Graphing Y = X^2 - 16x + 48: Vertex, Intercepts Explained

Understanding the key points of a quadratic function's graph, such as the vertex, x-intercepts, and y-intercept, is fundamental in mathematics. These points provide crucial information about the parabola's shape, position, and behavior. Today, we're going to dive deep into the quadratic equation $y = x^2 - 16x + 48$, and unravel its graphical secrets. By the end of this article, you'll be a pro at identifying these essential features, making it easier to sketch and interpret parabolas in any context. So, let's get started on this mathematical journey!

Unveiling the Vertex: The Turning Point of the Parabola

The vertex is arguably the most important point on a parabola. It represents the minimum or maximum value of the quadratic function. For a parabola that opens upwards (like ours, since the coefficient of $x^2$ is positive), the vertex is the lowest point. If the parabola opened downwards, the vertex would be the highest point. Finding the vertex involves a couple of steps, and there are different methods to achieve this. One common approach is to use the formula for the x-coordinate of the vertex, which is given by $-b/(2a)$ for a quadratic equation in the standard form $y = ax^2 + bx + c$. In our equation, $y = x^2 - 16x + 48$, we have $a = 1$, $b = -16$, and $c = 48$. Plugging these values into the formula, we get: $x = -(-16) / (2 * 1) = 16 / 2 = 8$. Once we have the x-coordinate of the vertex, we substitute this value back into the original equation to find the corresponding y-coordinate. So, $y = (8)^2 - 16(8) + 48 = 64 - 128 + 48 = -16$. Therefore, the vertex of the parabola $y = x^2 - 16x + 48$ is at the point (8, -16). This point is critical because it tells us where the function reaches its minimum value and signifies the turning point of the graph. The symmetry of the parabola also revolves around a vertical line passing through the vertex, known as the axis of symmetry. This axis of symmetry is represented by the equation $x = 8$ in our case. Understanding the vertex helps us visualize the entire shape and range of the quadratic function, providing a cornerstone for further analysis. It's the peak or the valley of the parabolic landscape, and knowing its coordinates gives us a significant anchor point.

Pinpointing the X-Intercepts: Where the Parabola Crosses the Horizontal Axis

The x-intercepts, also known as the roots or zeros of the quadratic equation, are the points where the graph of the parabola intersects the x-axis. At these points, the y-coordinate is always zero. To find the x-intercepts for $y = x^2 - 16x + 48$, we set $y = 0$ and solve the resulting quadratic equation: $x^2 - 16x + 48 = 0$. There are several methods to solve quadratic equations: factoring, completing the square, or using the quadratic formula. Factoring is often the quickest if the equation is easily factorable. We look for two numbers that multiply to 48 (the constant term) and add up to -16 (the coefficient of the x term). After some thought, we can identify that -4 and -12 fit these criteria: $(-4) * (-12) = 48$ and $(-4) + (-12) = -16$. So, we can factor the equation as $(x - 4)(x - 12) = 0$. For this product to be zero, at least one of the factors must be zero. Setting each factor to zero gives us: $x - 4 = 0 ightarrow x = 4$ and $x - 12 = 0 ightarrow x = 12$. Thus, the x-intercepts of the parabola $y = x^2 - 16x + 48$ are at the points (4, 0) and (12, 0). These intercepts are vital because they indicate the specific x-values where the function's output is zero. They are crucial for understanding the domain over which the function is positive or negative, and they provide additional anchor points for sketching the graph. If we were to use the quadratic formula, $x = [-b ± \sqrt{(b^2 - 4ac)}] / (2a)$, we would substitute $a=1, b=-16, c=48$ and arrive at the same x-intercepts, confirming our factored results. The symmetry of the parabola is also evident here, as the x-coordinate of the vertex (x=8) is exactly in the middle of the two x-intercepts (4 and 12). This relationship reinforces our understanding of the parabola's balanced structure.

Discovering the Y-Intercept: Where the Parabola Crosses the Vertical Axis

The y-intercept is the point where the graph of the parabola crosses the y-axis. At this point, the x-coordinate is always zero. To find the y-intercept for $y = x^2 - 16x + 48$, we simply substitute $x = 0$ into the equation. Let's do that: $y = (0)^2 - 16(0) + 48 = 0 - 0 + 48 = 48$. Therefore, the y-intercept of the parabola $y = x^2 - 16x + 48$ is at the point (0, 48). The y-intercept is a straightforward point to find and is often one of the first points identified when sketching a graph. It tells us the initial value of the function when the input is zero. In the standard form of a quadratic equation, $y = ax^2 + bx + c$, the y-intercept is always equal to the constant term, $c$. In our equation, $c = 48$, which directly corresponds to our calculated y-intercept of (0, 48). This makes identifying the y-intercept incredibly easy once the equation is in standard form. It's the point where the function begins its journey from the vertical axis, setting the stage for the rest of the parabolic curve. This point, along with the vertex and x-intercepts, provides a comprehensive set of reference points for accurately visualizing the graph of the quadratic function.

Putting It All Together: Sketching the Parabola

Now that we have identified the key points – the vertex at (8, -16), the x-intercepts at (4, 0) and (12, 0), and the y-intercept at (0, 48) – we can begin to sketch the graph of $y = x^2 - 16x + 48$. Since the coefficient of $x^2$ (which is $a=1$) is positive, the parabola opens upwards. We would first plot the vertex (8, -16), which is the lowest point. Then, we plot the x-intercepts (4, 0) and (12, 0) on the x-axis. Finally, we plot the y-intercept (0, 48) on the y-axis. Because parabolas are symmetrical, we can also find another point by reflecting the y-intercept across the axis of symmetry ($x = 8$). The y-intercept is 8 units to the left of the axis of symmetry (0 vs 8). So, its reflection will be 8 units to the right of the axis of symmetry, at $x = 8 + 8 = 16$. The y-value remains the same, so we have an additional point at (16, 48). By connecting these points with a smooth, U-shaped curve, keeping in mind the upward direction and the symmetry, we can accurately represent the graph of the quadratic function. The more points you plot, the more accurate your sketch will be, but these key points provide a solid foundation for understanding the function's behavior and its graphical representation. The interplay between the vertex, intercepts, and axis of symmetry creates a predictable and elegant shape that is characteristic of all quadratic functions.

Conclusion: Mastering Quadratic Graphs

In summary, by systematically finding the vertex, x-intercepts, and y-intercept, we gain a profound understanding of the graphical representation of the quadratic equation $y = x^2 - 16x + 48$. The vertex (8, -16) tells us the minimum point and axis of symmetry. The x-intercepts (4, 0) and (12, 0) show where the graph crosses the x-axis, indicating the roots of the equation. The y-intercept (0, 48) reveals where the graph crosses the y-axis, its starting point on that axis. These points are not just arbitrary coordinates; they are the landmarks that define the parabola's form and position. Mastering the identification of these key features is a critical skill in algebra and calculus, essential for solving a wide range of problems in mathematics, physics, engineering, and economics. The ability to translate an algebraic equation into a visual representation unlocks deeper insights into the relationships between variables and the behavior of functions. For those interested in further exploring the fascinating world of quadratic functions and their properties, I highly recommend visiting **

Desmos Graphing Calculator**. It's an excellent tool for visualizing graphs and experimenting with different functions in real-time, helping to solidify your understanding. Another fantastic resource is Khan Academy's section on quadratic functions, which offers comprehensive lessons and practice exercises.