Convert $y=4(x+3)^2+5$ To Standard Form.Enter Your Answer In The Box:$y=$ \$\square$[/tex\]

Understanding the Problem

In this problem, we are given a quadratic function in the form of $y=4(x+3)^2+5$ and we need to convert it to standard form. The standard form of a quadratic function is $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants.

The Process of Converting to Standard Form

To convert the given quadratic function to standard form, we need to follow these steps:

Step 1: Expand the Squared Term

The first step is to expand the squared term $(x+3)^2$. This can be done using the formula $(a+b)^2=a2+2ab+b^2$. In this case, $a=x$ and $b=3$.

(x+3)^2 = x^2 + 2 \cdot x \cdot 3 + 3^2

Step 2: Simplify the Expression

Now, we can simplify the expression by evaluating the terms.

(x+3)^2 = x^2 + 6x + 9

Step 3: Multiply the Coefficient

The next step is to multiply the coefficient $4$ with the expanded and simplified expression.

4(x+3)^2 = 4(x^2 + 6x + 9)

Step 4: Distribute the Coefficient

Now, we can distribute the coefficient $4$ to each term in the expression.

4(x+3)^2 = 4x^2 + 24x + 36

Step 5: Add the Constant Term

Finally, we can add the constant term $5$ to the expression.

y = 4x^2 + 24x + 36 + 5

Step 6: Simplify the Expression

Now, we can simplify the expression by combining like terms.

y = 4x^2 + 24x + 41

The Final Answer

Therefore, the quadratic function $y=4(x+3)^2+5$ in standard form is $y=4x^2+24x+41$.

Conclusion

Understanding the Problem

In the previous article, we learned how to convert a quadratic function from vertex form to standard form. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is $y=ax^2+bx+c$, where $a$, $b$, and $c$ are constants.

Q: Why do we need to convert a quadratic function to standard form?

A: We need to convert a quadratic function to standard form because it makes it easier to analyze and solve the equation. In standard form, we can easily identify the coefficients $a$, $b$, and $c$, which are essential in solving the equation.

Q: How do I know if a quadratic function is in vertex form or standard form?

A: To determine if a quadratic function is in vertex form or standard form, look for the presence of a squared term. If the squared term is in the form of $(x-h)^2$, then it is in vertex form. If the squared term is in the form of $x^2$, then it is in standard form.

Q: Can I convert a quadratic function from vertex form to standard form using a calculator?

A: Yes, you can convert a quadratic function from vertex form to standard form using a calculator. Most graphing calculators have a built-in function to convert a quadratic function from vertex form to standard form.

Q: What are the steps to convert a quadratic function from vertex form to standard form?

A: The steps to convert a quadratic function from vertex form to standard form are:

1. Expand the squared term.
2. Simplify the expression.
3. Multiply the coefficient.
4. Distribute the coefficient.
5. Add the constant term.
6. Simplify the expression.

Q: Can I convert a quadratic function from standard form to vertex form?

A: Yes, you can convert a quadratic function from standard form to vertex form. To do this, you need to complete the square.

Q: What is completing the square?

A: Completing the square is a process of rewriting a quadratic function in vertex form by adding and subtracting a constant term.

Q: How do I complete the square?

A: To complete the square, follow these steps:

1. Write the quadratic function in standard form.
2. Identify the coefficient of the squared term.
3. Add and subtract the square of half the coefficient of the squared term.
4. Simplify the expression.

Q: Can I use completing the square to convert a quadratic function from standard form to vertex form?

A: Yes, you can use completing the square to convert a quadratic function from standard form to vertex form.

Conclusion

In this article, we answered some frequently asked questions related to converting a quadratic function from vertex form to standard form. We also discussed the steps to convert a quadratic function from vertex form to standard form and how to complete the square to convert a quadratic function from standard form to vertex form.