The Question: Identifying the X-Intercept Point for the Quadratic Function: Given that f(x) = (x - 4)(x + 2), what is the intercept where the graph crosses the x-axis?

The Question: Identifying the X-Intercept Point for the Quadratic Function: Given that f(x) = (x - 4)(x + 2), what is the intercept where the graph crosses the x-axis?

The quadratic function f(x) = (x - 4)(x + 2) is a valuable tool in understanding real-world phenomena that exhibit quadratic behaviour. This function, with its two factors, belongs to the family of polynomial functions known as quadratic functions.

In the realm of analytic geometry, points are represented by ordered pairs or triples, while lines are described using linear equations. Quadratic functions, on the other hand, are of degree 2, with the general form being f(x) = ax^2 + bx + c. The highest power of x in the equation f(x) = (x - 4)(x + 2) is 2, making it a quadratic function.

One of the key characteristics of a quadratic function is its x-intercepts, the points where the graph intersects the x-axis. By setting the function equal to zero, we can determine the x-intercepts or zeros of a quadratic function. In the case of f(x) = (x - 4)(x + 2), this leads us to the solutions x = 4 and x = -2. Thus, the x-intercepts are at x = 4 and x = -2, or as points on the coordinate plane, (4,0) and (-2,0).

The behaviour of the function can be visualized on a graph, providing valuable insights into its properties. For instance, the vertex of a parabola represents the minimum or maximum point of the function depending on its concavity. The coefficient "a" in a quadratic function determines whether the parabola opens upwards or downwards. If "a" is positive, the parabola opens upwards; if "a" is negative, it opens downwards.

Quadratic functions are widely used in various fields such as physics, engineering, economics, and computer science. They provide valuable insights into real-world phenomena by modeling relationships between variables that exhibit quadratic behaviour. For example, the function f(x) = (x - 4)(x + 2) can be used to model projectile motion and parabolic arcs.

Analytic geometry, the branch of mathematics that combines algebraic techniques with geometric concepts, plays a crucial role in understanding and working with functions like f(x) = (x - 4)(x + 2). By analyzing the equation further, we can determine other important characteristics such as the vertex and the axis of symmetry, providing a deeper understanding of the function's behaviour.

[1]: The x-intercepts of the quadratic function ( f(x) = (x - 4)(x + 2) ) are the values of (x) that make ( f(x) = 0 ). [2]: By the Zero Product Property, either (x - 4 = 0 \quad \text{or} \quad x + 2 = 0) which gives (x = 4 \quad \text{or} \quad x = -2) [3]: Thus, the x-intercepts are at (x = 4) and (x = -2), or as points on the coordinate plane, ( (4,0) ) and ( (-2,0) )

[1]: Consistently mastering the properties of quadratic functions, such as the x-intercepts, aids in exploring education-and-self-development in mathematics and related fields.[2]: The learning process becomes more meaningful when these concepts are applied to real-world problems, like using the function f(x) = (x - 4)(x + 2) to model parabolic arcs.

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