Graphing The Quadratic Function Y=x^2-6x+10

Hey guys! Today, we're diving deep into graphing quadratic functions, specifically the function y = x² - 6x + 10. This might seem intimidating at first, but trust me, we'll break it down step by step, and you'll be graphing parabolas like a pro in no time! We'll focus on identifying key points, including the vertex and a couple of other points, to plot a clear and accurate graph.

Understanding Quadratic Functions

First, let's refresh our understanding of quadratic functions. Quadratic functions are polynomial functions of the form f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a symmetrical U-shaped curve. The sign of the coefficient a determines whether the parabola opens upwards (a > 0) or downwards (a < 0). In our case, a = 1, which is positive, so our parabola will open upwards, meaning it has a minimum point.

Now, why is understanding quadratic functions important? Well, they pop up everywhere in the real world! Think about the trajectory of a ball thrown in the air, the shape of suspension bridge cables, or even the design of satellite dishes. Knowing how to graph and analyze these functions gives you a powerful tool for understanding and modeling these situations. We are going to cover some important concepts, such as vertex form, axis of symmetry, and how to find additional points to make your graph super accurate.

Think of a parabola like a smiling face (if it opens upwards) or a frowning face (if it opens downwards). The most important feature of this face is the vertex, which is either the lowest point (minimum) or the highest point (maximum) on the curve. The axis of symmetry is like an invisible line that cuts the parabola perfectly in half, passing right through the vertex. Because parabolas are symmetrical, points on one side of the axis of symmetry have corresponding points on the other side, making graphing much easier!

Step 1: Finding the Vertex

The vertex is the most crucial point for graphing a parabola. It's the turning point of the curve and the easiest way to kickstart the graph. There are a couple of ways to find the vertex. One method involves completing the square, which transforms the quadratic equation into vertex form, y = a(x - h)² + k, where (h, k) represents the vertex. However, for a quicker approach, we can use the formula x = -b / 2a to find the x-coordinate of the vertex. Once we have the x-coordinate, we can plug it back into the original equation to find the y-coordinate.

Let's apply this to our function, y = x² - 6x + 10. Here, a = 1 and b = -6. Using the formula, we get:

x = -(-6) / (2 * 1) = 6 / 2 = 3

So, the x-coordinate of the vertex is 3. Now, let's substitute x = 3 back into the equation to find the y-coordinate:

y = (3)² - 6(3) + 10 = 9 - 18 + 10 = 1

Therefore, the vertex of the parabola is the point (3, 1). This is the lowest point on our graph since the parabola opens upwards. Go ahead and plot this point on your graph – it's the foundation of our parabola!

Why is the vertex so important, you ask? Well, it's the turning point of the parabola, the single point where the curve changes direction. It gives us the minimum (or maximum) value of the function and acts as the anchor around which the rest of the graph is built. Plus, knowing the vertex helps us determine the axis of symmetry, which we'll use to find other points easily.

Step 2: Identifying 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 simply x = h, where h is the x-coordinate of the vertex. In our case, the vertex is (3, 1), so the axis of symmetry is the line x = 3. This line is like a mirror; whatever happens on one side of the parabola is mirrored on the other side.

Draw a dashed vertical line on your graph at x = 3. This is your axis of symmetry. It's not part of the parabola itself, but it's a crucial guide for plotting points. Remember, every point on the parabola has a corresponding point on the opposite side of this line, at the same distance away. This symmetry makes graphing so much easier!

Understanding the axis of symmetry not only helps us visualize the parabola’s symmetry but also significantly reduces the amount of calculation needed. Instead of calculating points on both sides of the vertex, we can calculate points on one side and then mirror them across the axis of symmetry to find their counterparts. This technique is super handy for efficient graphing.

Step 3: Finding Additional Points

To get a good idea of the parabola's shape, we need to plot at least two more points besides the vertex. The key here is to choose x-values that are easy to work with and are on either side of the vertex. Symmetry is our friend here! We can pick an x-value, calculate the corresponding y-value, and then use the axis of symmetry to find another point automatically.

Let's choose x = 2. This value is one unit to the left of the axis of symmetry (x = 3). Now, we'll substitute x = 2 into the equation:

y = (2)² - 6(2) + 10 = 4 - 12 + 10 = 2

So, the point (2, 2) is on the parabola. Plot this point on your graph.

Now, thanks to symmetry, we know there's another point on the parabola that's one unit to the right of the axis of symmetry. That means the corresponding x-value is x = 4. And guess what? We don't even need to calculate the y-value! Because of symmetry, it will be the same as the y-value for x = 2, which is 2. So, the point (4, 2) is also on the parabola. Plot this point as well.

Let's find one more pair of points for extra precision. Let's choose x = 1, which is two units to the left of the axis of symmetry. Substituting into the equation:

y = (1)² - 6(1) + 10 = 1 - 6 + 10 = 5

So, the point (1, 5) is on the parabola. Plot it!

Now, use symmetry again! Two units to the right of the axis of symmetry is x = 5. The corresponding point will be (5, 5). Plot this point too.

By strategically choosing x-values and using the symmetry of the parabola, we can quickly find several points. These points provide a clear outline of the parabola’s shape, making it easier to sketch accurately. Remember, the more points you plot, the more precise your graph will be!

Step 4: Plotting and Graphing

Now for the fun part! You should have the following points plotted on your graph:

  • Vertex: (3, 1)
  • (2, 2) and (4, 2)
  • (1, 5) and (5, 5)

These five points give us a pretty good picture of the parabola's shape. Now, carefully draw a smooth, U-shaped curve through these points. Make sure the curve is symmetrical about the axis of symmetry (x = 3). Extend the parabola upwards on both sides, indicating that it continues infinitely.

And there you have it! You've successfully graphed the quadratic function y = x² - 6x + 10. It opens upwards, has a minimum at the vertex (3, 1), and is symmetrical about the line x = 3.

When you’re drawing the parabola, aim for a smooth, continuous curve. Avoid sharp corners or straight lines. The parabola should gradually curve away from the vertex, maintaining its symmetrical shape. If you find your points aren’t lining up in a smooth curve, double-check your calculations – a small error can significantly affect the graph’s appearance.

Step 5: Analyze the Graph

Once you've graphed the quadratic function, you can analyze it to understand its properties better. Here are a few things to consider:

  • Vertex: As we discussed, the vertex (3, 1) represents the minimum point of the function. The minimum value of y is 1.
  • Axis of Symmetry: The line x = 3 divides the parabola into two symmetrical halves.
  • Y-intercept: To find the y-intercept, set x = 0 in the original equation: y = (0)² - 6(0) + 10 = 10 So, the y-intercept is (0, 10).
  • X-intercepts (Zeros): To find the x-intercepts, set y = 0 and solve for x: 0 = x² - 6x + 10 This quadratic equation doesn't factor easily, and using the quadratic formula (x = (-b ± √(b² - 4ac)) / 2a) gives us: x = (6 ± √((-6)² - 4 * 1 * 10)) / 2 x = (6 ± √(-4)) / 2 Since the discriminant (the value inside the square root) is negative, there are no real x-intercepts. This means the parabola does not cross the x-axis, which we can confirm from our graph. The fact that there are no real roots means the parabola sits entirely above the x-axis.

Analyzing the graph helps us connect the algebraic representation of the quadratic function to its geometric representation. This connection is crucial for solving real-world problems using quadratic models. Understanding the vertex, intercepts, and symmetry allows us to make predictions and interpretations about the function’s behavior.

Conclusion

Graphing quadratic functions might seem tricky at first, but by breaking it down into steps – finding the vertex, identifying the axis of symmetry, plotting additional points, and drawing the curve – you can master this skill. Remember, practice makes perfect, so try graphing different quadratic functions to build your confidence.

So, the next time you encounter a quadratic function, you'll be ready to tackle it head-on and create a beautiful parabola! Keep practicing, and you'll become a quadratic graphing whiz in no time. You've got this! Remember, understanding these graphs opens up a world of applications, from physics to engineering, making it a truly valuable skill to have.