%5E2%2B5.jpeg "(x-2)^2+5")
Exploring the Expression (x-2)^2 + 5
This expression represents a quadratic function, which is a function that can be written in the form ax^2 + bx + c, where a, b, and c are constants. Let's break down its features and how it behaves.
Understanding the Components
- (x-2)^2: This part represents a perfect square trinomial. When expanded, it becomes x^2 - 4x + 4.
- +5: This is a constant term, meaning it does not change with the value of x.
Simplifying the Expression
We can simplify the expression by expanding the squared term:
(x-2)^2 + 5 = x^2 - 4x + 4 + 5 = x^2 - 4x + 9
Key Features of the Quadratic Function
- Vertex: The vertex is the point where the parabola reaches its minimum or maximum value. In this case, the vertex is at (2, 5). This can be determined by finding the value of x that minimizes the expression, which is x = 2, and then substituting it back into the expression to get the y-coordinate, y = 5.
- Axis of Symmetry: The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. The equation of the axis of symmetry is x = 2, which is the x-coordinate of the vertex.
- Concavity: Since the coefficient of the x^2 term is positive (1), the parabola opens upwards. This means it has a minimum value.
Graphical Representation
The graph of the function (x-2)^2 + 5 is a parabola that opens upwards, with its vertex at (2, 5). It is symmetrical about the line x = 2.
Applications
Quadratic functions have numerous applications in various fields, including:
- Physics: Modeling projectile motion, gravitational force, and energy.
- Engineering: Designing bridges, structures, and other objects that require understanding of curves and shapes.
- Economics: Analyzing market trends, supply and demand, and cost optimization.
Understanding the properties and behavior of quadratic functions like (x-2)^2 + 5 is crucial for solving various problems in mathematics, science, and engineering.