Match Each Quadratic Function Given In Factored Form With Its Equivalent Standard Form Listed On The Left.A. F ( X ) = X 2 − 11 X − 12 F(x)=x^2-11x-12 F ( X ) = X 2 − 11 X − 12 B. F ( X ) = X 2 − 4 X − 12 F(x)=x^2-4x-12 F ( X ) = X 2 − 4 X − 12 C. F ( X ) = X 2 + X − 12 F(x)=x^2+x-12 F ( X ) = X 2 + X − 12 D. F ( X ) = X 2 − X − 12 F(x)=x^2-x-12 F ( X ) = X 2 − X − 12 Factored Forms:1.

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to convert them from factored form to standard form is crucial for solving various mathematical problems. In this article, we will explore the process of matching quadratic functions given in factored form with their equivalent standard form.

What are Quadratic Functions?

Quadratic functions are polynomial functions of degree two, which means the highest power of the variable (usually x) is two. They can be represented in two forms: factored form and standard form. The factored form of a quadratic function is expressed as a product of two binomials, while the standard form is expressed as a single polynomial.

Factored Form to Standard Form

To convert a quadratic function from factored form to standard form, we need to multiply the two binomials. This can be done using the distributive property, which states that a(b + c) = ab + ac.

Example 1: Converting Factored Form to Standard Form

Let's consider the quadratic function f(x) = (x + 3)(x - 4). To convert it to standard form, we need to multiply the two binomials:

f(x) = (x + 3)(x - 4) = x(x - 4) + 3(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12

Example 2: Converting Factored Form to Standard Form

Now, let's consider the quadratic function f(x) = (x - 2)(x + 6). To convert it to standard form, we need to multiply the two binomials:

f(x) = (x - 2)(x + 6) = x(x + 6) - 2(x + 6) = x^2 + 6x - 2x - 12 = x^2 + 4x - 12

Example 3: Converting Factored Form to Standard Form

Let's consider the quadratic function f(x) = (x + 2)(x - 3). To convert it to standard form, we need to multiply the two binomials:

f(x) = (x + 2)(x - 3) = x(x - 3) + 2(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6

Example 4: Converting Factored Form to Standard Form

Now, let's consider the quadratic function f(x) = (x - 1)(x + 12). To convert it to standard form, we need to multiply the two binomials:

f(x) = (x - 1)(x + 12) = x(x + 12) - 1(x + 12) = x^2 + 12x - x - 12 = x^2 + 11x - 12

Matching Quadratic Functions

Now that we have converted the quadratic functions from factored form to standard form, let's match them with the equivalent standard form listed on the left.

A. $f(x)=x^2-11x-12$

To match this quadratic function, we need to find the factored form that corresponds to it. Let's consider the factored forms listed above and multiply them to see which one matches:

• (x + 3)(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12 (does not match)
• (x - 2)(x + 6) = x^2 + 6x - 2x - 12 = x^2 + 4x - 12 (does not match)
• (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 (does not match)
• (x - 1)(x + 12) = x^2 + 12x - x - 12 = x^2 + 11x - 12 (matches)

Therefore, the factored form of the quadratic function f(x) = x^2 - 11x - 12 is (x - 1)(x + 12).

B. $f(x)=x^2-4x-12$

To match this quadratic function, we need to find the factored form that corresponds to it. Let's consider the factored forms listed above and multiply them to see which one matches:

• (x + 3)(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12 (does not match)
• (x - 2)(x + 6) = x^2 + 6x - 2x - 12 = x^2 + 4x - 12 (does not match)
• (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 (does not match)
• (x - 1)(x + 12) = x^2 + 12x - x - 12 = x^2 + 11x - 12 (does not match)

However, we can try to factor the quadratic function f(x) = x^2 - 4x - 12 as (x - 6)(x + 2).

C. $f(x)=x^2+x-12$

To match this quadratic function, we need to find the factored form that corresponds to it. Let's consider the factored forms listed above and multiply them to see which one matches:

• (x + 3)(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12 (does not match)
• (x - 2)(x + 6) = x^2 + 6x - 2x - 12 = x^2 + 4x - 12 (does not match)
• (x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 (does not match)
• (x - 1)(x + 12) = x^2 + 12x - x - 12 = x^2 + 11x - 12 (does not match)

However, we can try to factor the quadratic function f(x) = x^2 + x - 12 as (x + 4)(x - 3).

D. $f(x)=x^2-x-12$

To match this quadratic function, we need to find the factored form that corresponds to it. Let's consider the factored forms listed above and multiply them to see which one matches:

• (x + 3)(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12 (matches)

Therefore, the factored form of the quadratic function f(x) = x^2 - x - 12 is (x + 3)(x - 4).

Conclusion

In this article, we have explored the process of matching quadratic functions given in factored form with their equivalent standard form. We have also provided examples of how to convert quadratic functions from factored form to standard form using the distributive property. By following these steps, we can easily match quadratic functions with their equivalent standard form.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Factoring Quadratic Expressions" by Purplemath
• [3] "Standard Form of a Quadratic Function" by Mathway

Additional Resources

• [1] "Quadratic Functions" by Khan Academy
• [2] "Factoring Quadratic Expressions" by IXL
• [3] "Standard Form of a Quadratic Function" by Wolfram Alpha
  Quadratic Functions: Factored Form to Standard Form - Q&A ===========================================================

Introduction

In our previous article, we explored the process of matching quadratic functions given in factored form with their equivalent standard form. We also provided examples of how to convert quadratic functions from factored form to standard form using the distributive property. In this article, we will answer some frequently asked questions (FAQs) related to quadratic functions and their conversion from factored form to standard form.

Q&A

Q: What is the difference between factored form and standard form of a quadratic function?

A: The factored form of a quadratic function is expressed as a product of two binomials, while the standard form is expressed as a single polynomial.

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

A: To convert a quadratic function from factored form to standard form, you need to multiply the two binomials using the distributive property.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a(b + c) = ab + ac.

Q: Can I factor a quadratic function that is not in the form (x + a)(x + b)?

A: Yes, you can factor a quadratic function that is not in the form (x + a)(x + b) by using other factoring techniques such as grouping or using the quadratic formula.

Q: How do I determine if a quadratic function is in factored form or standard form?

A: To determine if a quadratic function is in factored form or standard form, you need to look at the expression. If it is a product of two binomials, it is in factored form. If it is a single polynomial, it is in standard form.

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

A: Yes, you can convert a quadratic function from standard form to factored form by factoring the polynomial.

Q: What are some common mistakes to avoid when converting quadratic functions from factored form to standard form?

A: Some common mistakes to avoid when converting quadratic functions from factored form to standard form include:

• Not using the distributive property correctly
• Not multiplying the binomials correctly
• Not simplifying the expression correctly

Q: How do I check if my answer is correct when converting a quadratic function from factored form to standard form?

A: To check if your answer is correct, you need to multiply the two binomials and simplify the expression to see if it matches the original quadratic function.

Conclusion

In this article, we have answered some frequently asked questions related to quadratic functions and their conversion from factored form to standard form. We hope that this article has provided you with a better understanding of quadratic functions and how to convert them from factored form to standard form.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Factoring Quadratic Expressions" by Purplemath
• [3] "Standard Form of a Quadratic Function" by Mathway

Additional Resources

• [1] "Quadratic Functions" by Khan Academy
• [2] "Factoring Quadratic Expressions" by IXL
• [3] "Standard Form of a Quadratic Function" by Wolfram Alpha