Analyzing The Parabola F(x) = 3(x-4)^2 + 5 Determining Axis Of Symmetry, Y-intercept, And Vertex

In this article, we will delve into the characteristics of the parabola defined by the quadratic function f(x) = 3(x - 4)² + 5. Understanding the properties of parabolas is crucial in various fields, from mathematics and physics to engineering and computer graphics. We will analyze the given function to determine its axis of symmetry, y-intercept, direction of opening, and vertex. By examining these key features, we can gain a comprehensive understanding of the parabola's behavior and its graphical representation.

Unveiling the Vertex Form of a Quadratic Equation

To effectively analyze the parabola, we must first recognize that the given function, f(x) = 3(x - 4)² + 5, is presented in vertex form. The vertex form of a quadratic equation is expressed as f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola and 'a' determines the direction and stretch of the parabola. This form is particularly useful because it directly reveals the vertex, which is a critical point for understanding the parabola's overall shape and position in the coordinate plane.

In our case, by comparing f(x) = 3(x - 4)² + 5 with the general vertex form, we can easily identify the values of a, h, and k. We see that a = 3, h = 4, and k = 5. These values provide us with immediate insights into the parabola's characteristics. The vertex, as we'll discuss further, is a key feature that dictates the parabola's symmetry and its minimum or maximum point. Furthermore, the value of 'a' is crucial in determining whether the parabola opens upwards or downwards, and how stretched or compressed it is compared to the basic parabola y = x².

Understanding the vertex form is not just about plugging in values; it's about recognizing the inherent structure of quadratic functions and how each parameter influences the graph. This form allows us to quickly sketch the parabola and understand its behavior without needing to resort to more complex methods like completing the square or using the quadratic formula. It's a powerful tool for anyone studying quadratic functions and their applications.

Axis of Symmetry: The Parabola's Mirror

The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves. It's like a mirror running down the center of the parabola, where the reflection of any point on one side will have a corresponding point on the other side. The equation of the axis of symmetry is always of the form x = h, where h is the x-coordinate of the vertex. This direct relationship between the vertex and the axis of symmetry makes it easy to determine the axis once the vertex is known.

In our function, f(x) = 3(x - 4)² + 5, we identified the vertex as (4, 5). Therefore, the axis of symmetry is the vertical line x = 4. This means that the parabola is perfectly symmetrical around the line x = 4. Any point on the parabola to the left of this line will have a corresponding point at the same height to the right of the line, and vice versa. The axis of symmetry is a fundamental characteristic of parabolas and is essential for understanding their graphical representation.

Understanding the axis of symmetry is not only crucial for sketching the parabola accurately but also for solving various problems related to quadratic functions. For example, if you know one x-intercept of the parabola and the axis of symmetry, you can easily find the other x-intercept. Similarly, the axis of symmetry helps in finding the maximum or minimum value of the quadratic function, as it passes through the vertex, which represents the extreme point of the parabola. In essence, the axis of symmetry provides a framework for understanding the symmetrical nature of parabolas and their behavior.

The Y-intercept: Where the Parabola Crosses the Y-axis

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 and evaluate f(0). This gives us the y-coordinate of the point where the parabola crosses the y-axis. The y-intercept is an important feature of the parabola as it provides another key point for sketching the graph and understanding the function's behavior.

Let's calculate the y-intercept for f(x) = 3(x - 4)² + 5. Substituting x = 0, we get:

f(0) = 3(0 - 4)² + 5 f(0) = 3(-4)² + 5 f(0) = 3(16) + 5 f(0) = 48 + 5 f(0) = 53

Therefore, the y-intercept is the point (0, 53). This indicates that the parabola intersects the y-axis at the point where y is equal to 53. The y-intercept provides valuable information about the parabola's position and its behavior as x approaches zero. It's a useful reference point for sketching the parabola and understanding its relationship with the coordinate axes.

The y-intercept, along with other key features like the vertex and axis of symmetry, helps in building a complete picture of the parabola's graph. It's not just a single point; it's a piece of the puzzle that reveals the overall shape and position of the parabola in the coordinate plane. Understanding how to find and interpret the y-intercept is a fundamental skill in analyzing quadratic functions.

Direction of Opening: Upwards or Downwards?

The direction of opening of a parabola is determined by the coefficient 'a' in the vertex form f(x) = a(x - h)² + k. If a > 0, the parabola opens upwards, resembling a U-shape. This means that the vertex is the minimum point of the parabola. Conversely, if a < 0, the parabola opens downwards, resembling an inverted U-shape, and the vertex is the maximum point. The sign of 'a' is a crucial indicator of the parabola's overall shape and its extreme point.

In our function, f(x) = 3(x - 4)² + 5, the coefficient a is 3, which is a positive number (a = 3 > 0). Therefore, the parabola opens upwards. This tells us that the vertex of the parabola will be the lowest point on the graph, representing the minimum value of the function. The positive value of 'a' confirms that as x moves away from the vertex in either direction, the values of f(x) will increase, creating the characteristic upward-opening shape.

The direction of opening is a fundamental aspect of understanding the parabola's behavior. It not only helps in sketching the graph but also in solving problems involving optimization, where we need to find the minimum or maximum value of a quadratic function. The sign of 'a' provides a quick and easy way to determine whether the parabola has a minimum or maximum point, making it a valuable tool in the analysis of quadratic functions.

The Vertex: The Parabola's Turning Point

The vertex is the point where the parabola changes direction. It is either the minimum point (if the parabola opens upwards) or the maximum point (if the parabola opens downwards). In the vertex form of a quadratic equation, f(x) = a(x - h)² + k, the vertex is represented by the coordinates (h, k). The vertex is a crucial feature of the parabola as it provides a reference point for understanding the parabola's position and its extreme value.

For the given function, f(x) = 3(x - 4)² + 5, we can directly identify the vertex by comparing it with the vertex form. We see that h = 4 and k = 5. Therefore, the vertex of the parabola is the point (4, 5). This means that the parabola's turning point is at the coordinates (4, 5). Since we have already determined that the parabola opens upwards, the vertex (4, 5) represents the minimum point of the function.

The vertex is not just a point; it's a key feature that dictates the parabola's behavior. It's the point around which the parabola is symmetrical, and it represents the extreme value of the function. The vertex, along with the direction of opening, provides a complete picture of the parabola's shape and position in the coordinate plane. Understanding how to identify and interpret the vertex is a fundamental skill in analyzing quadratic functions and their applications.

Evaluating the Given Statements

Now that we have analyzed the properties of the parabola defined by f(x) = 3(x - 4)² + 5, let's evaluate the given statements:

• The axis of symmetry is x = -4: This statement is false. We determined that the axis of symmetry is x = 4, which is a vertical line passing through the x-coordinate of the vertex.
• The y-intercept is 0: This statement is false. We calculated the y-intercept to be (0, 53), meaning the parabola intersects the y-axis at y = 53, not at 0.
• The parabola opens down: This statement is false. Since the coefficient a is positive (a = 3), the parabola opens upwards, not downwards.
• The vertex is (4, 5): This statement is true. We identified the vertex as (4, 5) by comparing the given function with the vertex form of a quadratic equation.

Conclusion: The True Statement About the Parabola

In conclusion, after thoroughly analyzing the properties of the parabola defined by the function f(x) = 3(x - 4)² + 5, we have determined that only one of the given statements is true. The correct statement is: The vertex is (4, 5). This comprehensive analysis demonstrates the importance of understanding the vertex form of a quadratic equation and how it reveals key characteristics of the parabola, such as the axis of symmetry, y-intercept, direction of opening, and the vertex itself. By mastering these concepts, one can effectively analyze and interpret quadratic functions and their graphical representations.