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

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In the realm of mathematics, quadratic functions hold a significant place due to their widespread applications in various fields, including physics, engineering, and economics. A quadratic function is a polynomial function of degree two, generally expressed in the form g(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. 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 real-world problems and gaining a deeper appreciation for mathematical concepts.

This article delves into the analysis of a specific quadratic function, g(x) = 4x² - 16x - 64, exploring its coefficients, vertex, and graphical representation. By dissecting this function, we aim to enhance your understanding of quadratic functions and their applications. We will also discuss the general form of a quadratic function, the significance of its coefficients, and the method for determining the vertex, a critical point on the parabola.

Identifying Coefficients in g(x) = 4x² - 16x - 64

The given quadratic function, g(x) = 4x² - 16x - 64, is in the standard form g(x) = ax² + bx + c. To analyze this function, we first need to identify the coefficients a, b, and c. These coefficients play a crucial role in determining the shape, position, and other characteristics of the parabola.

  • a: The coefficient of the x² term determines the direction and width of the parabola. In our function, g(x) = 4x² - 16x - 64, the coefficient a is 4. Since a is positive, the parabola opens upwards. The larger the value of a, the narrower the parabola.
  • b: The coefficient of the x term influences the position of the parabola's axis of symmetry and vertex. In our function, the coefficient b is -16. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
  • c: The constant term c represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. In our function, the constant term c is -64. This means the parabola intersects the y-axis at the point (0, -64).

In summary, for the quadratic function g(x) = 4x² - 16x - 64:

  • a = 4
  • b = -16
  • c = -64

These coefficients provide valuable information about the parabola's shape and position on the coordinate plane. Now, let's move on to determining the vertex of the parabola, a crucial point that reveals the maximum or minimum value of the function.

Determining the Vertex of the Parabola

The vertex of a parabola is a significant point that represents either the minimum or maximum value of the quadratic function. For a parabola that opens upwards (a > 0), the vertex is the minimum point, while for a parabola that opens downwards (a < 0), the vertex is the maximum point. The vertex is also located on the axis of symmetry, which divides the parabola into two symmetrical halves.

For a quadratic function in the standard form g(x) = ax² + bx + c, the vertex can be found using the following formula:

Vertex = (-b/2a, g(-b/2a))

This formula provides the x-coordinate and y-coordinate of the vertex. The x-coordinate is calculated as -b/2a, and the y-coordinate is found by substituting the x-coordinate back into the original function, g(x).

Let's apply this formula to our function, g(x) = 4x² - 16x - 64:

  1. Find the x-coordinate of the vertex:

    x = -b/2a = -(-16) / (2 * 4) = 16 / 8 = 2

  2. Find the y-coordinate of the vertex:

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

Therefore, the vertex of the parabola represented by the function g(x) = 4x² - 16x - 64 is (2, -80). This point represents the minimum value of the function since the parabola opens upwards (a > 0).

Understanding the vertex is essential for various applications, such as finding the maximum height of a projectile or the minimum cost in a business scenario. The vertex provides critical information about the extreme values of the quadratic function.

Detailed Explanation of the Vertex Formula

To fully grasp the significance of the vertex formula (-b/2a, g(-b/2a)), it's beneficial to understand its derivation and the underlying mathematical principles. The formula arises from the process of completing the square, a technique used to rewrite a quadratic function in vertex form.

The vertex form of a quadratic function is given by:

g(x) = a(x - h)² + k

where (h, k) represents the vertex of the parabola. By converting the standard form g(x) = ax² + bx + c into vertex form, we can directly identify the vertex coordinates.

The process of completing the square involves manipulating the quadratic expression to create a perfect square trinomial. This is achieved by adding and subtracting a specific term, which is derived from the coefficient of the x term (b).

Let's outline the steps involved in completing the square for the general quadratic function g(x) = ax² + bx + c:

  1. Factor out the coefficient 'a' from the x² and x terms:

    g(x) = a(x² + (b/a)x) + c

  2. Take half of the coefficient of the x term (b/a), square it, and add and subtract it inside the parentheses:

    Half of (b/a) is (b/2a), and squaring it gives (b²/4a²).

    g(x) = a(x² + (b/a)x + (b²/4a²) - (b²/4a²)) + c

  3. Rewrite the expression inside the parentheses as a perfect square trinomial:

    g(x) = a((x + b/2a)²) - a(b²/4a²) + c

  4. Simplify the expression:

    g(x) = a(x + b/2a)² - b²/4a + c

  5. Rewrite the constant term to have a common denominator:

    g(x) = a(x + b/2a)² - (b² - 4ac) / 4a

Now, comparing this with the vertex form g(x) = a(x - h)² + k, we can identify the vertex coordinates:

  • h = -b/2a
  • k = - (b² - 4ac) / 4a

Notice that the x-coordinate of the vertex, h, is indeed -b/2a, as stated in the vertex formula. The y-coordinate, k, can also be expressed as g(-b/2a) by substituting -b/2a back into the original function.

Therefore, the vertex formula (-b/2a, g(-b/2a)) is a direct result of completing the square and rewriting the quadratic function in vertex form. This formula provides a convenient way to determine the vertex without having to go through the entire process of completing the square each time.

Graphical Representation of g(x) = 4x² - 16x - 64

Visualizing the quadratic function g(x) = 4x² - 16x - 64 as a graph provides a deeper understanding of its properties and behavior. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. Several key features of the parabola can be identified from the function's equation and the calculations we've performed.

  1. Direction of Opening: As we determined earlier, the coefficient a = 4 is positive, indicating that the parabola opens upwards. This means the vertex represents the minimum point of the function.

  2. Vertex: We calculated the vertex to be (2, -80). This point is the lowest point on the parabola and lies on the axis of symmetry.

  3. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation of the axis of symmetry is x = -b/2a, which in our case is x = 2.

  4. Y-intercept: The y-intercept is the point where the parabola intersects the y-axis. This occurs when x = 0. Substituting x = 0 into the function, we get g(0) = 4(0)² - 16(0) - 64 = -64. So, the y-intercept is (0, -64).

  5. X-intercepts (Roots): The x-intercepts, also known as roots or zeros, 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

    This quadratic equation does not factor easily, so we can use the quadratic formula to find the roots:

    x = (-b ± √(b² - 4ac)) / 2a

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

    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

    So, the x-intercepts are approximately x ≈ 6.47 and x ≈ -2.47. These are the points where the parabola crosses the x-axis.

By plotting these key points (vertex, y-intercept, and x-intercepts) and considering the direction of opening, we can sketch the graph of the parabola. The graph visually represents the function's behavior, showing its minimum value at the vertex and its intercepts with the axes.

Applications of Quadratic Functions

Quadratic functions have numerous applications in various fields, making their study essential in mathematics and beyond. Here are a few examples of how quadratic functions are used in real-world scenarios:

  1. Physics: Projectile motion is a classic example of a quadratic function in physics. The path of a projectile, such as a ball thrown into the air, can be modeled by a parabola. The vertex of the parabola represents the maximum height reached by the projectile, and the x-intercepts represent the points where the projectile lands.

  2. Engineering: Quadratic functions are used in structural engineering to design arches and bridges. The parabolic shape of an arch provides structural stability and can withstand significant loads. Engineers use quadratic equations to calculate the optimal dimensions and shape of arches and bridges.

  3. Economics: Quadratic functions can be used to model cost, revenue, and profit in business applications. For example, the cost function for producing a certain number of items might be a quadratic function. By finding the vertex of the cost function, businesses can determine the production level that minimizes costs.

  4. Optimization Problems: Many optimization problems involve finding the maximum or minimum value of a function. Quadratic functions are particularly useful in solving optimization problems where the function has a parabolic shape. The vertex of the parabola represents the optimal solution.

  5. Curve Fitting: Quadratic functions can be used to fit curves to data points. This is useful in various fields, such as statistics and data analysis. By finding the quadratic function that best fits a set of data points, we can model the relationship between variables and make predictions.

These are just a few examples of the many applications of quadratic functions. Their ability to model parabolic shapes and represent extreme values makes them a valuable tool in various disciplines.

Conclusion

In this article, we have explored the quadratic function g(x) = 4x² - 16x - 64, dissecting its coefficients, determining its vertex, and discussing its graphical representation. We identified the coefficients a, b, and c, and used them to calculate the vertex using the formula (-b/2a, g(-b/2a)). We also discussed the significance of the vertex as the minimum point of the parabola and its location on the axis of symmetry.

Furthermore, we delved into the derivation of the vertex formula through the process of completing the square, providing a deeper understanding of its mathematical foundation. We explored the graphical representation of the function, identifying key features such as the direction of opening, vertex, axis of symmetry, y-intercept, and x-intercepts.

Finally, we highlighted the numerous applications of quadratic functions in various fields, including physics, engineering, economics, and optimization problems. This demonstrates the practical significance of understanding quadratic functions and their properties.

By mastering the concepts and techniques discussed in this article, you can confidently analyze quadratic functions, solve related problems, and appreciate their role in various real-world applications. Quadratic functions are a fundamental building block in mathematics, and a strong understanding of them will pave the way for further exploration of more advanced mathematical concepts.