Graph Of F(x) = X² - 2x + 1: Explained!

Jan 14, 2026 by Tim Redaksi 40 views

Alright guys, let's break down the graph of the function f(x) = x² - 2x + 1. This is a quadratic function, and understanding its graph is super important in algebra and calculus. We're going to cover everything from the basic shape to key features like the vertex, axis of symmetry, and how to plot it accurately. So, grab your pencils, and let’s get started!

Understanding the Basic Shape

First off, recognizing that f(x) = x² - 2x + 1 is a quadratic function is crucial. Quadratic functions always have a U-shaped graph, which we call a parabola. The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants. In our case, a = 1, b = -2, and c = 1. Since a is positive (1 > 0), the parabola opens upwards. If a were negative, the parabola would open downwards.

Knowing that the parabola opens upwards tells us a lot. It means the function has a minimum value, which occurs at the vertex of the parabola. The vertex is the turning point of the graph. Think of it as the lowest point on the U-shape. Understanding the direction of the parabola helps us predict the overall behavior of the function. As x moves away from the vertex in either direction, the f(x) values will increase, heading towards positive infinity.

Moreover, the coefficient a also affects the 'width' of the parabola. If |a| is large, the parabola is narrower, and if |a| is small, the parabola is wider. In our case, a = 1, which means the parabola has a standard width, not stretched or compressed too much. This basic understanding sets the stage for finding more specific details about the graph.

Finding the Vertex

The vertex of the parabola is a key feature we need to find. The vertex is the point where the parabola changes direction. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex, often denoted as h, can be found using the formula: h = -b / (2a). Once we have h, we can find the y-coordinate of the vertex, k, by plugging h back into the function: k = f(h). The vertex is then the point (h, k).

Let's apply this to our function f(x) = x² - 2x + 1. Here, a = 1 and b = -2. So, the x-coordinate of the vertex is:

h = -(-2) / (2 * 1) = 2 / 2 = 1

Now, let's find the y-coordinate by plugging h = 1 into the function:

k = f(1) = (1)² - 2(1) + 1 = 1 - 2 + 1 = 0

So, the vertex of the parabola is (1, 0). This is a crucial point because it tells us the minimum value of the function is 0, and it occurs when x = 1. The vertex also helps us understand the symmetry of the parabola.

Understanding how to find the vertex is super useful. It not only gives us a specific point on the graph but also helps us visualize the overall shape and position of the parabola on the coordinate plane. Knowing the vertex is often the first step in sketching an accurate graph of a quadratic function.

Determining the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex. In our case, the vertex is (1, 0), so the axis of symmetry is the vertical line x = 1.

The axis of symmetry is like a mirror for the parabola. If you were to fold the graph along this line, the two halves would perfectly overlap. This symmetry is a fundamental property of quadratic functions and can help us plot the graph more easily. For example, if we know a point on one side of the axis of symmetry, we can easily find a corresponding point on the other side.

To illustrate, let’s consider a point on the graph. If we take x = 0, which is one unit to the left of the axis of symmetry (x = 1), we have:

f(0) = (0)² - 2(0) + 1 = 1

So, the point (0, 1) is on the graph. Due to the symmetry, there must be a corresponding point one unit to the right of the axis of symmetry, which is x = 2. Let’s verify:

f(2) = (2)² - 2(2) + 1 = 4 - 4 + 1 = 1

Indeed, the point (2, 1) is also on the graph. This demonstrates how the axis of symmetry helps us find symmetrical points and sketch the parabola more accurately.

Plotting Additional Points

While the vertex and axis of symmetry give us a good start, plotting additional points can help us refine our graph. To do this, we simply choose some values of x and calculate the corresponding values of f(x). It’s often useful to pick points on both sides of the vertex to take advantage of the symmetry.

We already know the vertex is (1, 0) and we’ve found that (0, 1) and (2, 1) are on the graph. Let’s pick a couple more points. How about x = -1 and x = 3?

For x = -1:

f(-1) = (-1)² - 2(-1) + 1 = 1 + 2 + 1 = 4

So, the point (-1, 4) is on the graph.

For x = 3:

f(3) = (3)² - 2(3) + 1 = 9 - 6 + 1 = 4

So, the point (3, 4) is on the graph. Notice that these points are also symmetrical with respect to the axis of symmetry. Now we have the points (-1, 4), (0, 1), (1, 0), (2, 1), and (3, 4).

With these points, we can sketch a more accurate graph of the function. Remember to draw a smooth, U-shaped curve through the points, reflecting the symmetrical nature of the parabola. The more points you plot, the more precise your graph will be.

Factoring and Finding Roots

Another important aspect of understanding the graph of f(x) = x² - 2x + 1 is to find its roots, also known as x-intercepts. The roots are the values of x for which f(x) = 0. In other words, they are the points where the parabola intersects the x-axis.

To find the roots, we need to solve the equation x² - 2x + 1 = 0. Notice that the quadratic expression can be factored: x² - 2x + 1 = (x - 1)(x - 1) = (x - 1)². Setting this equal to zero gives us:

(x - 1)² = 0

Taking the square root of both sides, we get:

x - 1 = 0

So, x = 1. This means the function has only one root, x = 1. This is a repeated root, which indicates that the vertex of the parabola lies on the x-axis. In this case, the vertex (1, 0) is also the x-intercept.

Because the parabola only touches the x-axis at one point, it doesn’t cross it. This is consistent with our earlier observation that the parabola opens upwards and has a minimum value of 0 at x = 1. Understanding the roots confirms our understanding of the graph’s behavior around the x-axis.

Putting It All Together

Okay, let’s recap and put everything together to get a clear picture of the graph of f(x) = x² - 2x + 1.

Basic Shape: The function is a quadratic, so its graph is a parabola. Since the coefficient of x² is positive, the parabola opens upwards.
Vertex: We found the vertex to be (1, 0). This is the minimum point of the graph and lies on the x-axis.
Axis of Symmetry: The axis of symmetry is the vertical line x = 1, which passes through the vertex.
Additional Points: We plotted the points (-1, 4), (0, 1), (2, 1), and (3, 4) to help sketch the curve.
Roots: The function has one repeated root at x = 1, which is also the x-coordinate of the vertex.

With all this information, you can confidently sketch the graph of f(x) = x² - 2x + 1. Start by plotting the vertex, draw the axis of symmetry, and then plot the additional points. Finally, draw a smooth, U-shaped curve through the points, ensuring it’s symmetrical about the axis of symmetry.

Conclusion

So there you have it! We've thoroughly examined the graph of the quadratic function f(x) = x² - 2x + 1. From understanding its basic shape as an upward-opening parabola to finding key features like the vertex, axis of symmetry, and roots, you now have a solid grasp of how to analyze and sketch quadratic functions. Remember, practice makes perfect, so keep plotting those parabolas and honing your skills. You've got this!