Standard Form of the Quadratic Function

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Standard Form of the Quadratic Function

The standard form of the quadratic function is:
Y=ax2+bx+cY=ax^2+bx+c

For example:
Y=4x2+3x+15Y=4x^2+3x+15

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Test yourself on standard representation!

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Choose the correct algebraic expression based on the parameters:

\( a=-3,b=3,c=7 \)

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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 XX vertex.
  • Let's find the YY vertex.
  • Let's place in the vertex form template the X X vertex instead of PP, the YY vertex instead of CC and the aa instead of aa.

How do you go from standard form to factored form?

  • Let's find the points of intersection of the parabola with the xx 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 X2X^2 we will place it in the factoring formula before locating the intersection points there, as follows:

y=a×(xt)×(xk) 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=1,b=1,c=1 a=-1,b=-1,c=-1

Video Solution

Step-by-Step Solution

The goal is to express the quadratic equation y=ax2+bx+c y = ax^2 + bx + c using the given parameters a=1 a = -1 , b=1 b = -1 , and c=1 c = -1 .

First, substitute the values of a a , b b , and c c into the standard form:

  • Substituting a=1 a = -1 , the term becomes x2 -x^2 .
  • Substituting b=1 b = -1 , the term becomes x -x .
  • Substituting c=1 c = -1 , the term remains 1-1.

Combine these terms to form the full expression:


y=x2x1 y = -x^2 - x - 1

Therefore, the algebraic expression for the parameters a=1 a = -1 , b=1 b = -1 , and c=1 c = -1 is: x2x1 -x^2 - x - 1 .

Comparing with the given choices, the correct choice is option 4: x2x1 -x^2-x-1

Answer

x2x1 -x^2-x-1

Exercise #2

Create an algebraic expression based on the following parameters:

a=1,b=2,c=5 a=-1,b=-2,c=-5

Video Solution

Step-by-Step Solution

To create the algebraic expression for the quadratic function given the parameters, we follow these steps:

  • Step 1: Identify the values to substitute into the equation. Here, we have a=1 a = -1 , b=2 b = -2 , and c=5 c = -5 .
  • Step 2: Use the standard quadratic equation format y=ax2+bx+c y = ax^2 + bx + c .
  • Step 3: Substitute the known values into the equation:

Substituting these values, we get:
y=(1)x2+(2)x+(5) y = (-1)x^2 + (-2)x + (-5)

Simplify this expression:
This simplifies to x22x5-x^2 - 2x - 5.

Therefore, the algebraic expression is x22x5 -x^2 - 2x - 5 .

Answer

x22x5 -x^2-2x-5

Exercise #3

Create an algebraic expression based on the following parameters:

a=4,b=2,c=5 a=4,b=2,c=5

Video Solution

Step-by-Step Solution

To derive the algebraic expression based on the parameters given, we follow these steps:

  • Step 1: Recognize the given parameters: a=4 a = 4 , b=2 b = 2 , and c=5 c = 5 .
  • Step 2: Acknowledge that the standard form for a quadratic expression is y=ax2+bx+c y = ax^2 + bx + c .
  • Step 3: Substitute the given parameter values into this quadratic expression.

Now, let's implement these steps to form the quadratic expression:
Step 1: The given parameters are a=4 a = 4 , b=2 b = 2 , and c=5 c = 5 .
Step 2: Our basis is the quadratic form y=ax2+bx+c y = ax^2 + bx + c .
Step 3: Substituting the given values, we find:

y=4x2+2x+5 y = 4x^2 + 2x + 5

This substitution provides us with the quadratic expression y=4x2+2x+5 y = 4x^2 + 2x + 5 , fulfilling the problem's requirements.

Therefore, the correct algebraic expression is 4x2+2x+5 4x^2 + 2x + 5 .

Answer

4x2+2x+5 4x^2+2x+5

Exercise #4

Create an algebraic expression based on the following parameters:

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

Video Solution

Step-by-Step Solution

To determine the algebraic expression, we start with the standard quadratic function:

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

Given the values:

  • a=1 a = -1
  • b=1 b = 1
  • c=0 c = 0

We substitute these into the formula:

y=(1)x2+1x+0 y = (-1)x^2 + 1x + 0

Simplifying the expression gives:

y=x2+x y = -x^2 + x

Thus, the algebraic expression, when these parameters are substituted, is:

The solution to the problem is x2+x \boxed{-x^2 + x} .

Answer

x2+x -x^2+x

Exercise #5

Create an algebraic expression based on the following parameters:

a=1,b=1,c=0 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=ax2+bx+c y = ax^2 + bx + c .
  • Substitute a=1 a = 1 , b=1 b = 1 , and c=0 c = 0 into the equation.

Substituting these values, the expression becomes:

y=1x2+1x+0 y = 1 \cdot x^2 + 1 \cdot x + 0 .

This simplifies to:

y=x2+x y = x^2 + x .

Therefore, the algebraic expression, based on the given parameters, is x2+x x^2 + x .

Answer

x2+x x^2+x

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