Graphing Quadratic Functions Y=x²-6x+10 A Comprehensive Guide
In the realm of mathematics, quadratic functions hold a significant place, forming the basis for various applications in physics, engineering, and economics. Understanding how to graph these functions is crucial for visualizing their behavior and extracting valuable information. This article delves into the process of graphing the quadratic function y=x²-6x+10, providing a step-by-step guide suitable for students and enthusiasts alike.
Understanding Quadratic Functions
Quadratic functions are polynomial functions of degree two, generally expressed in the form y=ax²+bx+c, where a, b, and c are constants and a≠0. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The parabola's orientation (opening upwards or downwards) depends on the sign of the coefficient 'a'. If a>0, the parabola opens upwards, and if a<0, it opens downwards. The vertex of the parabola represents the minimum or maximum point of the function, depending on the parabola's orientation. Graphing quadratic functions involves identifying key features such as the vertex, axis of symmetry, intercepts, and additional points to accurately sketch the parabola.
The vertex is a crucial point in understanding the behavior of a quadratic function. It represents the extremum of the function, either the minimum value if the parabola opens upwards or the maximum value if it opens downwards. The x-coordinate of the vertex can be found using the formula -b/2a, where a and b are the coefficients in the quadratic equation. Once the x-coordinate is known, the y-coordinate can be found by substituting the x-coordinate back into the original equation. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Its equation is given by x=-b/2a, which is the same as the x-coordinate of the vertex. Understanding the vertex and axis of symmetry is fundamental in sketching the graph of a quadratic function, as they provide a reference point and a line of symmetry for the parabola.
Intercepts play a vital role in accurately graphing quadratic functions. The y-intercept is the point where the parabola intersects the y-axis, and it can be found by setting x=0 in the quadratic equation. This point provides a clear indication of the function's value when the input is zero. The x-intercepts, also known as the roots or zeros of the quadratic function, are the points where the parabola intersects the x-axis. These can be found by setting y=0 in the quadratic equation and solving for x. The x-intercepts represent the values of x for which the function's output is zero, and they offer insights into the function's behavior around the x-axis. Depending on the discriminant (b²-4ac) of the quadratic equation, there may be two distinct x-intercepts, one x-intercept (a repeated root), or no real x-intercepts. Identifying these intercepts helps in accurately positioning and shaping the parabola on the coordinate plane.
Step-by-Step Guide to Graphing y=x²-6x+10
Now, let's apply our understanding to graph the specific quadratic function y=x²-6x+10. This section will provide a detailed, step-by-step approach, ensuring clarity and accuracy in the graphing process.
Step 1: Identify the Coefficients
The first step is to identify the coefficients a, b, and c in the quadratic equation. Comparing y=x²-6x+10 with the general form y=ax²+bx+c, we find:
- a = 1
- b = -6
- c = 10
Identifying these coefficients is crucial as they are used in subsequent steps to determine the vertex, axis of symmetry, and other key features of the parabola.
Step 2: Find the Vertex
The vertex is the most crucial point for graphing a parabola. To find the vertex, we use the formula (-b/2a, f(-b/2a)), where f(x) represents the quadratic function. First, we calculate the x-coordinate of the vertex:
x = -b / 2a = -(-6) / (2 * 1) = 6 / 2 = 3
Now, we substitute x = 3 back into the original equation to find the y-coordinate of the vertex:
y = (3)² - 6(3) + 10 = 9 - 18 + 10 = 1
Therefore, the vertex of the parabola is (3, 1). This point represents the minimum value of the function since the coefficient 'a' is positive (a=1), indicating that the parabola opens upwards.
Step 3: Determine the 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 given by x = -b/2a, which is the same as the x-coordinate of the vertex. In this case, the axis of symmetry is:
x = 3
This vertical line serves as a mirror, reflecting one side of the parabola onto the other, and it is a key element in sketching the graph accurately.
Step 4: Find the y-intercept
The y-intercept is the point where the parabola intersects the y-axis. To find the y-intercept, we set x = 0 in the equation:
y = (0)² - 6(0) + 10 = 10
So, the y-intercept is (0, 10). This point provides us with another reference point on the graph, helping to determine the parabola's position on the coordinate plane.
Step 5: Find the x-intercept(s) (if they exist)
The x-intercepts, also known as the roots or zeros of the quadratic function, are the points where the parabola intersects the x-axis. To find the x-intercepts, we set y = 0 in the equation and solve for x:
0 = x² - 6x + 10
We can use the quadratic formula to solve for x:
x = [-b ± √(b² - 4ac)] / 2a
Plugging in the values a = 1, b = -6, and c = 10, we get:
x = [6 ± √((-6)² - 4 * 1 * 10)] / (2 * 1)
x = [6 ± √(36 - 40)] / 2
x = [6 ± √(-4)] / 2
Since the discriminant (b² - 4ac) is negative (-4), there are no real x-intercepts. This means the parabola does not intersect the x-axis. This information is crucial as it tells us that the entire parabola lies above the x-axis.
Step 6: Find Additional Points
To sketch the parabola more accurately, it's helpful to find additional points. We can choose x-values that are close to the vertex and calculate the corresponding y-values. Given that the vertex is at x=3, let's choose x=2 and x=4:
For x = 2:
y = (2)² - 6(2) + 10 = 4 - 12 + 10 = 2
So, the point (2, 2) is on the parabola.
For x = 4:
y = (4)² - 6(4) + 10 = 16 - 24 + 10 = 2
So, the point (4, 2) is also on the parabola. These additional points, along with the vertex and y-intercept, provide a more detailed outline for sketching the curve.
Step 7: Sketch the Graph
Now that we have all the necessary information, we can sketch the graph of the quadratic function y = x² - 6x + 10. Plot the following points on the coordinate plane:
- Vertex: (3, 1)
- y-intercept: (0, 10)
- Additional points: (2, 2) and (4, 2)
Draw a smooth, symmetrical U-shaped curve (parabola) that passes through these points. The vertex (3, 1) is the lowest point on the graph, and the parabola opens upwards because the coefficient 'a' is positive. The axis of symmetry is the vertical line x = 3. Since there are no x-intercepts, the parabola does not cross the x-axis.
By connecting the points smoothly, you'll obtain a clear and accurate representation of the quadratic function y = x² - 6x + 10. This visual representation allows for a deeper understanding of the function's behavior and characteristics.
Conclusion
Graphing quadratic functions like y=x²-6x+10 involves a systematic approach. By identifying the coefficients, finding the vertex and axis of symmetry, determining the intercepts, and plotting additional points, we can accurately sketch the parabola. This process not only helps visualize the function's behavior but also provides valuable insights into its properties. Mastering the technique of graphing quadratic functions is essential for various mathematical applications and problem-solving scenarios. Remember to practice these steps with different quadratic equations to enhance your understanding and proficiency. Understanding quadratic functions and their graphs is a cornerstone of algebra, and this guide provides a solid foundation for further exploration in mathematics. By following the steps outlined in this article, you can confidently graph quadratic functions and unlock a deeper understanding of their mathematical properties. From finding the vertex to determining intercepts, each step contributes to a comprehensive understanding of the parabola's shape and position on the coordinate plane. Quadratic functions are not just abstract equations; they are powerful tools for modeling real-world phenomena, and the ability to graph them is a valuable skill in various fields. Keep practicing and exploring the world of quadratic functions to enhance your mathematical prowess.
