Finding The X-Intercept Of Quadratic Function F(x) = (x-4)(x+2)

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This comprehensive guide delves into the concept of x-intercepts, specifically within the context of quadratic functions. We will dissect the given function, f(x) = (x-4)(x+2), and employ a step-by-step approach to identify its x-intercepts. By understanding the fundamental principles behind quadratic equations and their graphical representations, we can effectively determine the points where the parabola intersects the x-axis. This exploration will not only provide the solution to this specific problem but also equip you with the knowledge to tackle similar challenges in the realm of algebra and beyond.

Understanding X-Intercepts

X-intercepts, also known as roots or zeros of a function, are the points where the graph of the function intersects the x-axis. At these points, the y-coordinate (or the function's value, f(x)) is always zero. Finding x-intercepts is a crucial skill in algebra and calculus, as it helps us understand the behavior of functions and solve equations. In the context of a quadratic function, the x-intercepts represent the solutions to the quadratic equation when set equal to zero. These points hold significant importance in various applications, including modeling projectile motion, optimizing areas, and analyzing growth patterns.

To find the x-intercepts, we set the function equal to zero and solve for x. This process involves identifying the values of x that make the function's output zero, effectively pinpointing where the graph crosses the x-axis. The number of x-intercepts a function has can vary, with quadratic functions typically having zero, one, or two x-intercepts. The nature and location of these intercepts provide valuable insights into the function's properties and its relationship to the x-axis.

Analyzing the Quadratic Function f(x) = (x-4)(x+2)

The given quadratic function is f(x) = (x-4)(x+2). This function is presented in its factored form, which makes identifying the x-intercepts relatively straightforward. The factored form of a quadratic function provides a direct representation of its roots, allowing us to quickly determine the values of x that make the function equal to zero. This form is particularly useful when solving quadratic equations and analyzing the behavior of the corresponding parabola.

To understand the function better, let's expand it into the standard form: f(x) = x² - 2x - 8. The standard form of a quadratic function, ax² + bx + c, provides a different perspective on the function's properties, such as its y-intercept and vertex. However, the factored form is more convenient for finding the x-intercepts. By setting the factored form equal to zero, we can easily identify the values of x that satisfy the equation. These values correspond to the points where the parabola intersects the x-axis, giving us the x-intercepts of the function.

Finding the X-Intercepts

To find the x-intercepts, we set f(x) = 0 and solve for x:

(x-4)(x+2) = 0

This equation is satisfied when either (x-4) = 0 or (x+2) = 0. This principle, known as the zero-product property, is fundamental to solving equations in factored form. It states that if the product of two factors is zero, then at least one of the factors must be zero. Applying this property allows us to break down the original equation into simpler equations, making it easier to find the solutions.

Solving each equation separately:

  • x - 4 = 0 => x = 4
  • x + 2 = 0 => x = -2

Therefore, the x-intercepts are x = 4 and x = -2. These values represent the points where the parabola intersects the x-axis. They are crucial for understanding the graph of the quadratic function and its behavior. Knowing the x-intercepts allows us to sketch the parabola, determine its axis of symmetry, and identify the intervals where the function is positive or negative.

Expressing X-Intercepts as Coordinates

The x-intercepts are points on the coordinate plane where the graph intersects the x-axis. These points have the form (x, 0), where x is the value we found in the previous step. Therefore, the x-intercepts of the given function are:

  • (4, 0)
  • (-2, 0)

These coordinates provide a precise location for the points where the parabola crosses the x-axis. They are essential for accurately graphing the quadratic function and visualizing its behavior. By plotting these points on a coordinate plane, we can begin to sketch the parabola and understand its overall shape and orientation.

Identifying the Correct Option

Now, let's look at the given options:

A. (-4, 0)

B. (-2, 0)

C. (0, 2)

D. (4, -2)

Comparing these options with our calculated x-intercepts, we can see that option B (-2, 0) is one of the x-intercepts, and while option A (-4, 0) looks similar, we found that the positive root was 4 not -4, so it is incorrect. D (4,-2) is incorrect, as x-intercepts have a y-value of 0. The remaining x-intercept is not present as one of the answer options, and therefore the correct answer is B. (-2, 0).

Additional Insights into Quadratic Functions

Quadratic functions are powerful tools for modeling a variety of real-world phenomena, from the trajectory of a projectile to the shape of a suspension bridge. Understanding their properties, including x-intercepts, vertex, and axis of symmetry, is essential for applying them effectively.

  • Vertex: The vertex of a parabola is the point where the function reaches its maximum or minimum value. Its x-coordinate can be found using the formula x = -b / 2a, where a and b are the coefficients of the quadratic function in standard form (ax² + bx + c). The vertex plays a crucial role in determining the overall shape and orientation of the parabola. It represents the turning point of the function, where it changes from increasing to decreasing or vice versa.
  • Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. Its equation is x = -b / 2a, the same as the x-coordinate of the vertex. The axis of symmetry provides a visual reference for the symmetry of the parabola and helps in sketching its graph. It also aids in understanding the relationship between the two x-intercepts, which are equidistant from the axis of symmetry.
  • Discriminant: The discriminant, b² - 4ac, provides information about the number and nature of the x-intercepts. If the discriminant is positive, there are two distinct real roots (two x-intercepts). If it's zero, there is one real root (one x-intercept). If it's negative, there are no real roots (no x-intercepts). The discriminant is a valuable tool for quickly assessing the number of x-intercepts without explicitly solving the quadratic equation. It provides insights into the behavior of the parabola and its relationship to the x-axis.

Conclusion

In conclusion, the x-intercepts of the quadratic function f(x) = (x-4)(x+2) are (-2, 0) and (4, 0), with option B being the correct answer in the choices provided. By understanding the concept of x-intercepts and applying the zero-product property, we successfully identified the points where the parabola intersects the x-axis. This knowledge empowers us to analyze quadratic functions, solve equations, and model real-world scenarios. Remember, mastering these fundamental concepts is crucial for success in algebra and beyond. Keep practicing, and you'll become proficient in working with quadratic functions and their applications.

Key Takeaways:

  • X-intercepts are points where the graph of a function intersects the x-axis.
  • To find x-intercepts, set f(x) = 0 and solve for x.
  • The factored form of a quadratic function makes finding x-intercepts easier.
  • The zero-product property is essential for solving equations in factored form.
  • Understanding x-intercepts helps in graphing quadratic functions and analyzing their behavior.