Graphing F(x) = (x - 3)^2 A Step-by-Step Guide
Understanding quadratic functions and their graphical representations is crucial in mathematics. Quadratic functions, which take the form f(x) = ax² + bx + c, create parabolas when graphed. This article focuses on the specific quadratic function f(x) = (x - 3)², exploring its properties and how to accurately represent it graphically. We will delve into the key features of this function, including its vertex, axis of symmetry, and how transformations affect its graph. By the end of this guide, you will have a clear understanding of how to identify the graph of f(x) = (x - 3)² and how to graph quadratic functions in general.
The function f(x) = (x - 3)² is a transformation of the basic quadratic function f(x) = x². The "- 3" inside the parenthesis causes a horizontal shift. Specifically, it shifts the graph 3 units to the right. This horizontal shift is a critical element in understanding the graph of this particular function. To fully grasp this concept, let's first revisit the basics of quadratic functions and parabolas. A parabola is a U-shaped curve, and its standard form is given by f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. The vertex is the point where the parabola changes direction, either the minimum or maximum point of the curve. In our case, f(x) = (x - 3)², we can see that a = 1, h = 3, and k = 0. This tells us that the vertex of the parabola is at the point (3, 0). The value of 'a' determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0). Since a = 1 in our function, the parabola opens upwards. Furthermore, the axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. For f(x) = (x - 3)², the axis of symmetry is the vertical line x = 3.
Key Features of the Function f(x) = (x - 3)²
To accurately graph f(x) = (x - 3)², we need to identify its key features. These features include the vertex, axis of symmetry, and x and y-intercepts. As previously mentioned, the vertex is the turning point of the parabola. For f(x) = (x - 3)², the vertex is at (3, 0). This is because the function is in the form f(x) = (x - h)² + k, where (h, k) is the vertex. The value of h is 3, and the value of k is 0, hence the vertex (3, 0). The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation for the axis of symmetry is x = h, which in this case is x = 3. This means that the parabola is symmetrical about the vertical line x = 3. To find the x-intercepts, we set f(x) = 0 and solve for x. So, (x - 3)² = 0. Taking the square root of both sides, we get x - 3 = 0, which gives us x = 3. This means there is only one x-intercept, which is at the point (3, 0). This also coincides with the vertex, indicating that the parabola touches the x-axis at its vertex. To find the y-intercept, we set x = 0 and evaluate f(0). So, f(0) = (0 - 3)² = (-3)² = 9. This means the y-intercept is at the point (0, 9). These key features – the vertex at (3, 0), the axis of symmetry at x = 3, the x-intercept at (3, 0), and the y-intercept at (0, 9) – provide a solid foundation for graphing the function. Understanding how these features are derived and how they relate to the equation is crucial for accurately representing the function graphically.
Determining the Vertex and Axis of Symmetry
The vertex and axis of symmetry are fundamental in graphing quadratic functions. In the function f(x) = (x - 3)², the vertex form of a quadratic function, f(x) = a(x - h)² + k, is readily apparent. Here, a = 1, h = 3, and k = 0. The vertex, represented by the coordinates (h, k), is therefore (3, 0). The 'h' value represents the horizontal shift of the parabola from the standard form f(x) = x², while the 'k' value represents the vertical shift. In this case, the graph of f(x) = x² is shifted 3 units to the right and 0 units vertically, resulting in the vertex at (3, 0). The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation for the axis of symmetry is x = h, which in this case is x = 3. This means that the parabola is symmetrical about the vertical line x = 3. The vertex is always located on the axis of symmetry, and it is the point where the parabola changes direction. If 'a' is positive, as it is in this case (a = 1), the parabola opens upwards, and the vertex represents the minimum point of the function. If 'a' were negative, the parabola would open downwards, and the vertex would represent the maximum point. Understanding the relationship between the vertex form of the quadratic function and the graphical representation of the parabola is essential for quickly and accurately sketching the graph. By identifying the vertex and axis of symmetry, we can establish a framework for the parabola's shape and position on the coordinate plane.
Finding the Intercepts
Intercepts are the points where the graph of a function intersects the x-axis and y-axis. These points provide valuable information about the function's behavior and are essential for graphing. To find the x-intercepts, we set f(x) = 0 and solve for x. For the function f(x) = (x - 3)², setting f(x) = 0 gives us (x - 3)² = 0. Taking the square root of both sides, we get x - 3 = 0, which yields x = 3. This means there is only one x-intercept, which is at the point (3, 0). Since the vertex of the parabola is also at (3, 0), this indicates that the parabola touches the x-axis at its vertex. This is a special case where the parabola has only one x-intercept, which occurs when the discriminant (b² - 4ac) of the quadratic equation is equal to zero. To find the y-intercept, we set x = 0 and evaluate f(0). For f(x) = (x - 3)², we have f(0) = (0 - 3)² = (-3)² = 9. This means the y-intercept is at the point (0, 9). The y-intercept is the point where the graph crosses the y-axis, and it provides information about the function's value when x is zero. Having both the x-intercept and the y-intercept allows us to plot these points on the coordinate plane and use them as reference points when sketching the graph of the parabola. These intercepts, along with the vertex and axis of symmetry, provide a comprehensive understanding of the parabola's position and shape.
Graphing f(x) = (x - 3)² Step-by-Step
Graphing a quadratic function like f(x) = (x - 3)² can be done systematically by following a few key steps. First, identify the vertex. As we discussed earlier, the vertex of f(x) = (x - 3)² is (3, 0). Plot this point on the coordinate plane. This point serves as the central point around which the parabola will be drawn. Next, determine the axis of symmetry. For this function, the axis of symmetry is the vertical line x = 3. Imagine this line running through the vertex; it will divide the parabola into two symmetrical halves. Now, find the intercepts. We already found that the x-intercept is (3, 0), which is also the vertex. The y-intercept is (0, 9). Plot the y-intercept on the coordinate plane. To get a better sense of the parabola's shape, it's helpful to find a few additional points. Choose x-values on either side of the vertex and calculate the corresponding f(x) values. For example, let's choose x = 1 and x = 5. When x = 1, f(1) = (1 - 3)² = (-2)² = 4. So, the point (1, 4) is on the graph. When x = 5, f(5) = (5 - 3)² = (2)² = 4. So, the point (5, 4) is also on the graph. Notice that these points are symmetrical about the axis of symmetry. Plot these additional points on the coordinate plane. Finally, sketch the parabola by connecting the points with a smooth, U-shaped curve. The curve should be symmetrical about the axis of symmetry and pass through the vertex, intercepts, and additional points. The resulting graph is a parabola that opens upwards, with its vertex at (3, 0) and symmetric about the line x = 3. By following these steps, you can accurately graph the quadratic function f(x) = (x - 3)² and understand its graphical representation.
Plotting the Vertex and Axis of Symmetry
Plotting the vertex and axis of symmetry is the crucial first step in graphing the quadratic function f(x) = (x - 3)². The vertex, as we determined, is at the point (3, 0). Locate this point on the coordinate plane and mark it clearly. The vertex serves as the anchor point for the parabola, indicating its lowest (or highest) point. The axis of symmetry is the vertical line x = 3. To represent this on the graph, draw a dashed vertical line through the point x = 3. This line divides the coordinate plane into two symmetrical halves and serves as a mirror for the parabola. The parabola will be symmetrical about this line, meaning that for every point on one side of the axis of symmetry, there will be a corresponding point on the other side at the same distance from the axis. Plotting these two elements provides a clear visual framework for the parabola. The vertex gives the location of the turning point, and the axis of symmetry establishes the line around which the parabola is reflected. Together, they form the foundation for the graph's shape and position. Once the vertex and axis of symmetry are plotted, we can proceed to find other key points, such as intercepts, and then sketch the parabola with confidence. This systematic approach ensures accuracy and a clear understanding of the function's graphical representation.
Sketching the Parabola
After plotting the key points, the final step is to sketch the parabola. Remember, a parabola is a smooth, U-shaped curve. Start by drawing a curve that passes through the vertex (3, 0). Since the coefficient 'a' in f(x) = (x - 3)² is positive (a = 1), the parabola opens upwards. This means the curve will extend upwards from the vertex. Use the intercepts and additional points you plotted to guide the shape of the curve. The y-intercept is at (0, 9), so the parabola must pass through this point. The additional points (1, 4) and (5, 4) also help define the parabola's shape. Draw a smooth curve that connects these points, ensuring that the curve is symmetrical about the axis of symmetry (x = 3). The left and right sides of the parabola should mirror each other across the axis of symmetry. As you sketch the parabola, pay attention to the curvature. The curve should be smooth and gradual, not sharp or angular. The parabola should also extend indefinitely upwards, indicating that the function's values increase as x moves away from the vertex in either direction. Once you have sketched the parabola, review it to ensure it accurately represents the function f(x) = (x - 3)². The vertex should be at (3, 0), the axis of symmetry should be x = 3, the y-intercept should be at (0, 9), and the parabola should open upwards. If all these conditions are met, you have successfully graphed the function. This process of plotting key points and then sketching a smooth curve is a fundamental technique in graphing quadratic functions and provides a visual representation of the function's behavior.
Transformations and the Graph
The function f(x) = (x - 3)² is a transformation of the basic quadratic function f(x) = x². Understanding transformations is crucial for quickly and accurately graphing functions. The "- 3" inside the parenthesis represents a horizontal shift. Specifically, it shifts the graph 3 units to the right. This is a fundamental concept in function transformations: replacing x with (x - h) shifts the graph h units to the right, while replacing x with (x + h) shifts the graph h units to the left. In this case, the graph of f(x) = x² is shifted 3 units to the right to obtain the graph of f(x) = (x - 3)². The vertex of f(x) = x² is at (0, 0), and the vertex of f(x) = (x - 3)² is at (3, 0), illustrating this horizontal shift. There are other types of transformations as well, such as vertical shifts, stretches, and reflections. A vertical shift occurs when a constant is added or subtracted outside the function. For example, f(x) = (x - 3)² + 2 would shift the graph 2 units upwards. Stretches and compressions occur when the function is multiplied by a constant. For example, f(x) = 2(x - 3)² would vertically stretch the graph, making it narrower, while f(x) = 0.5(x - 3)² would vertically compress the graph, making it wider. Reflections occur when the function is multiplied by -1. For example, f(x) = -(x - 3)² would reflect the graph across the x-axis, causing the parabola to open downwards. By understanding these transformations, you can quickly visualize and graph a wide variety of quadratic functions without having to plot numerous points. The ability to recognize and apply transformations is a powerful tool in mathematics and is essential for understanding the behavior of functions.
Horizontal Shifts
Horizontal shifts are a fundamental type of transformation that affects the graph of a function. In the case of f(x) = (x - 3)², the "- 3" inside the parenthesis causes a horizontal shift. To understand this, it's important to remember the general rule: replacing x with (x - h) shifts the graph h units to the right, and replacing x with (x + h) shifts the graph h units to the left. In our function, h = 3, so the graph of f(x) = x² is shifted 3 units to the right. This means that every point on the original graph is moved 3 units to the right on the new graph. For example, the vertex of f(x) = x² is at (0, 0), and after the horizontal shift, the vertex of f(x) = (x - 3)² is at (3, 0). The entire parabola is translated 3 units to the right along the x-axis. Horizontal shifts do not change the shape or orientation of the parabola; they simply move it to a different location on the coordinate plane. The axis of symmetry also shifts along with the vertex. In f(x) = x², the axis of symmetry is the y-axis (x = 0), and in f(x) = (x - 3)², the axis of symmetry is the vertical line x = 3. Understanding horizontal shifts is essential for graphing quadratic functions and other types of functions as well. By recognizing the pattern of replacing x with (x - h) or (x + h), you can quickly determine the direction and magnitude of the shift and accurately graph the transformed function. This knowledge is a valuable tool in mathematical analysis and problem-solving.
Understanding the Effect of the "- 3"
The presence of the "- 3" inside the parenthesis in f(x) = (x - 3)² has a specific and predictable effect on the graph of the function. This "- 3" causes a horizontal shift, moving the entire graph 3 units to the right compared to the graph of the basic quadratic function f(x) = x². To understand why this happens, consider what the function is doing. It's taking the input x, subtracting 3 from it, and then squaring the result. This means that to get the same y-value as f(x) = x², you need to use an x-value that is 3 units larger. For example, in f(x) = x², to get a y-value of 0, you need x = 0. In f(x) = (x - 3)², to get a y-value of 0, you need (x - 3) = 0, which means x = 3. This shows that the entire graph is shifted 3 units to the right. The "- 3" effectively delays the function's response, requiring a larger x-value to achieve the same output. This horizontal shift is a fundamental transformation in function graphing. It allows us to manipulate the position of a graph without changing its shape or orientation. The "- 3" is not just a random number; it's a parameter that directly controls the horizontal translation of the parabola. By understanding this effect, you can quickly visualize and graph functions of this form, making it a valuable skill in mathematical analysis.
Conclusion
In conclusion, graphing the quadratic function f(x) = (x - 3)² involves understanding its key features, including the vertex, axis of symmetry, intercepts, and transformations. The vertex (3, 0) and axis of symmetry x = 3 provide the foundational structure for the parabola. The intercepts, particularly the y-intercept at (0, 9), offer additional points to guide the sketch. The horizontal shift of 3 units to the right, caused by the "- 3" inside the parenthesis, is a crucial transformation that positions the parabola on the coordinate plane. By systematically identifying these elements and plotting them, we can accurately graph the function and gain a visual understanding of its behavior. The ability to graph quadratic functions is a fundamental skill in mathematics, with applications in various fields, including physics, engineering, and economics. This guide has provided a comprehensive approach to graphing f(x) = (x - 3)², emphasizing the importance of understanding the underlying concepts and techniques. By mastering these skills, you can confidently tackle more complex graphing problems and deepen your understanding of mathematical functions.
