Graphing F(x) = (1/2)x² + 2x - 6 A Step-by-Step Guide

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Hey guys! Today, we're diving deep into the world of quadratic functions, specifically focusing on how to graph the function f(x) = (1/2)x² + 2x - 6. This might seem intimidating at first, but trust me, we'll break it down step by step so you can confidently tackle any quadratic graph. We'll explore key features like the vertex, axis of symmetry, intercepts, and how the leading coefficient influences the shape of the parabola. Understanding these elements is crucial for accurately plotting the graph and interpreting the function's behavior. So, grab your pencils and paper, and let's get started on this mathematical adventure!

Understanding Quadratic Functions

Before we jump into graphing, let's solidify our understanding of quadratic functions. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (in our case, x) is 2. The general form of a quadratic function is 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. Now, let's relate this back to our specific function, f(x) = (1/2)x² + 2x - 6. Here, a = 1/2, b = 2, and c = -6. The value of a plays a significant role in determining the parabola's shape and direction. A positive a (like our 1/2) indicates that the parabola opens upwards, while a negative a would mean it opens downwards. The larger the absolute value of a, the narrower the parabola. Conversely, a smaller absolute value makes the parabola wider. This understanding of the coefficient 'a' is fundamental in visualizing the graph even before plotting any points. We'll delve deeper into how to use these coefficients to accurately sketch the graph in the following sections. Understanding the general form and the influence of 'a', 'b', and 'c' is the bedrock for mastering quadratic functions.

Finding the Vertex: The Heart of the Parabola

The vertex is arguably the most crucial point when graphing a parabola. It's the turning point of the graph – the minimum point if the parabola opens upwards (like ours) or the maximum point if it opens downwards. The vertex gives us a central reference point around which the rest of the graph is symmetrical. To find the vertex, we first need to determine its x-coordinate. There's a handy formula for this: x = -b / 2a. Remember our function f(x) = (1/2)x² + 2x - 6? We identified a = 1/2 and b = 2. Plugging these values into the formula, we get x = -2 / (2 * (1/2)) = -2 / 1 = -2. So, the x-coordinate of our vertex is -2. To find the y-coordinate, we simply substitute this x value back into our function: f(-2) = (1/2)(-2)² + 2(-2) - 6 = (1/2)(4) - 4 - 6 = 2 - 4 - 6 = -8. Therefore, the vertex of our parabola is at the point (-2, -8). This point is the lowest point on our graph, and it's the cornerstone for drawing the rest of the parabola. Knowing the vertex not only helps in plotting the graph but also provides valuable information about the function's range and minimum value. The vertex is truly the heart of the parabola, and mastering its calculation is key to understanding the function's behavior.

Determining the Axis of Symmetry

The axis of symmetry is an imaginary vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Think of it as a mirror; whatever is on one side of the line is mirrored on the other side. This symmetry makes graphing much easier because once we plot points on one side of the axis, we can easily reflect them to the other side. The equation of the axis of symmetry is simply x = the x-coordinate of the vertex. We already found that the x-coordinate of our vertex is -2. So, the axis of symmetry for f(x) = (1/2)x² + 2x - 6 is the vertical line x = -2. This line acts as a guide for drawing the parabola, ensuring that it's balanced and symmetrical. Any point on the parabola has a corresponding point on the opposite side of the axis of symmetry, at the same y-value. For example, if we find a point at x = -1, which is one unit to the right of the axis of symmetry, there will be a corresponding point at x = -3, one unit to the left of the axis of symmetry, with the same y-value. This symmetrical property significantly reduces the amount of calculation needed to plot the graph accurately. Understanding and utilizing the axis of symmetry is a smart way to simplify the graphing process and gain a deeper understanding of the parabola's characteristics.

Finding the Intercepts: Where the Parabola Meets the Axes

Intercepts are the points where the parabola intersects the x-axis (x-intercepts) and the y-axis (y-intercept). These points provide additional key locations for accurately plotting the graph. The y-intercept is the easiest to find. It's the point where x = 0. So, we substitute x = 0 into our function: f(0) = (1/2)(0)² + 2(0) - 6 = -6. This means the y-intercept is at the point (0, -6). For the x-intercepts, we need to find the values of x where f(x) = 0. This involves solving the quadratic equation (1/2)x² + 2x - 6 = 0. To make things easier, let's multiply the entire equation by 2 to get rid of the fraction: x² + 4x - 12 = 0. Now we can try to factor this quadratic. We're looking for two numbers that multiply to -12 and add up to 4. These numbers are 6 and -2. So, we can factor the equation as (x + 6)(x - 2) = 0. Setting each factor equal to zero gives us the solutions x = -6 and x = 2. Therefore, the x-intercepts are at the points (-6, 0) and (2, 0). By finding both the y-intercept and the x-intercepts, we've pinpointed three crucial points on our parabola. These intercepts, along with the vertex and the axis of symmetry, give us a solid framework for drawing an accurate graph. Remember, intercepts are not just points on the graph; they represent the solutions to the equation f(x) = 0 (x-intercepts) and the value of the function when x = 0 (y-intercept), providing valuable insights into the function's behavior.

Plotting the Graph: Putting it All Together

Now, for the exciting part – plotting the graph! We've gathered all the essential information: the vertex (-2, -8), the axis of symmetry x = -2, the y-intercept (0, -6), and the x-intercepts (-6, 0) and (2, 0). Let's start by drawing the coordinate plane and plotting these points. First, mark the vertex (-2, -8). This is the lowest point of our parabola. Next, draw the axis of symmetry as a dashed vertical line through x = -2. Now, plot the y-intercept (0, -6) and the x-intercepts (-6, 0) and (2, 0). Remember, the parabola is symmetrical about the axis of symmetry. This means for every point on one side of the axis, there's a corresponding point on the other side. We already have the y-intercept (0, -6), which is 2 units to the right of the axis of symmetry. So, there must be a corresponding point 2 units to the left of the axis of symmetry, at (-4, -6). This gives us another point on the parabola. Now, we have enough points to sketch a smooth U-shaped curve. Start at one x-intercept, pass through the plotted points, make sure the vertex is the turning point, and then continue smoothly through the other x-intercept. The parabola should be symmetrical about the axis of symmetry. If the curve looks uneven or asymmetrical, double-check your points and make sure they're plotted correctly. Plotting the graph is the visual culmination of all our hard work. It allows us to see the function's behavior in a clear and intuitive way. The shape, position, and intercepts of the parabola tell us a story about the quadratic function, and mastering the plotting process allows us to read that story with confidence.

Conclusion: Mastering Quadratic Graphs

So, there you have it! We've successfully navigated the process of graphing the quadratic function f(x) = (1/2)x² + 2x - 6. We started by understanding the general form of a quadratic function and the significance of its coefficients. Then, we meticulously found the vertex, the axis of symmetry, and the intercepts. Finally, we put all the pieces together to plot the graph, creating a visual representation of the function's behavior. The key takeaways from this exercise are the importance of the vertex as the central point, the axis of symmetry for ensuring balance, and the intercepts for pinpointing where the parabola crosses the axes. By mastering these elements, you can confidently graph any quadratic function. Remember, practice makes perfect! The more you graph quadratic functions, the more intuitive the process will become. You'll start to recognize patterns and relationships between the equation and the graph, making you a true quadratic graph master. So, keep exploring, keep practicing, and keep graphing! Understanding quadratic functions is a fundamental skill in mathematics, and it opens doors to more advanced concepts. Keep up the great work, guys!