Standard Form of the Quadratic Function

The standard form of the quadratic function is:
$Y=ax^2+bx+c$

For example:
$Y=4x^2+3x+15$

Detailed explanation

Start practice

Test yourself on standard representation!

Create an algebraic expression based on the following parameters:

$$ a=3,b=6,c=9 $$

Practice more now

How do you go from standard form to vertex form?

We need to find the vertex of the parabola using the formula to find the $X$ vertex.
Let's find the $Y$ vertex.
Let's place in the vertex form template the $X$ vertex instead of $P$ , the $Y$ vertex instead of $C$ and the $a$ instead of $a$ .

How do you go from standard form to factored form?

Let's find the points of intersection of the parabola with the $x$ axis.
Let's place it in the factored form template.

Look!
If we were to realize that in the standard form there is a coefficient for $X^2$ we will place it in the factoring formula before locating the intersection points there, as follows:

$y=a\times(x-t)\times(x-k)$

Examples and exercises with solutions of the Standard form of the quadratic function

Exercise #1

Create an algebraic expression based on the following parameters:

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow the steps outlined:

Step 1: Identify the given values for the quadratic function's parameters: $a = 2$ , $b = \frac{1}{2}$ , and $c = 4$ .
Step 2: Apply these values to the standard quadratic form $y = ax^2 + bx + c$ .
Step 3: Substitute the values to construct the algebraic expression.

Now, let's proceed with these steps:

Given the standard form of a quadratic expression $y = ax^2 + bx + c$ :

Substituting the values, we obtain:

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

Therefore, the correct algebraic expression for the quadratic function is $2x^2 + \frac{1}{2}x + 4$ .

Answer

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

Exercise #2

Create an algebraic expression based on the following parameters:

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

Video Solution

Step-by-Step Solution

To solve this problem, we will derive the algebraic expression step-by-step:

Step 1: Identify the given information:
The problem states $a = 2$ , $b = 0$ , and $c = 4$ .

Step 2: Write the standard quadratic expression:
The general form is $y = ax^2 + bx + c$ .

Step 3: Substitute the given values into the expression:
Replace $a$ with 2, $b$ with 0, and $c$ with 4:
$y = 2x^2 + 0x + 4$ .

Step 4: Simplify the expression:
Since $0x$ is zero, the expression simplifies to:
$y = 2x^2 + 4$ .

Thus, the algebraic expression based on the given parameters is $2x^2 + 4$ .

The correct answer is: $2x^2 + 4$ (Choice 1).

Answer

$2x^2+4$

Exercise #3

Create an algebraic expression based on the following parameters:

$a=2,b=4,c=8$

Video Solution

Step-by-Step Solution

To solve this problem, we need to form an algebraic expression for a quadratic function using given parameters.

We start by recalling the standard form of a quadratic function: $( ax^2 + bx + c )$ . In this expression:

$a$ is the coefficient of $x^2$
$b$ is the coefficient of $x$
$c$ is the constant term

Given the values are $a = 2$ , $b = 4$ , and $c = 8$ , we substitute these into the standard form equation:

$ax^2 + bx + c = 2x^2 + 4x + 8$

This yields the algebraic expression for the quadratic function.

The correct expression, given all calculations and simplifications, is $2x^2 + 4x + 8$ .

Referring to the choices provided, the correct choice is:

: $( 2x^2 + 4x + 8 )$

Therefore, the solution to the problem is $\boxed{2x^2 + 4x + 8}$ .

Answer

$2x^2+4x+8$

Exercise #4

Create an algebraic expression based on the following parameters:

$a=3,b=0,c=-3$

Video Solution

Step-by-Step Solution

To solve the problem of creating an algebraic expression with the given parameters, we will proceed as follows:

Step 1: Identify the given coefficients for the quadratic function, which are $a = 3$ , $b = 0$ , and $c = -3$ .
Step 2: Substitute these values into the standard quadratic expression $y = ax^2 + bx + c$ .

Through substitution, the expression becomes:

$y = 3x^2 + 0x - 3$

We can further simplify this expression:

$y = 3x^2 - 3$

Thus, the algebraic expression with the given parameters is $y = 3x^2 - 3$ .

The correct answer corresponds to choice number 1: $3x^2-3$ .

Therefore, the solution to the problem is

$y = 3x^2 - 3$

Answer

$3x^2-3$

Exercise #5

Choose the correct algebraic expression based on the parameters:

$a=-3,b=3,c=7$

Video Solution

Step-by-Step Solution

To solve this problem, we will substitute the given values into the standard quadratic form:

Step 1: Identify the formula to use. We need the standard form of a quadratic function, which is $y = ax^2 + bx + c$ .
Step 2: Substitute the given parameters into the formula:
$a = -3$ , $b = 3$ , and $c = 7$ .
Step 3: Perform the substitution:
Substituting in, we get $y = -3x^2 + 3x + 7$ .