Graphing Inequalities: A Visual Guide

Hey guys! Today, we're diving into the world of graphing inequalities, specifically focusing on how to sketch the graph of $y ge x^2 + 2x - 8$ and, most importantly, shade the correct region. Graphing inequalities might seem tricky at first, but with a step-by-step approach and a little practice, you'll become a pro in no time. So, grab your graph paper (or fire up your favorite graphing software), and let's get started!

Step 1: Understand the Inequality

Before we jump into graphing, let's make sure we understand what the inequality $y ge x^2 + 2x - 8$ means. In simple terms, it tells us that the y-values we're interested in are greater than or equal to the values of the quadratic expression $x^2 + 2x - 8$. This "or equal to" part is crucial because it tells us something about the boundary line (or curve in this case) of our shaded region. If it was just "greater than", we'd have a dashed line, but since it includes "or equal to", we'll have a solid line. This solid line indicates that the points on the curve are also part of the solution set. Remember this point; it’s important for the final graph.

Step 2: Graph the Boundary Curve

To graph the inequality, we first need to graph the equation $y = x^2 + 2x - 8$. This is a quadratic equation, which means its graph will be a parabola. To sketch the parabola, we can find the vertex, the x-intercepts, and a few additional points to get a good idea of its shape. Let's start by finding the x-intercepts.

Finding the x-intercepts

The x-intercepts are the points where the parabola crosses the x-axis, meaning $y = 0$. So, we need to solve the equation $x^2 + 2x - 8 = 0$. This is a quadratic equation that we can solve by factoring. Factoring the quadratic, we get $(x + 4)(x - 2) = 0$. Setting each factor equal to zero gives us the solutions $x = -4$ and $x = 2$. Therefore, the x-intercepts are $(-4, 0)$ and $(2, 0)$. These are critical points to plot on our graph. The x-intercepts give us a good foundation when graphing the parabola, so make sure that you are solving the quadratic equation correctly.

Finding the Vertex

The vertex is the highest or lowest point on the parabola. For a parabola in the form $y = ax^2 + bx + c$, the x-coordinate of the vertex is given by $x = -b / (2a)$. In our case, $a = 1$ and $b = 2$, so the x-coordinate of the vertex is $x = -2 / (2 * 1) = -1$. To find the y-coordinate of the vertex, we plug this x-value back into the equation: $y = (-1)^2 + 2(-1) - 8 = 1 - 2 - 8 = -9$. Thus, the vertex is $(-1, -9)$. The vertex will act as the minimum point on the graph, since the 'a' value in the original equation is positive.

Plotting Additional Points (If Needed)

With the x-intercepts and the vertex, we have a good start, but plotting a few additional points can help us refine the shape of the parabola. Let's pick a couple of x-values and find the corresponding y-values. For example, if $x = -5$, then $y = (-5)^2 + 2(-5) - 8 = 25 - 10 - 8 = 7$. So, we have the point $(-5, 7)$. If $x = 3$, then $y = (3)^2 + 2(3) - 8 = 9 + 6 - 8 = 7$. So, we have the point $(3, 7)$. Plotting these points along with the x-intercepts and vertex, we can sketch the parabola. Remember to connect these points with a smooth curve, not straight lines.

Drawing the Parabola

Now that we have the x-intercepts $(-4, 0)$ and $(2, 0)$, the vertex $(-1, -9)$, and a couple of additional points like $(-5, 7)$ and $(3, 7)$, we can draw the parabola. Since our inequality includes "or equal to," we draw a solid parabola. If it were just "greater than," we would draw a dashed parabola to indicate that the points on the curve are not included in the solution. A dashed line means that any point on the line is not a solution to the inequality.

Step 3: Determine the Shaded Region

Now comes the crucial part: shading the correct region. The inequality $y ge x^2 + 2x - 8$ tells us that we want the region where the y-values are greater than or equal to the values on the parabola. To determine which region to shade (inside or outside the parabola), we can use a test point. A test point is simply a point that is not on the parabola. The easiest test point is usually the origin $(0, 0)$, as long as the parabola doesn't pass through it.

Using a Test Point

Let's use the test point $(0, 0)$. We plug $x = 0$ and $y = 0$ into the inequality: $0 ge (0)^2 + 2(0) - 8$, which simplifies to $0 ge -8$. Is this true? Yes, $0$ is indeed greater than or equal to $-8$. Since the test point $(0, 0)$ satisfies the inequality, we shade the region that contains $(0, 0)$. In this case, that's the region above the parabola. Make sure that you are choosing a test point that is not on the actual line, otherwise you will not be able to test which region to shade.

Shading the Region

So, we shade the area above (outside) the parabola. This shaded region represents all the points $(x, y)$ that satisfy the inequality $y ge x^2 + 2x - 8$. Grab your pencil or marker and shade liberally! The darker you make the shade, the easier it will be to see what your final solution is.

Step 4: Finalize the Graph

To finalize the graph, make sure the following are clear:

• The solid parabola representing $y = x^2 + 2x - 8$
• The shaded region above the parabola, indicating the solution to the inequality
• Clearly labeled x and y axes
• Important points like the vertex and intercepts labeled on the graph

Tips for Accuracy

• Use graph paper or graphing software for accurate plotting.
• Double-check your calculations for intercepts and the vertex.
• Choose a test point that's easy to work with (like $(0, 0)$) and not on the boundary curve.
• If the test point satisfies the inequality, shade the region containing the test point; otherwise, shade the other region.

Common Mistakes to Avoid

• Using a dashed line when it should be solid (or vice versa).
• Incorrectly calculating the vertex or intercepts.
• Choosing a test point on the boundary curve.
• Shading the wrong region.

Practice Makes Perfect

Graphing inequalities is a skill that improves with practice. Try graphing other inequalities, both linear and quadratic, to build your confidence. Play around with different test points and see how they help you determine the correct shaded region. Remember, the goal is to visualize the solution set of the inequality on the coordinate plane. You can try different values for the quadratic equation to gain more experience.

Conclusion

And there you have it! We've successfully sketched the graph of $y ge x^2 + 2x - 8$ and shaded the appropriate region. Remember, understanding the inequality, accurately graphing the boundary curve, and using a test point are the key steps to success. Keep practicing, and you'll become a master of graphing inequalities in no time! Good job, you have reached the end of this guide! Remember to practice this skill for future math related skills.