Exploring the Expression (x-3)^2 + 5
This article will delve into the expression (x-3)^2 + 5, examining its components, its graphical representation, and its potential uses in various mathematical contexts.
Understanding the Components
The expression (x-3)^2 + 5 is a quadratic expression, meaning it involves a variable raised to the power of two. Let's break down each component:
- (x-3)^2: This represents the square of the binomial (x-3). Squaring a binomial means multiplying it by itself: (x-3) * (x-3). Expanding this using the distributive property gives us x^2 - 6x + 9.
- + 5: This is a constant term, meaning it's a numerical value that does not change.
Therefore, the entire expression can be written as: x^2 - 6x + 14.
The Graph of the Expression
The expression (x-3)^2 + 5 represents a parabola when graphed. Here's why:
- x^2: This term determines the parabolic shape.
- -6x + 14: This part shifts and positions the parabola on the coordinate plane.
The vertex of the parabola, the lowest point, is located at the point (3, 5). This is because the expression is in vertex form: (x - h)^2 + k, where (h, k) represents the vertex.
Applications in Mathematics
The expression (x-3)^2 + 5 has various applications in mathematics, including:
- Finding the minimum value: The expression's minimum value is 5, occurring at x = 3. This is useful in optimization problems where we need to find the smallest possible value of a function.
- Solving quadratic equations: By setting the expression equal to zero, we get a quadratic equation. This equation can then be solved using methods like factoring, completing the square, or the quadratic formula.
- Modeling real-world phenomena: Quadratic expressions can be used to model various real-world situations, such as the trajectory of a projectile or the shape of a bridge.
Conclusion
The expression (x-3)^2 + 5, while seemingly simple, offers a rich understanding of quadratic expressions, their graphical representation, and their diverse applications in mathematics. By dissecting its components and exploring its properties, we gain valuable insights into the broader world of algebra and its role in solving real-world problems.