Factored Form of Quadratic Functions

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Understanding Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. 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.

Factored Form of Quadratic Functions

The factored form of a quadratic function is a way of expressing the function as a product of two binomials. This form is useful for solving quadratic equations and graphing quadratic functions. The factored form of a quadratic function is given by f(x) = (x - r)(x - s), where r and s are the roots of the quadratic equation.

Vertex of a Quadratic Function

The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. The vertex is also the minimum or maximum point of the graph, depending on the direction of the parabola. In this article, we will focus on finding the vertex of a quadratic function in factored form.

Which Function Has a Vertex at the Origin?

To determine which function has a vertex at the origin, we need to analyze each option given.

Option A: $f(x) = (x-4)(x+4)$

The factored form of this function is (x - 4)(x + 4). To find the vertex, we need to find the roots of the function, which are the values of x that make the function equal to zero. In this case, the roots are x = 4 and x = -4. The vertex is the midpoint of the roots, which is (4 - 4)/2 = 0. However, the vertex is not at the origin, but rather at (0, 16).

Option B: $f(x) = -x^2$

The factored form of this function is not given, but we can rewrite it as f(x) = -(x^2). The vertex of this function is at the origin, since the function is symmetric about the y-axis and the coefficient of x^2 is negative.

Option C: $f(x) = (x+4)^2$

The factored form of this function is (x + 4)^2. To find the vertex, we need to find the root of the function, which is the value of x that makes the function equal to zero. In this case, the root is x = -4. The vertex is the point (x, f(x)) = (-4, 0).

Option D: $f(x) = x(x-4)$

The factored form of this function is x(x - 4). To find the vertex, we need to find the roots of the function, which are the values of x that make the function equal to zero. In this case, the roots are x = 0 and x = 4. The vertex is the midpoint of the roots, which is (0 + 4)/2 = 2. However, the vertex is not at the origin, but rather at (2, 0).

Conclusion

In conclusion, the function that has a vertex at the origin is option B: $f(x) = -x^2$ . This function is symmetric about the y-axis and has a negative coefficient of x^2, which means it opens downwards and has a vertex at the origin.

Key Takeaways

Quadratic functions can be expressed in factored form as f(x) = (x - r)(x - s), where r and s are the roots of the quadratic equation.
The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa.
The vertex of a quadratic function in factored form can be found by finding the roots of the function and taking the midpoint of the roots.
The function $f(x) = -x^2$ has a vertex at the origin.

References

Larson, R., & Hostetler, R. P. (2015). College Algebra. Cengage Learning.
Sullivan, M. (2016). College Algebra. Pearson Education.

About the Author

The author of this article is a mathematics educator with a passion for teaching and learning. They have a strong background in mathematics and have taught various courses, including algebra, geometry, and calculus. They are committed to providing high-quality educational resources to students and educators alike.