The standard representation of the quadratic function looks like this:

The standard representation of the quadratic function looks like this:

$y=ax^2+bx+c$

When:

$a$ –

- The coefficient of $X^2$
- determines whether the parabola will be a maximum or minimum parabola (sad or smiling) and how open or closed it will be.
- $a$ must be different from $0$.
- If $a$ is positive – the parabola is a minimum parabola – smiling
- If $a$ is negative – the parabola is a maximum parabola – sad
- The larger $a$ is – the narrower the function will be, and vice versa.

$b$ -

- The coefficient of $X$
- can be any negative or positive number

$c$ –

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

**Let's see an example:**

We are given the function:

$y=5x^2+2x+7$

What can we conclude?

$A=5$ therefore the parabola is smiling

$C=7$ therefore the function intersects the $Y$-axis at the point $(0,7)$

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

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:

$Y=a(X-p)^2+c$

When:

$P$ - represents the $X$ value of the vertex.

$C$ - represents the $Y$ value of the vertex.

**For example:**

In the function

$Y=6(X-5)^2+2$

The vertex of the parabola is:

$(2,5)$

**Note-**

The vertex formula is structured such that there is always a – before the $P$, meaning $X$ vertex, but this does not necessarily mean that the $X$ vertex is negative.

If the parabola has a negative $X$ vertex, a $+$ will appear before the $P$ in the formula because $–$ times $–$ equals $+$.

For example:

For example:

In the function

$Y=6(X+3)^2+8$

The vertex of the parabola is:

$(-3,8)$

There is a $+$ before the $3$ in the formula, so it is $-3$.

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

The product form shows multiplication between $2$ expressions. With the product form, we can easily determine the points of intersection of the function with the $X$-axis.

The product form of the quadratic function looks like this:

$y=(x-t)*(x-k)$

where

$t$ and $k$ are the $2$ points of intersection of the parabola with the $X$-axis.

As follows: $(t,0) (k,0)$

Let's see an example of the product form to understand better:

$y=(x-6)*(x+5)$

We can determine that:

The points of intersection with the $X$-axis are:

$(6,0)$

$( -5,0)$

Note - Since in the original template there is a minus before \(k\) and $t$, we can infer that if there is a plus before one of them, it is negative, hence $-5$ and not $5$.

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

Sometimes we encounter quadratic equations that are missing terms such as $BX$ or $C$ and quadratic equations with denominators.

Quadratic equations with missing terms are quadratic equations where $c$ or $b$ are equal to $0$.

**When we have an incomplete equation where b=0:**

We equate the constant term to the term with $x^2$

and solve for $X$. Note that a square root has two answers (negative and positive).

Let's see an example:

$x^2-16=0$

We equate the constant term to $x^2$

and get:

$X^2=16$

$X=4,-4$

When we have an incomplete equation where $c=0$:

we can immediately determine that the parabola intersects the $Y$ axis when $Y=0$

we will factor out the common factor and find the roots of the equation.

For example:

$x^2-16x=0$

We factor out a common factor and get:

$X(X-16)=0$

The factors that zero the equation are –

$X=0,16$

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

Create an algebraic expression based on the following parameters:

\( a=2,b=2,c=2 \)

NULL

Test your knowledge

Question 1

Create an algebraic expression based on the following parameters:

\( a=2,b=\frac{1}{2},c=4 \)

Question 2

Create an algebraic expression based on the following parameters:

\( a=2,b=0,c=4 \)

Question 3

Create an algebraic expression based on the following parameters:

\( a=2,b=4,c=8 \)

Find the standard representation of the following function

$f(x)=(x-2)(x+5)$

We will use the expanded distributive property.

(*a*+1)⋆(*b*+2) = *ab*+2*a*+*b*+2

Let's substitute the given values and solve:

(x-2)(x+5) =

x²-2x+5x+-2*5=

x²+3x-10

And that's the solution!

$f(x)+x^2+3x-10$

Determine the points of intersection of the function

$y=(x-5)(x+5)$

With the X

In order to find the point of the intersection with the X-axis, we first need to establish that Y=0.

0 = (x-5)(x+5)

When we have an equation of this type, we know that one of these parentheses must be equal to 0, so we begin by checking the possible options.

x-5 = 0

x = 5

x+5 = 0

x = -5

That is, we have two points of intersection with the x-axis, when we discover their x points, and the y point is already known to us (0, as we placed it):

(5,0)(-5,0)

This is the solution!

$(5,0),(-5,0)$

Create an algebraic expression based on the following parameters:

$a=2,b=2,c=2$

$2x^2+2x+2$

Create an algebraic expression based on the following parameters:

$a=2,b=\frac{1}{2},c=4$

$2x^2+\frac{1}{2}x+4$

Create an algebraic expression based on the following parameters:

$a=2,b=0,c=4$

$2x^2+4$

Related Subjects

- Quadratice Equations and Systems of Quadraric Equations
- Quadratic Equations System - Algebraic and Graphical Solution
- Solution of a system of equations - one of them is linear and the other quadratic
- Intersection between two parabolas
- Word Problems
- Properties of the roots of quadratic equations - Vieta's formulas
- Ways to represent a quadratic function
- Various Forms of the Quadratic Function
- Standard Form of the Quadratic Function
- Vertex form of the quadratic equation
- Factored form of the quadratic function
- The quadratic function
- Quadratic Inequality
- Parabola
- Symmetry in a parabola
- Plotting the Quadratic Function Using Parameters a, b and c
- Finding the Zeros of a Parabola
- Methods for Solving a Quadratic Function
- Completing the square in a quadratic equation
- Squared Trinomial
- The quadratic equation
- Families of Parabolas
- The functions y=x²
- Family of Parabolas y=x²+c: Vertical Shift
- Family of Parabolas y=(x-p)²
- Family of Parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)