What Is The First Step In Writing $f(x)=6x^2-42x+5$ In Vertex Form?A. Factor 6 Out Of The First Two Terms.B. Write The Function In Standard Form.C. Factor 6 Out Of Each Term.D. Write The Trinomial As A Binomial Squared.

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What is the First Step in Writing f(x)=6x2−42x+5f(x)=6x^2-42x+5 in Vertex Form?

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+kf(x)=a(x-h)^2+k, where (h,k)(h,k) is the vertex of the parabola. To write a quadratic function in vertex form, we need to complete the square. However, before we can do that, we need to make sure the function is in the correct form.

Step 1: Write the Function in Standard Form

The first step in writing f(x)=6x2−42x+5f(x)=6x^2-42x+5 in vertex form is to write the function in standard form. Standard form is the form f(x)=ax2+bx+cf(x)=ax^2+bx+c, where aa, bb, and cc are constants. In this case, the function is already in standard form, so we don't need to do anything.

Step 2: Factor Out the Leading Coefficient

The next step is to factor out the leading coefficient, which is 6 in this case. Factoring out the leading coefficient will make it easier to complete the square.

# Factoring Out the Leading Coefficient

## Step 1: Factor Out 6

$f(x) = 6(x^2 - 7x) + 5$

## Step 2: Simplify the Expression

$f(x) = 6(x^2 - 7x) + 5$
$f(x) = 6(x^2 - 7x + \frac{49}{4}) - 6(\frac{49}{4}) + 5$
$f(x) = 6(x - \frac{7}{2})^2 - \frac{147}{2} + 5$
$f(x) = 6(x - \frac{7}{2})^2 - \frac{139}{2}$

Conclusion

In conclusion, the first step in writing f(x)=6x2−42x+5f(x)=6x^2-42x+5 in vertex form is to write the function in standard form. However, in this case, the function is already in standard form, so we need to factor out the leading coefficient. By factoring out the leading coefficient, we can make it easier to complete the square and write the function in vertex form.

Key Takeaways

  • The first step in writing a quadratic function in vertex form is to write the function in standard form.
  • If the function is not in standard form, we need to rewrite it in standard form.
  • Factoring out the leading coefficient will make it easier to complete the square and write the function in vertex form.

Common Mistakes

  • Not writing the function in standard form before attempting to write it in vertex form.
  • Not factoring out the leading coefficient, which can make it difficult to complete the square.

Real-World Applications

  • Writing a quadratic function in vertex form can be useful in a variety of real-world applications, such as modeling the trajectory of a projectile or the motion of an object under the influence of gravity.
  • Vertex form can also be used to find the maximum or minimum value of a quadratic function, which can be useful in a variety of fields, including economics and engineering.

Practice Problems

  • Write the following quadratic function in vertex form: f(x)=2x2+12x+10f(x) = 2x^2 + 12x + 10
  • Write the following quadratic function in vertex form: f(x)=3x2−9x+2f(x) = 3x^2 - 9x + 2

Solutions

  • f(x)=2(x+3)2−13f(x) = 2(x + 3)^2 - 13
  • f(x)=3(x−1)2−5f(x) = 3(x - 1)^2 - 5

Conclusion

In conclusion, writing a quadratic function in vertex form requires a series of steps, including writing the function in standard form and factoring out the leading coefficient. By following these steps, we can write a quadratic function in vertex form and use it to solve a variety of problems.
Q&A: Writing Quadratic Functions in Vertex Form

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

A: The vertex form of a quadratic function is f(x)=a(x−h)2+kf(x)=a(x-h)^2+k, where (h,k)(h,k) is the vertex of the parabola.

Q: Why is it important to write a quadratic function in vertex form?

A: Writing a quadratic function in vertex form can be useful in a variety of real-world applications, such as modeling the trajectory of a projectile or the motion of an object under the influence of gravity. It can also be used to find the maximum or minimum value of a quadratic function, which can be useful in a variety of fields, including economics and engineering.

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

A: The first step in writing a quadratic function in vertex form is to write the function in standard form. Standard form is the form f(x)=ax2+bx+cf(x)=ax^2+bx+c, where aa, bb, and cc are constants.

Q: How do I write a quadratic function in standard form?

A: To write a quadratic function in standard form, you need to rewrite the function in the form f(x)=ax2+bx+cf(x)=ax^2+bx+c. This can be done by rearranging the terms of the function.

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

A: The next step in writing a quadratic function in vertex form is to factor out the leading coefficient. Factoring out the leading coefficient will make it easier to complete the square.

Q: How do I factor out the leading coefficient?

A: To factor out the leading coefficient, you need to divide each term of the function by the leading coefficient. This will give you a new function that is in the form f(x)=a(x2+bx+c)f(x)=a(x^2+bx+c).

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

A: The final step in writing a quadratic function in vertex form is to complete the square. Completing the square involves rewriting the function in the form f(x)=a(x−h)2+kf(x)=a(x-h)^2+k, where (h,k)(h,k) is the vertex of the parabola.

Q: How do I complete the square?

A: To complete the square, you need to take the coefficient of the xx term and divide it by 2. You then need to square this value and add it to both sides of the equation. This will give you a new function that is in the form f(x)=a(x−h)2+kf(x)=a(x-h)^2+k.

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

A: Some common mistakes to avoid when writing a quadratic function in vertex form include not writing the function in standard form before attempting to write it in vertex form, and not factoring out the leading coefficient.

Q: What are some real-world applications of writing a quadratic function in vertex form?

A: Writing a quadratic function in vertex form can be useful in a variety of real-world applications, such as modeling the trajectory of a projectile or the motion of an object under the influence of gravity. It can also be used to find the maximum or minimum value of a quadratic function, which can be useful in a variety of fields, including economics and engineering.

Q: How can I practice writing quadratic functions in vertex form?

A: You can practice writing quadratic functions in vertex form by working through a series of examples and exercises. You can also try writing your own quadratic functions in vertex form and then checking your work to make sure it is correct.

Q: What are some resources that can help me learn how to write quadratic functions in vertex form?

A: There are a variety of resources that can help you learn how to write quadratic functions in vertex form, including textbooks, online tutorials, and practice problems. You can also try working with a tutor or teacher who can provide you with one-on-one instruction and feedback.

Conclusion

In conclusion, writing a quadratic function in vertex form requires a series of steps, including writing the function in standard form and factoring out the leading coefficient. By following these steps and practicing writing quadratic functions in vertex form, you can develop the skills and knowledge you need to succeed in a variety of fields, including mathematics, science, and engineering.