The quadratic relation y = 3(x + 1)^2 - 2 can be rewritten in standard form as y = _____.

Understand the Problem

The question is asking us to convert the quadratic relation given in vertex form into standard form. The standard form of a quadratic equation is typically written as y = ax^2 + bx + c. We will expand the vertex form and combine the like terms to derive the standard form.

Answer

$$ y = ax^2 - 2ahx + (ah^2 + k) $$

Steps to Solve

Identify the vertex form of the quadratic equation

The vertex form of a quadratic equation is given as:

$$ y = a(x - h)^2 + k $$

where $(h, k)$ is the vertex of the parabola.

Expand the squared binomial

To convert the equation from vertex form to standard form, we will first expand the squared term.

Using the formula $(x - h)^2 = x^2 - 2hx + h^2$, we substitute this into the equation:

$$ y = a(x - h)^2 + k $$ becomes:

$$ y = a(x^2 - 2hx + h^2) + k $$

Distribute the coefficient 'a'

Now we will distribute the coefficient 'a' across the terms:

$$ y = ax^2 - 2ahx + ah^2 + k $$

Combine the constant terms

Finally, we combine the constant terms $ah^2 + k$ to express the equation in standard form:

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

where

$$ b = -2ah $$ and $$ c = ah^2 + k $$

The standard form of the quadratic equation will be

$$ y = ax^2 - 2ahx + (ah^2 + k) $$

More Information

The transformation from vertex form to standard form makes it easier to identify the coefficients of the quadratic equation, which can help in analyzing its properties such as its vertex, direction, and x-intercepts.

Tips

Not expanding the squared term correctly: Ensure you use the formula correctly.
Forgetting to distribute 'a': Always remember to apply the coefficient to each term in the expanded expression.
Incorrectly combining constant terms: Make sure to combine constant terms accurately to prevent errors in the final equation.

AI-generated content may contain errors. Please verify critical information

Thank you for voting!