Sketching Regions & Finding Minimum Y Value

Alright, guys, let's dive into a fun problem where we're going to sketch a region defined by inequalities and then hunt for the smallest y value within that region. It's like a mini treasure hunt, but with graphs! We'll break it down step-by-step so it’s super clear. Our goal is to sketch the region given by $y \leq 0$ and $y \geq x^2-x-6$, then find the smallest value for $y$ for the points $(x, y)$ in this region.

Understanding the Inequalities

Before we start sketching, let’s get a good handle on what each inequality means. This will make the sketching process much smoother. Understanding inequalities and their graphical representations is crucial in various fields like optimization, economics, and engineering. Being able to visualize these regions helps in solving real-world problems where constraints are defined by inequalities. For instance, in linear programming, you might need to find the feasible region defined by a set of linear inequalities to optimize a certain objective function. Similarly, in engineering design, constraints on physical parameters are often expressed as inequalities, and understanding the feasible region is essential for finding acceptable solutions. In economics, inequalities can represent budget constraints or production possibilities, and visualizing these helps in making informed decisions about resource allocation and production strategies. The ability to work with inequalities and visualize their solutions is therefore a fundamental skill that has broad applicability.

Inequality 1: $y \leq 0$

This one's pretty straightforward. It simply means we're looking at all the points where the y-coordinate is less than or equal to zero. In other words, we're focusing on the region at or below the x-axis. Think of the x-axis as the boundary, and we're shading everything below it. So, anything above the x-axis is out of bounds! Remember, the line $y = 0$ is included because of the "equal to" part of the inequality. Understanding this simple inequality is the first step to defining our region.

Inequality 2: $y \geq x^2 - x - 6$

This is where things get a little more interesting. We're dealing with a quadratic inequality. The equation $y = x^2 - x - 6$ represents a parabola. The inequality $y \geq x^2 - x - 6$ means we want the region above (or on) this parabola. To sketch this, we'll first need to understand the parabola itself. We need to find its roots, vertex, and general shape. Knowing these key features will help us accurately represent the parabola and shade the correct region. This inequality adds a curved boundary to our region, making the problem a bit more complex but also more fun!

Sketching the Region

Now, let's put these two inequalities together and sketch the region they define. This involves drawing the boundaries and shading the appropriate area. Visualizing the region is a key step in understanding the constraints of the problem. The intersection of the two regions defined by the individual inequalities will give us the final region we are interested in. This region contains all the points that satisfy both conditions simultaneously. Accurately sketching this region allows us to identify the points where we need to find the minimum value of y. The sketch will also give us a visual confirmation of our algebraic solutions, ensuring that we have correctly interpreted the inequalities.

Step 1: Sketch the Line $y = 0$

This is just the x-axis. Draw a horizontal line along the x-axis. Remember, since we have $y \leq 0$, we will eventually shade the region below this line.

Step 2: Sketch the Parabola $y = x^2 - x - 6$

To sketch the parabola, we need to find a few key points:

• Roots (x-intercepts): Set $y = 0$ and solve for x: $x^2 - x - 6 = 0$. Factoring, we get $(x - 3)(x + 2) = 0$, so the roots are $x = 3$ and $x = -2$. This means the parabola crosses the x-axis at these two points. These points are crucial for understanding the parabola's position on the graph.
• Vertex: The x-coordinate of the vertex is given by $x_v = -b / (2a)$, where $a = 1$ and $b = -1$ in our quadratic equation. So, $x_v = -(-1) / (2 * 1) = 1/2$. Now, find the y-coordinate of the vertex by plugging $x_v$ back into the equation: $y_v = (1/2)^2 - (1/2) - 6 = 1/4 - 1/2 - 6 = -25/4 = -6.25$. So, the vertex is at $(1/2, -6.25)$. The vertex is the lowest (or highest) point on the parabola and is essential for drawing an accurate curve.

Now, plot the roots and the vertex on your graph. Sketch the parabola passing through these points. Since the coefficient of $x^2$ is positive, the parabola opens upwards. This means it has a minimum point at the vertex. Make sure your parabola looks symmetrical and smooth.

Step 3: Shade the Region

Now, we need to shade the correct region based on our inequalities.

• For $y \leq 0$, shade the region below the x-axis.
• For $y \geq x^2 - x - 6$, shade the region above the parabola.

The region that satisfies both inequalities is the area that is both below the x-axis and above the parabola. This will be the area enclosed between the parabola and the x-axis, below the x-axis. This is our target region! Make sure the shading clearly indicates the area that satisfies both inequalities. The intersection of the shaded regions is the graphical representation of the solution set.

Finding the Smallest Value for y

Okay, we've got our region sketched. Now, where is the smallest y value in this region? Remember, we're looking for the lowest point within the shaded area. Think about it – where does the parabola reach its lowest point within our region?

The lowest point on the parabola is its vertex. We already found the vertex to be at $(1/2, -6.25)$. Since this point lies within our shaded region (it's below the x-axis and on or above the parabola), the smallest value for y in the region is the y-coordinate of the vertex. Therefore, the smallest value for y is $-6.25$ or $-25/4$.

Conclusion:

So, to wrap it all up, we sketched the region defined by $y \leq 0$ and $y \geq x^2 - x - 6$, and we found that the smallest value for y within that region is $-6.25$. Not too shabby, right? Remember, the key is to understand the inequalities, sketch the boundaries, shade the correct region, and then look for the extreme points within that region. Keep practicing, and you'll become a pro at these types of problems! Understanding how to solve and visualize these types of problems is key to succeeding in optimization and other mathematical fields.