Understanding Quadratic Functions
Quadratic functions are polynomial functions of degree two, typically expressed in the form:
- f(x) = ax² + bx + c, where:
- a, b, and c are constants
- a ≠ 0 (if a = 0, it becomes a linear function)
These functions graph as parabolas, which can open upwards or downwards depending on the sign of 'a'.
Characteristics of Quadratic Functions
- Vertex: The highest or lowest point of the parabola, determined by the formula x = -b/(2a).
- Axis of Symmetry: The vertical line that divides the parabola into two mirror images, given by the equation x = -b/(2a).
- Y-intercept: The point where the graph intersects the y-axis, found by evaluating f(0) = c.
- X-intercepts (Roots): Points where the graph intersects the x-axis, found using the quadratic formula.
Solving Quadratic Functions using the Quadratic Formula
The quadratic formula is a reliable method to find the roots of any quadratic equation. The formula is:
- x = (-b ± √(b² - 4ac)) / (2a)
Steps to use the formula:
- Identify coefficients: From the standard form, determine a, b, and c.
- Calculate the discriminant (D): D = b² - 4ac
- If D > 0: Two distinct real roots.
- If D = 0: One real root (repeated).
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- Substitute values into the formula: Plug in a, b, and c into the quadratic formula to find the solutions for x.
Conclusion
Quadratic functions are essential in various fields, including commerce, for modeling and predicting outcomes. Understanding how to solve them using the quadratic formula is a critical skill.