Graphing Quadratic Equations: A Comprehensive Guide

Hey everyone! Let's dive into graphing the quadratic equation $y = -x^2 + 4x + 4$. It might sound a bit intimidating at first, but trust me, it's totally manageable. We'll break it down step by step, so even if you're new to this, you'll be graphing like a pro in no time. This guide is designed to be super friendly and easy to follow, so let's get started!

Step 1: Calculate the y-values for Given x-values

Our first mission is to plot the solutions for the given x-values. Essentially, we're going to plug in each x-value into the equation and calculate the corresponding y-value. Think of it like a fun little puzzle. Let's start with the equation and the given x-values:

$y = -x^2 + 4x + 4$

x = -1, 0, 1, 2, 3, 4, 5

We'll create a table to keep things organized. This table will have two columns: one for x-values and one for the calculated y-values. Let's fill it out! Here's how to calculate each y-value:

• For x = -1: $y = -(-1)^2 + 4(-1) + 4 = -1 - 4 + 4 = -1$. So, when x = -1, y = -1.
• For x = 0: $y = -(0)^2 + 4(0) + 4 = 0 + 0 + 4 = 4$. When x = 0, y = 4.
• For x = 1: $y = -(1)^2 + 4(1) + 4 = -1 + 4 + 4 = 7$. When x = 1, y = 7.
• For x = 2: $y = -(2)^2 + 4(2) + 4 = -4 + 8 + 4 = 8$. When x = 2, y = 8.
• For x = 3: $y = -(3)^2 + 4(3) + 4 = -9 + 12 + 4 = 7$. When x = 3, y = 7.
• For x = 4: $y = -(4)^2 + 4(4) + 4 = -16 + 16 + 4 = 4$. When x = 4, y = 4.
• For x = 5: $y = -(5)^2 + 4(5) + 4 = -25 + 20 + 4 = -1$. When x = 5, y = -1.

Here's the completed table:

With these x and y values, we are ready to plot them on a graph. Remember, each pair of (x, y) represents a point on the graph. Plotting these points is the first step in visualizing our quadratic equation. Keep going; you're doing great!

Step 2: Plotting the Points on a Graph

Alright, now comes the fun part: let's plot those points on a graph! You can do this by hand on graph paper or use graphing software. Each pair of (x, y) values we calculated represents a point on the graph. Remember, the x-value tells you how far to move horizontally (left or right), and the y-value tells you how far to move vertically (up or down).

Let's plot each point:

• (-1, -1): Move 1 unit to the left on the x-axis and 1 unit down on the y-axis.
• (0, 4): Stay on the y-axis and move up 4 units.
• (1, 7): Move 1 unit to the right on the x-axis and 7 units up on the y-axis.
• (2, 8): Move 2 units to the right on the x-axis and 8 units up on the y-axis.
• (3, 7): Move 3 units to the right on the x-axis and 7 units up on the y-axis.
• (4, 4): Move 4 units to the right on the x-axis and 4 units up on the y-axis.
• (5, -1): Move 5 units to the right on the x-axis and 1 unit down on the y-axis.

If you're using graph paper, make sure to label your axes (x and y) and choose an appropriate scale. For example, you might let each square on the graph paper represent one unit. If you're using graphing software, you'll typically just enter the points, and the software will plot them for you. After you plot the points, they will show you the basic shape of the graph, which is a parabola. It's a U-shaped curve. This is the characteristic shape of all quadratic equations. The more points you have, the more accurately you can see the curve.

Step 3: Drawing the Parabola

Now, here comes the grand finale: drawing the parabola. Once you've plotted all your points, you'll see a pattern – a U-shaped curve. A parabola is a symmetrical curve, which means if you were to fold it in half, both sides would match perfectly. To draw the parabola, carefully connect the points you plotted. Start from the left-most point and smoothly curve through each point, ending at the right-most point. It is extremely important that the curve is smooth and does not have any sharp corners. A smooth curve is very important for the appearance and mathematical accuracy of your graph. Ensure the curve has a slight U-shape. This can be challenging at first, but don't worry – practice makes perfect!

If you have a lot of points, this task becomes easier because the points guide you to the shape of the parabola. If you're doing this by hand, try to make the curve as smooth as possible. If you're using graphing software, the software will usually draw the curve automatically for you. The resulting graph represents the quadratic equation $y = -x^2 + 4x + 4$. It's a visual representation of how the y-value changes as the x-value changes. The highest or lowest point on the parabola is called the vertex, and we'll talk about that in the next step. Well done, you have completed the graph, now you can understand the shape and characteristics of the parabola.

Step 4: Analyzing the Parabola (Optional)

Let's get a bit deeper and analyze the parabola we've drawn. This equation, $y = -x^2 + 4x + 4$, is a quadratic equation, and its graph is a parabola. Here's a quick analysis:

1. Vertex: The vertex is the highest or lowest point of the parabola. Since our equation has a negative coefficient in front of $x^2$ (-1), the parabola opens downwards, and the vertex is the highest point. In this case, the vertex is at the point (2, 8). You can find the x-coordinate of the vertex using the formula $x = -b / 2a$, where a and b are coefficients from the standard form $ax^2 + bx + c$. In our equation, $a = -1$ and $b = 4$, so $x = -4 / (2 * -1) = 2$. Then, plug x = 2 back into the equation to find y: $y = -(2)^2 + 4(2) + 4 = 8$. Therefore, the vertex is (2, 8).
2. Axis of Symmetry: The axis of symmetry is a vertical line that passes through the vertex. It divides the parabola into two symmetrical halves. For our equation, the axis of symmetry is the line x = 2 (the x-coordinate of the vertex).
3. Direction: Because the coefficient of the x² term is negative (-1), the parabola opens downwards. If the coefficient were positive, the parabola would open upwards.
4. Y-intercept: This is the point where the parabola intersects the y-axis. You can find it by setting x = 0 in the equation. In our case, when x = 0, y = 4. The y-intercept is (0, 4).
5. X-intercepts: These are the points where the parabola intersects the x-axis. You can find them by setting y = 0 and solving for x. For our equation, it is a little more complex because there is only one x-intercept, approximately. For this parabola, it will just touch the x-axis, at around x = -1.268. You can solve it using the quadratic formula: $x = (-b ± ext{sqrt}(b^2 - 4ac)) / 2a$. This analysis helps you to understand the behavior of the quadratic equation and its graph.

Conclusion

And that's a wrap, guys! We've successfully graphed the quadratic equation $y = -x^2 + 4x + 4$. We started by calculating y-values for different x-values, then plotted the points on a graph, drew the parabola, and even analyzed some key features like the vertex and axis of symmetry. Remember, practice makes perfect. The more you graph quadratic equations, the more comfortable and confident you will become. If you've enjoyed this guide, or have any questions, feel free to share them! Keep learning, keep practicing, and never be afraid to ask for help! Happy graphing! You're now well on your way to mastering quadratic equations and their graphs. Keep up the excellent work! Now, you're ready to tackle any quadratic equation that comes your way. Awesome work! You did it! Congratulations, and keep going.