation as a product

The product form shows multiplication between 22 expressions. With the product form, we can easily determine the points of intersection of the function with the XX-axis.
The product form of the quadratic function looks like this:
y=(x−t)∗(x−k)y=(x-t)*(x-k)
where
tt and kk are the 22 points of intersection of the parabola with the XX-axis.
As follows: (t,0)(k,0)(t,0) (k,0)
Let's see an example of the product form to understand better:
y=(x−6)∗(x+5)y=(x-6)*(x+5)
We can determine that:
The points of intersection with the XX-axis are:
(6,0) (6,0) 
 (−5,0) ( -5,0)
Note - Since in the original template there is a minus before \(k\) and tt, we can infer that if there is a plus before one of them, it is negative, hence −5-5 and not 55.

Click here to learn more about representing as a product of a quadratic function

Other different representations

Different representations of a quadratic function –

Sometimes we encounter quadratic equations that are missing terms such as BXBX or CC and quadratic equations with denominators.

Quadratic equations with missing terms are quadratic equations where cc or bb are equal to 00.

When we have an incomplete equation where b=0:
We equate the constant term to the term with x2x^2
and solve for XX. Note that a square root has two answers (negative and positive).
Let's see an example:
x2−16=0x^2-16=0
We equate the constant term to x2x^2
and get:
X2=16X^2=16
X=4,−4X=4,-4

When we have an incomplete equation where c=0c=0:
we can immediately determine that the parabola intersects the YY axis when Y=0Y=0
we will factor out the common factor and find the roots of the equation.

For example:
x2−16x=0x^2-16x=0

We factor out a common factor and get:
X(X−16)=0X(X-16)=0
The factors that zero the equation are –
X=0,16X=0,16

Quadratic equations with denominators – fractions

Sometimes we encounter quadratic equations with a fraction (numerator and denominator). To read the function more accurately, we need to get rid of the fraction.
To solve quadratic equations with denominators-
find the common denominator, multiply each term, and obtain an equation without a fraction. Then solve it completely normally and find the solutions.


Click here to learn more about different representations of a quadratic function

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Test yourself on ways of representing the quadratic function!

einstein

Create an algebraic expression based on the following parameters:

\( a=-1,b=-1,c=-1 \)

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