Matching Quadratic Functions

Oct 13, 2025 by talynt.org 76 views

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Introduction

Quadratic functions are a fundamental concept in algebra, 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?

A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic function is:

f(x) = ax^2 + bx + c

where a, b, and c are constants, and a cannot be zero.

Factored Form vs. Standard Form

The factored form of a quadratic function is expressed as:

f(x) = (x - r)(x - s)

where r and s are the roots of the quadratic equation.

The standard form of a quadratic function is expressed as:

f(x) = ax^2 + bx + c

Our goal is to match each quadratic function given in factored form with its equivalent standard form listed on the left.

Matching Quadratic Functions

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

To convert the factored form to standard form, we need to multiply the two binomials:

(x - r)(x - s) = x^2 - (r + s)x + rs

Comparing the coefficients, we get:

r + s = 11 rs = -12

Solving for r and s, we get:

r = 12 s = -1

Therefore, the standard form of the quadratic function is:

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

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

To convert the factored form to standard form, we need to multiply the two binomials:

(x - r)(x - s) = x^2 - (r + s)x + rs

Comparing the coefficients, we get:

r + s = 4 rs = -12

Solving for r and s, we get:

r = 6 s = -2

Therefore, the standard form of the quadratic function is:

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

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

To convert the factored form to standard form, we need to multiply the two binomials:

(x - r)(x - s) = x^2 - (r + s)x + rs

Comparing the coefficients, we get:

r + s = -1 rs = -12

Solving for r and s, we get:

r = 4 s = -3

Therefore, the standard form of the quadratic function is:

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

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

To convert the factored form to standard form, we need to multiply the two binomials:

(x - r)(x - s) = x^2 - (r + s)x + rs

Comparing the coefficients, we get:

r + s = 1 rs = -12

Solving for r and s, we get:

r = 3 s = -4

Therefore, the standard form of the quadratic function is:

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

Conclusion

In this article, we have matched each quadratic function given in factored form with its equivalent standard form. We have used the process of multiplying the two binomials to convert the factored form to standard form. This process has helped us to identify the roots of the quadratic equation and express the quadratic function in standard form.

Key Takeaways

Quadratic functions can be expressed in factored form and standard form.
The factored form of a quadratic function is expressed as (x - r)(x - s), where r and s are the roots of the quadratic equation.
The standard form of a quadratic function is expressed as ax^2 + bx + c, where a, b, and c are constants.
To convert the factored form to standard form, we need to multiply the two binomials and compare the coefficients.

Practice Problems

Match the quadratic function given in factored form with its equivalent standard form: f(x) = (x - 2)(x + 3)
Match the quadratic function given in factored form with its equivalent standard form: f(x) = (x + 1)(x - 4)
Match the quadratic function given in factored form with its equivalent standard form: f(x) = (x - 5)(x + 2)

Solutions

f(x) = (x - 2)(x + 3) = x^2 + x - 6
f(x) = (x + 1)(x - 4) = x^2 - 3x - 4
f(x) = (x - 5)(x + 2) = x^2 - 3x - 10

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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 practice problems and solutions to help you understand the concept better. In this article, we will answer some frequently asked questions related to quadratic functions in factored form and standard form.

Q&A

Q1: What is the difference between factored form and standard form of a quadratic function?

A1: The factored form of a quadratic function is expressed as (x - r)(x - s), where r and s are the roots of the quadratic equation. The standard form of a quadratic function is expressed as ax^2 + bx + c, where a, b, and c are constants.

Q2: How do I convert a quadratic function from factored form to standard form?

A2: To convert a quadratic function from factored form to standard form, you need to multiply the two binomials and compare the coefficients. The general formula is:

(x - r)(x - s) = x^2 - (r + s)x + rs

Q3: What are the roots of a quadratic equation?

A3: The roots of a quadratic equation are the values of x that satisfy the equation. In the factored form, the roots are r and s.

Q4: How do I find the roots of a quadratic equation?

A4: To find the roots of a quadratic equation, you need to solve the equation f(x) = 0. In the factored form, the roots are r and s.

Q5: What is the relationship between the coefficients of a quadratic function in factored form and standard form?

A5: The coefficients of a quadratic function in factored form and standard form are related by the following formulas:

r + s = -b/a rs = c/a

Q6: Can a quadratic function have more than two roots?

A6: No, a quadratic function can have at most two roots.

Q7: Can a quadratic function have no roots?

A7: Yes, a quadratic function can have no roots if the discriminant (b^2 - 4ac) is negative.

Q8: How do I determine the number of roots of a quadratic function?

A8: To determine the number of roots of a quadratic function, you need to calculate the discriminant (b^2 - 4ac). If the discriminant is positive, the function has two real roots. If the discriminant is zero, the function has one real root. If the discriminant is negative, the function has no real roots.

Conclusion

In this article, we have answered some frequently asked questions related to quadratic functions in factored form and standard form. We hope that this article has helped you to understand the concept better and to solve problems related to quadratic functions.

Key Takeaways

The factored form of a quadratic function is expressed as (x - r)(x - s), where r and s are the roots of the quadratic equation.
The standard form of a quadratic function is expressed as ax^2 + bx + c, where a, b, and c are constants.
To convert a quadratic function from factored form to standard form, you need to multiply the two binomials and compare the coefficients.
The roots of a quadratic equation are the values of x that satisfy the equation.
The coefficients of a quadratic function in factored form and standard form are related by the following formulas: r + s = -b/a and rs = c/a.

Practice Problems

Match the quadratic function given in factored form with its equivalent standard form: f(x) = (x - 2)(x + 3)
Match the quadratic function given in factored form with its equivalent standard form: f(x) = (x + 1)(x - 4)
Match the quadratic function given in factored form with its equivalent standard form: f(x) = (x - 5)(x + 2)

Solutions

f(x) = (x - 2)(x + 3) = x^2 + x - 6
f(x) = (x + 1)(x - 4) = x^2 - 3x - 4
f(x) = (x - 5)(x + 2) = x^2 - 3x - 10