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

Understanding quadratic equations is a fundamental concept in mathematics, and a crucial aspect of this understanding lies in the ability to analyze and interpret their graphs. In this article, we will delve into the specifics of graphing the quadratic equation y = x² - 16x + 48. Our primary focus will be on identifying key features of the graph, specifically the vertex, x-intercepts, and y-intercept. These points provide critical information about the behavior and position of the parabola, the characteristic U-shaped curve formed by quadratic equations. By meticulously examining each of these features, we can gain a comprehensive understanding of the graph's properties and its relationship to the equation itself.

Understanding Quadratic Equations and Their Graphs

Before diving into the specifics of y = x² - 16x + 48, let's briefly review the general form of a quadratic equation and the characteristics of its graph. A quadratic equation is typically expressed in the form y = ax² + bx + c, where 'a', 'b', and 'c' are constants and 'a' is not equal to zero. The graph of a quadratic equation is a parabola, which can open upwards (if 'a' is positive) or downwards (if 'a' is negative). The vertex of the parabola is the point where the curve changes direction – it's the minimum point if the parabola opens upwards and the maximum point if it opens downwards. Intercepts are the points where the parabola intersects the x-axis (x-intercepts) and the y-axis (y-intercept). These intercepts, along with the vertex, are key features that help us sketch and analyze the graph of a quadratic equation.

In the context of our equation, y = x² - 16x + 48, we can see that a = 1, b = -16, and c = 48. Since 'a' is positive, we know that the parabola will open upwards, indicating that it has a minimum point (the vertex). Now, let's move on to finding the specific coordinates of the vertex, x-intercepts, and y-intercept for this equation.

Finding the Vertex

The vertex of a parabola is a crucial point, representing either the minimum or maximum value of the quadratic function. For the equation y = x² - 16x + 48, the vertex can be found using a couple of different methods. One common method involves completing the square, which transforms the equation into vertex form. Another method utilizes the formula for the x-coordinate of the vertex, which is given by x = -b / 2a. Once we find the x-coordinate, we can substitute it back into the original equation to find the corresponding y-coordinate.

Let's use the formula x = -b / 2a to find the x-coordinate of the vertex. In our equation, a = 1 and b = -16, so we have:

x = -(-16) / (2 * 1) = 16 / 2 = 8

Now that we have the x-coordinate, which is 8, we can substitute it back into the equation to find the y-coordinate:

y = (8)² - 16(8) + 48 = 64 - 128 + 48 = -16

Therefore, the vertex of the parabola is at the point (8, -16). This point represents the minimum value of the function, as the parabola opens upwards. The vertex is a significant point as it helps define the symmetry and overall position of the parabola on the coordinate plane. Knowing the vertex allows us to better visualize the graph and understand the function's behavior.

Determining the x-intercepts

The x-intercepts of a graph are the points where the graph intersects the x-axis. At these points, the y-coordinate is always zero. To find the x-intercepts for the equation y = x² - 16x + 48, we need to set y equal to zero and solve the resulting quadratic equation:

0 = x² - 16x + 48

This quadratic equation can be solved using several methods, including factoring, completing the square, or using the quadratic formula. In this case, factoring is a straightforward approach. We need to find two numbers that multiply to 48 and add up to -16. These numbers are -4 and -12. Therefore, we can factor the equation as follows:

0 = (x - 4)(x - 12)

Setting each factor equal to zero gives us the solutions:

x - 4 = 0 => x = 4 x - 12 = 0 => x = 12

Thus, the x-intercepts are at the points (4, 0) and (12, 0). These points are where the parabola crosses the x-axis. The x-intercepts provide valuable information about the roots or solutions of the quadratic equation, which are the x-values that make the equation equal to zero. Understanding the x-intercepts is crucial for analyzing the intervals where the function is positive or negative.

Identifying the y-intercept

The y-intercept is the point where the graph intersects the y-axis. At this point, the x-coordinate is always zero. To find the y-intercept for the equation y = x² - 16x + 48, we simply substitute x = 0 into the equation:

y = (0)² - 16(0) + 48 y = 0 - 0 + 48 y = 48

Therefore, the y-intercept is at the point (0, 48). This is the point where the parabola crosses the y-axis. The y-intercept is particularly easy to find as it corresponds directly to the constant term in the quadratic equation (the 'c' value in the general form y = ax² + bx + c). In our case, the constant term is 48, which is the y-coordinate of the y-intercept.

Putting It All Together: Graphing the Parabola

Now that we have found the vertex, x-intercepts, and y-intercept, we can use this information to sketch the graph of the parabola y = x² - 16x + 48. We know the following:

• Vertex: (8, -16)
• x-intercepts: (4, 0) and (12, 0)
• y-intercept: (0, 48)

To sketch the graph, start by plotting these key points on the coordinate plane. The vertex (8, -16) is the lowest point on the parabola, as it opens upwards. The x-intercepts (4, 0) and (12, 0) are where the parabola crosses the x-axis, and the y-intercept (0, 48) is where it crosses the y-axis. Draw a smooth, U-shaped curve that passes through these points, ensuring that the parabola is symmetrical about the vertical line passing through the vertex (the axis of symmetry).

The shape and position of the parabola are now clearly defined by these points. The vertex indicates the minimum value of the function, and the intercepts show where the function's value is zero (x-intercepts) and the value when x is zero (y-intercept). This graphical representation provides a visual understanding of the equation y = x² - 16x + 48 and its behavior.

Significance of Key Features

The vertex, x-intercepts, and y-intercept are not just points on a graph; they hold significant meaning in the context of the quadratic equation and the function it represents. The vertex provides the minimum or maximum value of the function, which can be critical in various applications, such as optimization problems. For instance, if this equation represented the profit of a business, the vertex would indicate the point at which the profit is minimized (or maximized, if the parabola opened downwards).

The x-intercepts are the solutions or roots of the quadratic equation. They represent the values of x for which the function equals zero. These points are crucial in solving real-world problems, such as finding the time when a projectile hits the ground (assuming the equation represents the projectile's trajectory). The x-intercepts can also help determine the intervals where the function is positive or negative.

The y-intercept is the value of the function when x is zero. It represents the initial value in many practical situations. For example, if the equation represented the population growth of a species, the y-intercept would represent the initial population size. The y-intercept provides a starting point for understanding the behavior of the function.

Conclusion

In summary, analyzing the key features of the graph of y = x² - 16x + 48—the vertex, x-intercepts, and y-intercept—provides a comprehensive understanding of the quadratic equation and its graphical representation. By finding the vertex at (8, -16), the x-intercepts at (4, 0) and (12, 0), and the y-intercept at (0, 48), we can accurately sketch the parabola and interpret its behavior. These features are essential for understanding the roots, minimum/maximum values, and initial conditions represented by the equation.

Understanding these concepts is foundational in mathematics and has wide-ranging applications in various fields, including physics, engineering, economics, and computer science. The ability to analyze and interpret graphs of quadratic equations is a valuable skill that enhances problem-solving capabilities and provides a deeper insight into mathematical relationships.

For more in-depth information on quadratic functions and their graphs, consider exploring resources like Khan Academy's Algebra I course.