Quadratic Functions: Vertex and Y-Intercept

Quadratic Functions

Quadratic functions are among the most basic types of polynomial functions, with applications across many fields including mathematics, physics, and economics. They have the general form (f(x) = ax^2 + bx + c), where (a, b,) and (c) are constants and (a \neq 0). These functions exhibit exponential growth when graphed, which means they increase very rapidly over time. In this article, we will explore how to calculate the vertex and find the y-intercept of such functions.

Calculating Vertex

The vertex of a quadratic function is the point where the parabola reaches its maximum or minimum value, denoted by ((x_v,y_v)). To find the x-coordinate of the vertex, also known as the abscissa, you can use the formula [x_v = -\frac{b}{2a}]. For example, if we have the equation (f(x) = -x^2 + 8x - 9), (a=-1), (b=8), and (c=-9), so (x_v = -\frac{8}{2(-1)} = 4). The actual value of the corresponding function evaluation would be [f(4) = (-1)(4)^2 + 8(4) - 9 = -16 + 32 - 9 = 7] Therefore, the vertex of this quadratic function occurs at the point ((4,7)).

Finding Y-Intercept

To find the y-intercept of a quadratic function, set (x = 0) and solve for (y). This gives us (y_i = f(0) = a(0)^2 + b(0) + c = c). By plugging the coefficients from our earlier equation into the formula above, we get (y_i = -9). So, the y-intercept of our quadratic function is ((0,-9)).

Understanding the Concepts

Now let's delve deeper into these concepts. A quadratic function represents a parabolic curve. If the coefficient (a) is positive, it opens upward, reaching its maximum value at the vertex. On the other hand, if (a) is negative, the quadratic function opens downward, reaching its minimum value at the vertex. Each quadratic function has exactly one vertex.

In conclusion, a quadratic function can be described mathematically with the general expression (f(x) = ax^2 + bx + c), where (a), (b), and (c) are constants. We can determine the x-coordinate of the vertex using the formula (x_v = -\frac{b}{2a}) and the y-value at the vertex by evaluating the function at that x-value. Finally, the y-intercept can be found by setting (x = 0) and solving for (y).