Mastering The Basics: Quadratic Functions And Their Applications In Science And Math

Quadratic Function

f(x) = x²

A quadratic function is a type of function that involves the variable raised to the second power, such as f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can have different shapes and orientations depending on the values of a, b, and c.

The a value in a quadratic function determines the direction and width of the parabola: if a is positive, the parabola opens upwards and is wider; if a is negative, the parabola opens downwards and is narrower. The b value determines the position of the vertex (the highest or lowest point of the parabola) and the axis of symmetry (the line that divides the parabola into two equal halves). The c value is the y-intercept, i.e. the point where the parabola intersects the y-axis.

To find the vertex and axis of symmetry of a quadratic function, we can use the formula x = -b/2a for the x-coordinate of the vertex, and substitute it into the function to find the y-coordinate. The equation of the axis of symmetry is x = -b/2a.

In addition to the standard form of a quadratic function, there are other forms that are useful in different contexts, such as vertex form (f(x) = a(x-h)^2 + k, where (h,k) is the vertex), factored form (f(x) = a(x-r)(x-s), where r and s are the x-intercepts), and completing the square form (f(x) = a(x-p)^2 + q, where (p,q) is the vertex).

Quadratic functions have many applications in science, engineering, and mathematics, such as modeling the motion of projectiles, optimizing problem-solving scenarios, and approximating functions that are difficult to compute.

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