Graphing Y=x^2 + 10x + 16: Roots, Vertex & Symmetry

Hey guys! Let's dive into graphing quadratic equations, specifically the equation y = x² + 10x + 16. We're going to break it down step by step, making sure to plot five key points: the roots, the vertex, and a couple of extra points to get a good curve. Plus, we'll figure out the axis of symmetry. Ready to get started?

Finding the Roots

First off, let's talk about roots. In the context of a quadratic equation, the roots are the points where the parabola intersects the x-axis. Essentially, these are the x-values when y = 0. Finding these roots is crucial because they give us a foundational understanding of where our graph crosses the x-axis.

To find the roots, we need to set our equation to zero and solve for x:

0 = x² + 10x + 16

This looks like a job for factoring! We need to find two numbers that multiply to 16 and add up to 10. Those numbers are 2 and 8. So, we can rewrite our equation like this:

0 = (x + 2)(x + 8)

Now, we set each factor equal to zero:

x + 2 = 0 or x + 8 = 0

Solving these simple equations gives us our roots:

x = -2 or x = -8

Awesome! We've found our roots: (-2, 0) and (-8, 0). These are two of the five points we need to plot. Remember these, they're key to sketching our parabola accurately. Understanding the roots helps us visualize where the parabola will intersect the x-axis, which is a fundamental aspect of graphing quadratic equations. The roots act as anchors for our curve, giving us a clear sense of its horizontal placement on the coordinate plane.

Locating the Vertex

Alright, let's move on to finding the vertex. The vertex is the turning point of the parabola, either the minimum or maximum point. It's like the peak or the valley of our curve, and it’s super important for graphing.

There are a couple of ways to find the vertex, but let's use the formula that directly gives us the x-coordinate of the vertex:

x = -b / 2a

In our equation, y = x² + 10x + 16, a is the coefficient of the x² term (which is 1), and b is the coefficient of the x term (which is 10). Let's plug those values in:

x = -10 / (2 * 1) = -10 / 2 = -5

So, the x-coordinate of our vertex is -5. Now, to find the y-coordinate, we plug this value back into our original equation:

y = (-5)² + 10(-5) + 16 y = 25 - 50 + 16 y = -9

Fantastic! Our vertex is at the point (-5, -9). This is the lowest point on our parabola, since the coefficient of our x² term is positive (meaning the parabola opens upwards). The vertex is a critical point because it not only represents the minimum value of the function but also helps define the overall shape and position of the parabola. It's the point around which the parabola is symmetrical, making it a key reference for graphing.

Plotting Additional Points

We've got our roots and vertex, but to really nail the shape of the parabola, let's plot a couple more points. We want five points in total, so we need two more. A smart way to choose these points is to pick x-values that are symmetrical around the vertex. Since the x-coordinate of our vertex is -5, we can pick x-values like -4 and -6. These are each one unit away from -5, giving us a symmetrical spread.

Let's start with x = -4:

y = (-4)² + 10(-4) + 16 y = 16 - 40 + 16 y = -8

So, one of our additional points is (-4, -8).

Now let's try x = -6:

y = (-6)² + 10(-6) + 16 y = 36 - 60 + 16 y = -8

Look at that! We got the same y-value, -8, for x = -6. This gives us our fifth point: (-6, -8). This symmetry isn't a coincidence; it's a characteristic of parabolas. Picking points equidistant from the vertex ensures we capture the symmetrical nature of the curve. These additional points help us to more accurately draw the parabola, ensuring we capture its curvature and width.

Graphing the Parabola

Alright, we've got all the points we need! Let's plot them on a graph:

1. Roots: (-2, 0) and (-8, 0)
2. Vertex: (-5, -9)
3. Additional Points: (-4, -8) and (-6, -8)

Now, carefully connect these points with a smooth, U-shaped curve. Remember, a parabola is symmetrical, so make sure your curve looks balanced around the vertex. The curve should pass through all five points, gradually curving upwards as it moves away from the vertex. Take your time to ensure the curve is smooth and accurately reflects the quadratic equation.

As you draw, visualize how the roots anchor the parabola to the x-axis, and how the vertex dictates its lowest point. The additional points help to fill in the shape, creating a more accurate representation of the graph. Once you've connected the points, you should have a clear picture of the parabola defined by the equation y = x² + 10x + 16. This visual representation is incredibly powerful for understanding the behavior of the quadratic function.

Determining the Axis of Symmetry

Last but not least, let's find the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. It's like the spine of the parabola, and it makes graphing much easier because it highlights the symmetry.

Since the axis of symmetry is a vertical line passing through the vertex, its equation will be in the form x = constant. And guess what? That constant is just the x-coordinate of the vertex!

Our vertex is (-5, -9), so the equation of the axis of symmetry is:

x = -5

Simple as that! The axis of symmetry is a critical feature of a parabola, highlighting its symmetrical nature. It not only helps in graphing but also in understanding the properties of the quadratic function. The axis of symmetry visually confirms that the parabola is balanced around the vertex, making it a fundamental concept in the study of quadratic equations.

Conclusion

And there you have it! We've successfully graphed the equation y = x² + 10x + 16 by finding the roots, vertex, and a couple of extra points. We also determined the equation of the axis of symmetry. Graphing quadratic equations might seem tricky at first, but by breaking it down into these steps, it becomes much more manageable. Remember, the key is to understand the significance of the roots, vertex, and axis of symmetry in shaping the parabola. These elements provide a framework for understanding and visualizing quadratic functions.

Keep practicing, and you'll be graphing parabolas like a pro in no time! Understanding how to graph quadratic equations is a fundamental skill in algebra, and it opens the door to more advanced mathematical concepts. So, keep exploring and challenging yourself – you've got this!