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A quadratic function is a type of polynomial function specifically characterized by its highest exponent being 2. This means it has the general form: \[ f(x) = ax^2 + bx + c \]Quadratic functions graph to form a parabola, which is a U-shaped curve. These functions are widely used because they model many real-world scenarios, from projectile motion to economics.

• a is the coefficient of the squared term \( (x^2) \), which determines the width and the direction of the parabola.
• b is the coefficient of the linear term \( (x) \), affecting the position and orientation of the parabola.
• c is the constant term.

By changing the values of a, b, and c, you can manipulate the shape and position of the parabola.

The standard form of a quadratic function is essential for identifying various properties of the function, including the vertex. The general expression in standard form is: \[ f(x) = ax^2 + bx + c \]To understand the vertex of the parabola described by the quadratic function, the values of the coefficients a, b, and c are crucial.For example, given the quadratic function in the exercise: \[ f(x) = x^2 - 4 \]

• Here, a is 1, affecting the width and the direction of the parabola.
• b is 0, which means the parabola is symmetrical about the y-axis in this instance.
• c is -4, which represents the y-intercept of the parabola.

Recognizing and using this standard form enables us to complete tasks such as finding the vertex or roots of the function efficiently.

The vertex formula is a valuable tool for finding the vertex of a parabola described by a quadratic function in standard form \( ax^2 + bx + c \). It provides a straightforward way to determine the highest or lowest point on the parabola. The x-coordinate of the vertex can be found using: \[ x = -\frac{b}{2a} \]Using the vertex formula for the given function \( f(x) = x^2 - 4 \):

• We identify a as 1 and b as 0.

Substituting these values into the vertex formula, we get: \[ x = -\frac{0}{2(1)} = 0 \]With the y-coordinate found by substituting the x-value back into the original function: \[ f(0) = 0^2 - 4 = -4 \]So, the vertex is located at: \[ (0, -4) \]This point represents the minimum value of the function for the given example.