Analyzing G(x) = 4x² - 16x - 64 Finding Vertex And Coefficients

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In the realm of mathematics, quadratic functions hold a significant position due to their wide-ranging applications in various fields, including physics, engineering, and economics. A quadratic function is a polynomial function of degree two, typically expressed in the form g(x) = ax² + bx + c, where a, b, and c are constants, and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. Understanding the properties and characteristics of quadratic functions is crucial for solving mathematical problems and gaining insights into real-world phenomena.

Deconstructing the Quadratic Function g(x) = 4x² - 16x - 64

Let's delve into the specifics of the given quadratic function, g(x) = 4x² - 16x - 64. By comparing this function to the general form g(x) = ax² + bx + c, we can identify the values of the coefficients a, b, and c. In this case:

  • a = 4: This coefficient determines the parabola's concavity. Since 'a' is positive (4 > 0), the parabola opens upwards, indicating a minimum point.
  • b = -16: This coefficient influences the parabola's horizontal position and symmetry.
  • c = -64: This constant represents the y-intercept of the parabola, the point where the graph intersects the y-axis.

These coefficients play a pivotal role in shaping the parabola's characteristics and behavior. Understanding their significance allows us to analyze and interpret the function's properties effectively.

Unveiling the Vertex: The Heart of the Parabola

The vertex of a parabola is a critical point that marks either the minimum or maximum value of the quadratic function. For a parabola that opens upwards (a > 0), the vertex represents the minimum point, while for a parabola that opens downwards (a < 0), the vertex represents the maximum point. The vertex is also the point of symmetry for the parabola, dividing it into two identical halves.

The vertex's coordinates can be determined using the formula (-b/2a, g(-b/2a)). This formula provides a systematic approach to locating the vertex, which is essential for understanding the parabola's behavior and solving related problems.

Calculating the Vertex of g(x) = 4x² - 16x - 64

Now, let's apply the vertex formula to our function, g(x) = 4x² - 16x - 64. We already know that a = 4 and b = -16. Plugging these values into the formula, we get:

  • x-coordinate of the vertex: -b/2a = -(-16) / (2 * 4) = 16 / 8 = 2

To find the y-coordinate of the vertex, we substitute x = 2 into the function g(x):

  • g(2) = 4(2)² - 16(2) - 64 = 16 - 32 - 64 = -80

Therefore, the vertex of the parabola represented by g(x) = 4x² - 16x - 64 is (2, -80). This point signifies the minimum value of the function, as the parabola opens upwards.

Significance of the Vertex

The vertex holds immense significance in understanding the behavior and characteristics of a quadratic function. It provides valuable information about the function's minimum or maximum value, the axis of symmetry, and the overall shape of the parabola. Here's a breakdown of the vertex's key contributions:

  1. Minimum or Maximum Value: As mentioned earlier, the vertex represents the minimum value of the function when the parabola opens upwards (a > 0) and the maximum value when the parabola opens downwards (a < 0). This information is crucial for optimization problems, where we seek to find the smallest or largest possible value of a function.
  2. Axis of Symmetry: The vertex lies on the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves. The equation of the axis of symmetry is x = -b/2a, which is the x-coordinate of the vertex. This symmetry simplifies the analysis and graphing of quadratic functions.
  3. Graphing the Parabola: The vertex serves as a crucial reference point for graphing the parabola. Knowing the vertex's coordinates allows us to accurately plot the parabola and visualize its shape and behavior. Along with the vertex, other key points, such as the y-intercept and x-intercepts (if they exist), further enhance the accuracy of the graph.

Applications of Quadratic Functions and Vertex

Quadratic functions and the concept of the vertex have numerous applications in various fields. Here are a few prominent examples:

  1. Projectile Motion: The trajectory of a projectile, such as a ball thrown into the air, can be modeled using a quadratic function. The vertex of the parabola represents the projectile's maximum height and the time it takes to reach that height. This knowledge is essential in sports, engineering, and physics.
  2. Optimization Problems: Quadratic functions are frequently used in optimization problems, where we aim to find the maximum or minimum value of a certain quantity. For instance, a business might use a quadratic function to model the profit generated by selling a product and then use the vertex to determine the price that maximizes profit.
  3. Engineering Design: In engineering, quadratic functions are used to design various structures, such as bridges and arches. The parabolic shape of these structures provides optimal strength and stability. The vertex plays a crucial role in determining the structure's dimensions and load-bearing capacity.
  4. Curve Fitting: Quadratic functions can be used to fit curves to data points, allowing us to model and analyze trends in various phenomena. This technique is widely used in statistics, data analysis, and machine learning.

Finding Intercepts of g(x) = 4x² - 16x - 64

To further understand the behavior of the quadratic function g(x) = 4x² - 16x - 64, let's determine its intercepts, which are the points where the parabola intersects the x and y axes.

Y-intercept

The y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0. To find the y-intercept, we substitute x = 0 into the function:

g(0) = 4(0)² - 16(0) - 64 = -64

Therefore, the y-intercept is (0, -64).

X-intercepts

The x-intercepts, also known as the roots or zeros of the function, are the points where the parabola intersects the x-axis. This occurs when g(x) = 0. To find the x-intercepts, we need to solve the quadratic equation:

4x² - 16x - 64 = 0

We can simplify this equation by dividing both sides by 4:

x² - 4x - 16 = 0

Now, we can use the quadratic formula to solve for x:

x = [-b ± √(b² - 4ac)] / 2a

Where a = 1, b = -4, and c = -16.

Plugging these values into the quadratic formula, we get:

x = [4 ± √((-4)² - 4 * 1 * -16)] / (2 * 1)

x = [4 ± √(16 + 64)] / 2

x = [4 ± √80] / 2

x = [4 ± 4√5] / 2

x = 2 ± 2√5

Therefore, the x-intercepts are (2 + 2√5, 0) and (2 - 2√5, 0). These points represent the values of x where the function g(x) equals zero.

Conclusion

In summary, the quadratic function g(x) = 4x² - 16x - 64 is a parabola that opens upwards, with a vertex at (2, -80). The vertex represents the minimum value of the function and serves as a crucial point for understanding the parabola's behavior. The y-intercept is (0, -64), and the x-intercepts are (2 + 2√5, 0) and (2 - 2√5, 0). These intercepts provide further insights into the function's graph and its relationship with the coordinate axes.

Understanding quadratic functions and their properties is essential for solving mathematical problems and gaining insights into real-world phenomena. The vertex, intercepts, and coefficients all contribute to the function's unique characteristics and behavior. By mastering these concepts, we can effectively analyze and interpret quadratic functions in various applications. These applications span from projectile motion analysis to optimization problems in business and engineering, highlighting the broad relevance and utility of quadratic functions in diverse fields.

By understanding the concepts discussed, you can analyze quadratic functions with greater confidence and apply them effectively in various mathematical and real-world scenarios. The ability to deconstruct these functions, identify their key components, and interpret their behavior is a valuable skill in mathematics and beyond.