Graphical Analysis Of F(x) = -x² + 3x + 1 A Comprehensive Study

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Hey guys! Let's dive deep into understanding the function f(x) = -x² + 3x + 1 by performing a comprehensive graphical analysis. This means we'll explore its key features, from its overall shape to specific points and behaviors. Think of it as becoming detectives, uncovering all the secrets hidden within this equation. We will focus on identifying crucial elements such as intercepts, vertex, axis of symmetry, and the range, painting a complete picture of this quadratic function. Understanding these features will not only help you visualize the function but also provide a solid foundation for more advanced mathematical concepts. So, grab your thinking caps, and let's embark on this exciting mathematical journey!

Understanding Quadratic Functions

Before we jump into the specifics of our function, let's take a moment to review the basics of quadratic functions. A quadratic function is a polynomial function of degree two, generally represented in the form f(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 U-shaped curve that opens either upwards or downwards. The coefficient 'a' plays a crucial role in determining the parabola's shape and direction. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. This gives us an immediate visual clue about the function's behavior.

In our case, f(x) = -x² + 3x + 1, 'a' is -1, which tells us that the parabola will open downwards. This initial observation sets the stage for further analysis. Understanding the general form of quadratic functions and the significance of the coefficients is essential for interpreting their graphs effectively. Moreover, the vertex of the parabola represents either the maximum or minimum value of the function, a critical point to identify. The axis of symmetry, a vertical line passing through the vertex, divides the parabola into two symmetrical halves, simplifying our analysis further. So, with these foundational concepts in mind, let's proceed to analyze our specific function, f(x) = -x² + 3x + 1, in detail.

Key Features of f(x) = -x² + 3x + 1

Okay, let's break down the key features of our function, f(x) = -x² + 3x + 1, step by step. This will give us a solid understanding of its behavior and how it looks on a graph. We're going to look at the intercepts, the vertex, the axis of symmetry, and the range. Each of these elements provides valuable insights into the function's characteristics and its graphical representation. Think of them as puzzle pieces that, when put together, create the complete picture of the parabola.

1. Intercepts

  • Y-intercept: The y-intercept is where the parabola intersects the y-axis. This happens when x = 0. So, let's plug in x = 0 into our function: f(0) = -(0)² + 3(0) + 1 = 1. Therefore, the y-intercept is (0, 1). This point is relatively easy to find and gives us a starting point for sketching the graph. The y-intercept is a direct reflection of the constant term 'c' in the quadratic function, making it a convenient feature to identify.
  • X-intercepts: The x-intercepts are where the parabola intersects the x-axis. This happens when f(x) = 0. So, we need to solve the equation -x² + 3x + 1 = 0. Since this quadratic equation doesn't factor easily, we'll use the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). In our case, a = -1, b = 3, and c = 1. Plugging these values into the formula, we get: x = [-3 ± √(3² - 4(-1)(1))] / (2(-1)) = [-3 ± √(13)] / (-2). This gives us two x-intercepts: x₁ ≈ 3.30 and x₂ ≈ -0.30. These x-intercepts, often called roots or zeros of the function, are crucial for understanding where the parabola crosses the x-axis. They provide a sense of the function's distribution and symmetry.

2. Vertex

The vertex is the highest or lowest point on the parabola. Since our parabola opens downwards (because a = -1), the vertex will be the maximum point. The x-coordinate of the vertex can be found using the formula: xᵥ = -b / (2a). In our case, xᵥ = -3 / (2(-1)) = 1.5. Now, to find the y-coordinate of the vertex, we plug xᵥ back into our function: f(1.5) = -(1.5)² + 3(1.5) + 1 = 3.25. So, the vertex is (1.5, 3.25). The vertex is a critical point as it represents the turning point of the parabola. It helps define the function's maximum or minimum value and its overall shape.

3. Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is x = xᵥ. In our case, the axis of symmetry is x = 1.5. The axis of symmetry simplifies the graphing process by providing a line of reflection. It ensures that for every point on one side of the parabola, there is a corresponding point on the other side, making the graph symmetrical.

4. Range

The range is the set of all possible y-values that the function can take. Since our parabola opens downwards and has a maximum point at the vertex (1.5, 3.25), the range is y ≤ 3.25. In interval notation, this is (-∞, 3.25]. Understanding the range is essential for defining the function's output values. It tells us the limits of the function's vertical spread and helps us interpret its behavior.

Graphing f(x) = -x² + 3x + 1

Alright, now that we've identified all the key features, let's put them together to sketch the graph of f(x) = -x² + 3x + 1. This is where the magic happens – we'll visually represent the function and see how all the pieces fit together. Remember, graphing is a powerful tool for understanding functions, providing a visual representation of their behavior. By plotting the key points and understanding the overall shape, we can create an accurate representation of the parabola.

  1. Plot the Intercepts: We found the y-intercept to be (0, 1) and the x-intercepts to be approximately (3.30, 0) and (-0.30, 0). Mark these points on your graph. These intercepts give us a framework for the parabola's position relative to the axes. They act as anchors, guiding us in drawing the curve accurately.
  2. Plot the Vertex: The vertex is (1.5, 3.25). Plot this point – it's the highest point on our parabola. The vertex is the most crucial point, representing the peak or trough of the function. Its position determines the overall shape and orientation of the parabola.
  3. Draw the Axis of Symmetry: Draw a vertical dashed line at x = 1.5. This line helps us keep the parabola symmetrical. The axis of symmetry ensures that the graph is balanced, with each side mirroring the other. It's a visual aid for maintaining accuracy and symmetry.
  4. Sketch the Parabola: Now, sketch a smooth curve that passes through the intercepts and the vertex, opening downwards. Remember, the parabola is symmetrical about the axis of symmetry. Connect the points smoothly, creating the characteristic U-shape of a parabola. Pay attention to the curvature, ensuring it's smooth and symmetrical. The resulting graph visually represents the function, allowing us to observe its behavior and characteristics directly.

By following these steps, you'll have a clear visual representation of the function f(x) = -x² + 3x + 1. The graph not only confirms our calculations but also provides a deeper understanding of the function's behavior. We can see how the intercepts, vertex, and axis of symmetry work together to define the parabola's shape and position.

Analyzing the Graph

With the graph in front of us, we can now analyze the function's behavior more intuitively. We can observe the intervals where the function is increasing or decreasing, identify the maximum value, and understand the symmetry of the parabola. Graphical analysis complements algebraic calculations, providing a holistic understanding of the function.

  • Increasing and Decreasing Intervals: The function is increasing from (-∞, 1.5) and decreasing from (1.5, ∞). This is evident from the graph, where the parabola rises to the vertex and then falls. Identifying these intervals helps us understand the function's trend, showing where it's climbing and where it's descending.
  • Maximum Value: The maximum value of the function is 3.25, which occurs at the vertex (1.5, 3.25). This is the highest point the function reaches. The maximum value is a key feature, representing the upper limit of the function's output. It's directly linked to the vertex and provides a significant insight into the function's behavior.
  • Symmetry: The parabola is symmetrical about the line x = 1.5. This symmetry is a fundamental property of quadratic functions, making them visually appealing and mathematically elegant. Symmetry simplifies analysis, allowing us to predict behavior on one side of the graph based on the other.

Conclusion

So there you have it, guys! We've conducted a thorough graphical analysis of the function f(x) = -x² + 3x + 1. We've identified its key features, sketched its graph, and interpreted its behavior. By understanding the intercepts, vertex, axis of symmetry, and range, we've gained a comprehensive understanding of this quadratic function. Remember, graphical analysis is not just about plotting points; it's about understanding the story the function tells. It's about connecting the equation to its visual representation, gaining insights that would be harder to see from the equation alone. Keep practicing, and you'll become graphing pros in no time!

By understanding these elements, we not only grasp the visual representation but also gain a deeper insight into the function's behavior and characteristics. This comprehensive analysis equips us with the tools to tackle more complex functions and mathematical problems. So, keep exploring, keep graphing, and keep discovering the fascinating world of mathematics!