Graphing H(x)=(x-1)^2-9 Plot X-intercept, Y-intercept, Vertex, And Axis Of Symmetry

Jul 28, 2025 by ADMIN 84 views

Graphing Quadratic Functions: A Step-by-Step Guide to Plotting Key Features

Hey guys! Today, we're diving into the exciting world of graphing quadratic functions. Specifically, we'll be focusing on how to plot the x-intercept(s), y-intercept, vertex, and axis of symmetry of a quadratic function. These key features provide a comprehensive understanding of the parabola's shape and position on the coordinate plane. Let's take the function h(x) = (x - 1)² - 9 as our example and walk through the process step by step.

Understanding Quadratic Functions and Their Forms

Before we jump into plotting, let's quickly recap what quadratic functions are and the forms they can take. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (usually 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 ≠ 0. However, our function h(x) = (x - 1)² - 9 is presented in vertex form, which is incredibly useful for identifying the vertex and axis of symmetry directly. The vertex form of a quadratic function is:

f(x) = a(x - h)² + k

where (h, k) represents the vertex of the parabola. Recognizing the vertex form is a game-changer because it simplifies the process of plotting these key features. In our case, h(x) = (x - 1)² - 9 is already in vertex form, making our task much easier. By comparing it to the general vertex form, we can quickly identify that a = 1, h = 1, and k = -9. This immediately tells us that the vertex of our parabola is at the point (1, -9). Remember, the vertex is the turning point of the parabola, and it's a crucial point to plot. It's the minimum value if the parabola opens upwards (a > 0) or the maximum value if the parabola opens downwards (a < 0). Since a = 1 in our function, which is positive, we know our parabola opens upwards and the vertex is the minimum point.

Understanding the forms of quadratic functions is your first step in mastering graphing. The vertex form, in particular, provides a direct route to the vertex, which is a cornerstone for plotting the entire parabola. So, keep this form in mind, and you'll be well-equipped to tackle any quadratic function graphing challenge!

Finding the X-Intercept(s): Where the Parabola Crosses the X-Axis

Now, let's dive into finding the x-intercept(s), also known as the roots or zeros, of our function h(x) = (x - 1)² - 9. These are the points where the parabola intersects the x-axis, meaning the y-value (or h(x) in this case) is zero. To find the x-intercept(s), we set h(x) equal to zero and solve for x:

(x - 1)² - 9 = 0

There are a couple of ways we can solve this equation. One method is to expand the squared term, simplify, and then use the quadratic formula. However, a more efficient approach in this case is to use the square root property. First, we isolate the squared term:

(x - 1)² = 9

Next, we take the square root of both sides. Remember, when taking the square root, we need to consider both the positive and negative roots:

√(x - 1)² = ±√9

This simplifies to:

x - 1 = ±3

Now we have two separate equations to solve:

x - 1 = 3
x - 1 = -3

Solving the first equation, we add 1 to both sides:

x = 3 + 1

x = 4

Solving the second equation, we also add 1 to both sides:

x = -3 + 1

x = -2

So, we have found two x-intercepts: x = 4 and x = -2. These correspond to the points (4, 0) and (-2, 0) on the graph. Plotting these points is crucial as they give us a sense of the parabola's spread and position relative to the x-axis. Understanding how to find x-intercepts is a fundamental skill in graphing quadratic functions. It not only helps in sketching the parabola accurately but also provides valuable insights into the function's behavior and solutions. So, mastering this technique will significantly enhance your ability to analyze and graph quadratic functions.

Determining the Y-Intercept: Where the Parabola Meets the Y-Axis

Next up, let's find the y-intercept of the function h(x) = (x - 1)² - 9. The y-intercept is the point where the parabola intersects the y-axis. This occurs when x is equal to zero. So, to find the y-intercept, we simply substitute x = 0 into our function:

h(0) = (0 - 1)² - 9

Now, let's simplify:

h(0) = (-1)² - 9

h(0) = 1 - 9

h(0) = -8

Therefore, the y-intercept is -8. This corresponds to the point (0, -8) on the graph. Plotting the y-intercept gives us another key reference point for sketching the parabola. It tells us where the parabola crosses the vertical axis, which helps in understanding the parabola's vertical positioning. The y-intercept, along with the x-intercepts and vertex, provides a comprehensive set of points to guide our graph. It's a straightforward calculation, but it adds valuable information to our understanding of the function's behavior. So, remember to always find the y-intercept as part of your graphing process. It's a quick step that significantly enhances the accuracy of your parabola sketch.

Locating the Vertex: The Turning Point of the Parabola

Now, let's talk about the vertex of the parabola represented by the function h(x) = (x - 1)² - 9. As we discussed earlier, our function is in vertex form:

f(x) = a(x - h)² + k

where (h, k) is the vertex. By comparing our function to the vertex form, we can easily identify the coordinates of the vertex. In our case:

h(x) = (x - 1)² - 9

We can see that h = 1 and k = -9. Therefore, the vertex of our parabola is at the point (1, -9). The vertex is a crucial point because it's the turning point of the parabola. It's either the minimum value (if the parabola opens upwards, like ours does since a = 1 is positive) or the maximum value (if the parabola opens downwards). Plotting the vertex is essential as it forms the central point around which the parabola is symmetrical. It anchors the graph and helps us visualize the overall shape and position of the parabola. Furthermore, the vertex provides valuable information about the range of the function. Since our parabola opens upwards and the vertex is at (1, -9), we know that the minimum value of h(x) is -9, and the range of the function is all real numbers greater than or equal to -9. Understanding the vertex is fundamental to graphing quadratic functions. It's a direct link to the function's key characteristics and provides a solid foundation for sketching an accurate representation of the parabola.

Defining the Axis of Symmetry: The Parabola's Mirror Line

Finally, let's identify the axis of symmetry for our function h(x) = (x - 1)² - 9. The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. It's like a mirror line; if you were to fold the parabola along this line, the two halves would perfectly overlap. The equation of the axis of symmetry is always in the form x = h, where h is the x-coordinate of the vertex. Since we've already determined that the vertex of our parabola is at (1, -9), the x-coordinate of the vertex is 1. Therefore, the equation of the axis of symmetry is:

x = 1

This is a vertical line that passes through x = 1 on the coordinate plane. Drawing the axis of symmetry on our graph is incredibly helpful. It not only visually represents the symmetry of the parabola but also aids in plotting additional points. For example, if we have a point on the parabola at a certain distance to the left of the axis of symmetry, we know there's a corresponding point at the same distance to the right. The axis of symmetry simplifies the graphing process and reinforces our understanding of the parabola's symmetry. It's a key feature that ties together the vertex and the overall shape of the quadratic function. So, remember to always identify and draw the axis of symmetry when graphing parabolas. It's a powerful tool for accuracy and visualization.

Putting It All Together: Plotting the Graph

Okay, guys, we've done all the groundwork! We've found the x-intercepts, the y-intercept, the vertex, and the axis of symmetry for the function h(x) = (x - 1)² - 9. Now, it's time to put all these pieces together and plot the graph. Let's recap our findings:

X-intercepts: (4, 0) and (-2, 0)
Y-intercept: (0, -8)
Vertex: (1, -9)
Axis of symmetry: x = 1

First, we'll draw our coordinate plane with the x and y axes. Then, we'll plot the points we've identified: the two x-intercepts at (4, 0) and (-2, 0), the y-intercept at (0, -8), and the vertex at (1, -9). Next, we'll draw the axis of symmetry, which is the vertical line x = 1. This line should pass directly through the vertex. Now, we can start sketching the parabola. Remember, the parabola is symmetrical about the axis of symmetry. It opens upwards since the coefficient of the (x - 1)² term is positive (a = 1). Starting from the vertex, we'll draw a smooth curve that passes through the plotted points, making sure the curve is symmetrical about the axis of symmetry. The x-intercepts give us the points where the curve crosses the x-axis, and the y-intercept shows where it crosses the y-axis. The vertex is the lowest point on the graph, and the axis of symmetry provides the mirror image. As we sketch, we can also plot a couple of extra points to ensure accuracy. For instance, we can choose an x-value, plug it into the function, and find the corresponding y-value. Then, we can use the symmetry to plot the mirror image of that point on the other side of the axis of symmetry. By connecting all these points with a smooth, symmetrical curve, we'll have a well-defined graph of the quadratic function h(x) = (x - 1)² - 9. This step-by-step process, combining the key features we've calculated, allows us to accurately visualize the parabola and understand its behavior. So, go ahead and plot those points, draw the axis of symmetry, and sketch that parabola! You've got this!

Conclusion: Mastering Quadratic Function Graphing

Alright, guys, we've reached the end of our journey through graphing quadratic functions! We've covered all the essential steps, from understanding the different forms of quadratic equations to plotting the x-intercepts, y-intercept, vertex, and axis of symmetry. We've seen how each of these elements contributes to the overall shape and position of the parabola. By taking the function h(x) = (x - 1)² - 9 as our example, we've walked through a practical application of these concepts. We started by recognizing the vertex form of the equation, which immediately gave us the vertex coordinates. Then, we found the x-intercepts by setting h(x) to zero and solving for x, the y-intercept by setting x to zero and calculating h(0), and the axis of symmetry using the x-coordinate of the vertex. Finally, we combined all this information to sketch the graph, ensuring it was symmetrical and accurately represented the function. The key takeaway here is that graphing quadratic functions is a systematic process. By breaking it down into smaller steps and understanding the significance of each key feature, you can confidently tackle any quadratic function graphing challenge. Remember, practice makes perfect! The more you work with different quadratic functions, the more comfortable you'll become with the process. So, keep exploring, keep graphing, and keep honing your skills. You've got the tools and the knowledge to master this important concept in mathematics. Happy graphing!