What Is The First Step In Writing $f(x) = 3x^2 + 6x - 8$ In Vertex Form?A. Factor Out 3 From Each Term. B. Form A Perfect Square Trinomial By Keeping The Value Of The Function Equivalent. C. Write The Trinomial As A Binomial Squared. D.

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Understanding the Basics of Vertex Form

Vertex form is a way of expressing a quadratic function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is particularly useful for identifying the vertex and the direction of the parabola. To convert a quadratic function from standard form to vertex form, we need to follow a series of steps.

Step 1: Factor out the Leading Coefficient

The first step in writing a quadratic function in vertex form is to factor out the leading coefficient from each term. In the given function $f(x) = 3x^2 + 6x - 8$, the leading coefficient is 3. We can factor out 3 from each term as follows:

$f(x) = 3(x^2 + 2x - 8/3)$

By factoring out the leading coefficient, we have simplified the function and made it easier to work with.

Step 2: Complete the Square

The next step is to complete the square by keeping the value of the function equivalent. To do this, we need to add and subtract a constant term inside the parentheses. The constant term is calculated by taking half of the coefficient of the $x$ term and squaring it.

In this case, the coefficient of the $x$ term is 2. Half of 2 is 1, and 1 squared is 1. We add and subtract 1 inside the parentheses:

$f(x) = 3(x^2 + 2x + 1 - 1 - 8/3)$

Now, we can rewrite the expression inside the parentheses as a perfect square trinomial:

$f(x) = 3((x + 1)^2 - 1 - 8/3)$

Step 3: Simplify the Expression

We can simplify the expression by combining the constant terms:

$f(x) = 3((x + 1)^2 - 1 - 8/3)$

$f(x) = 3((x + 1)^2 - (1 + 8/3))$

$f(x) = 3((x + 1)^2 - (3 + 8)/3)$

$f(x) = 3((x + 1)^2 - 11/3)$

$f(x) = 3(x + 1)^2 - 11$

Step 4: Write the Trinomial as a Binomial Squared

We can rewrite the trinomial as a binomial squared by factoring out the coefficient of the squared term:

$f(x) = 3(x + 1)^2 - 11$

$f(x) = 3(x + 1)^2 - 11$

This is the final step in writing the quadratic function in vertex form.

Conclusion

In conclusion, the first step in writing a quadratic function in vertex form is to factor out the leading coefficient from each term. This simplifies the function and makes it easier to work with. The next steps involve completing the square and rewriting the trinomial as a binomial squared. By following these steps, we can convert a quadratic function from standard form to vertex form.

Key Takeaways

• Factor out the leading coefficient from each term.
• Complete the square by adding and subtracting a constant term inside the parentheses.
• Rewrite the trinomial as a binomial squared by factoring out the coefficient of the squared term.

Practice Problems

1. Write the quadratic function $f(x) = 2x^2 + 4x - 6$ in vertex form.
2. Write the quadratic function $f(x) = x^2 + 2x - 3$ in vertex form.
3. Write the quadratic function $f(x) = 4x^2 - 8x + 4$ in vertex form.

Answer Key

1. $f(x) = 2(x + 1)^2 - 5$
2. $f(x) = (x + 1)^2 - 4$
3. $f(x) = 4(x - 1)^2 + 0$

References

• [1] "Vertex Form of a Quadratic Function." Math Open Reference, mathopenref.com/quadratic-vertexform.html.
• [2] "Completing the Square." Khan Academy, khanacademy.org/math/algebra/advanced_algebra/completing-the-square/v/completing-the-square.
• [3] "Vertex Form of a Quadratic Function." Purplemath, purplemath.com/modules/vertexform.htm.
  Q&A: Writing Quadratic Functions in Vertex Form =====================================================

Frequently Asked Questions

Q: What is the first step in writing a quadratic function in vertex form?

A: The first step is to factor out the leading coefficient from each term. This simplifies the function and makes it easier to work with.

Q: How do I complete the square?

A: To complete the square, you need to add and subtract a constant term inside the parentheses. The constant term is calculated by taking half of the coefficient of the x term and squaring it.

Q: What is the difference between standard form and vertex form?

A: Standard form is the original form of a quadratic function, while vertex form is a way of expressing the function in the form f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Q: How do I rewrite a trinomial as a binomial squared?

A: To rewrite a trinomial as a binomial squared, you need to factor out the coefficient of the squared term.

Q: What is the significance of the vertex form of a quadratic function?

A: The vertex form of a quadratic function is useful for identifying the vertex and the direction of the parabola. It also makes it easier to graph the function.

Q: Can I use the vertex form to find the x-intercepts of a quadratic function?

A: No, the vertex form is not suitable for finding the x-intercepts of a quadratic function. You need to use the standard form or the factored form for that purpose.

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

A: To convert a quadratic function from standard form to vertex form, you need to follow the steps outlined above: factor out the leading coefficient, complete the square, and rewrite the trinomial as a binomial squared.

Q: What are some common mistakes to avoid when writing a quadratic function in vertex form?

A: Some common mistakes to avoid include:

• Not factoring out the leading coefficient correctly
• Not completing the square correctly
• Not rewriting the trinomial as a binomial squared correctly
• Not checking the function for errors

Q: How do I check my work when writing a quadratic function in vertex form?

A: To check your work, you need to:

• Verify that the function is in the correct form (f(x) = a(x - h)^2 + k)
• Check that the vertex is correct
• Check that the direction of the parabola is correct
• Check that the function is equivalent to the original function

Additional Resources

• [1] "Vertex Form of a Quadratic Function." Math Open Reference, mathopenref.com/quadratic-vertexform.html.
• [2] "Completing the Square." Khan Academy, khanacademy.org/math/algebra/advanced_algebra/completing-the-square/v/completing-the-square.
• [3] "Vertex Form of a Quadratic Function." Purplemath, purplemath.com/modules/vertexform.htm.

Practice Problems

1. Write the quadratic function f(x) = 2x^2 + 4x - 6 in vertex form.
2. Write the quadratic function f(x) = x^2 + 2x - 3 in vertex form.
3. Write the quadratic function f(x) = 4x^2 - 8x + 4 in vertex form.

Answer Key

1. f(x) = 2(x + 1)^2 - 5
2. f(x) = (x + 1)^2 - 4
3. f(x) = 4(x - 1)^2 + 0

Conclusion

Writing a quadratic function in vertex form requires careful attention to detail and a clear understanding of the steps involved. By following the steps outlined above and practicing with sample problems, you can become proficient in writing quadratic functions in vertex form. Remember to check your work carefully and verify that the function is equivalent to the original function.