Create an algebraic expression based on the following parameters: $$ a=\frac{1}{2},b=\frac{1}{2},c=\frac{1}{2} $$

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

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

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

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

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Test yourself on standard representation!

Create an algebraic expression based on the following parameters:

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

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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:

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

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Substitute the given values $a = 3$ , $b = 0$ , and $c = 0$ into the quadratic function formula $y = ax^2 + bx + c$ .
Step 2: Simplify the expression.

Let's execute these steps:

Step 1: Substitute the values into the formula:
$y = 3x^2 + 0x + 0$

Step 2: Simplify the expression:
Eliminate the terms with zero coefficients to get:
$y = 3x^2$

Thus, the algebraic expression for the quadratic function with $a = 3$ , $b = 0$ , and $c = 0$ is $3x^2$ .

Therefore, the correct choice from the options provided is choice 1: $3x^2$

Answer

$3x^2$

Exercise #2

Create an algebraic expression based on the following parameters:

$a=1,b=1,c=0$

Video Solution

Step-by-Step Solution

To determine the algebraic expression, we will substitute the given parameters into the standard form of the quadratic function:

The standard quadratic form is $y = ax^2 + bx + c$ .
Substitute $a = 1$ , $b = 1$ , and $c = 0$ into the equation.

Substituting these values, the expression becomes:

$y = 1 \cdot x^2 + 1 \cdot x + 0$ .

This simplifies to:

$y = x^2 + x$ .

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

Answer

$x^2+x$

Exercise #3

Create an algebraic expression based on the following parameters:

$a=-1,b=0,c=0$

Video Solution

Step-by-Step Solution

We begin by noting that the general form of a quadratic function is represented by the equation:

$y = ax^2 + bx + c$

Given the parameters $a = -1$ , $b = 0$ , and $c = 0$ , we substitute these values into the equation:

$y = (-1)x^2 + (0)x + 0$

Simplifying the expression, we get:

$y = -x^2$

Thus, the algebraic expression representing the given parameters is $-x^2$ .

The correct answer choice that corresponds to this expression is:

$-x^2$

Answer

$-x^2$

Exercise #4

Create an algebraic expression based on the following parameters:

$a=-1,b=-8,c=0$

Video Solution

Step-by-Step Solution

To solve this problem, we'll start by substituting the given parameters into the standard quadratic formula:

$y = ax^2 + bx + c$

The problem gives us the values:

$a = -1$
$b = -8$
$c = 0$

This means we need to replace $a$ , $b$ , and $c$ in the formula:

$y = (-1)x^2 + (-8)x + 0$

Simplifying this expression further:

The term with $a$ : (-1)x^2\) results in $-x^2$ .
The term with $b$ : (-8)x\) simplifies to $-8x$ .
The term with $c$ : $0$ contributes nothing to the expression, so it is omitted.

Thus, the final algebraic expression is:

$y = -x^2 - 8x$

Therefore, the algebraic expression based on the given parameters is

$-x^2 - 8x$ .

Answer

$-x^2-8x$

Exercise #5

Create an algebraic expression based on the following parameters:

$a=5,b=3,c=-4$

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Identify the general form of the quadratic expression.
Step 2: Substitute the given values $a = 5$ , $b = 3$ , and $c = -4$ into the quadratic form.
Step 3: Write down the resultant expression.

Now, let's work through each step:
Step 1: The general form of a quadratic expression is $ax^2 + bx + c$ .
Step 2: We are given $a = 5$ , $b = 3$ , and $c = -4$ . Substituting these into the expression, we get: