Problem 13 - Precalculus: Mathematical for Calculus

Question

Graph the inequality. $$x^{2}+y^{2} \leq 25$$

Step-by-Step Solution

Answer

Graph a circle of radius 5 centered at the origin and shade the interior.

Step 1: Understanding the Equation

The equation given is \(x^2 + y^2 \leq 25\). This equation represents a circle centered at the origin \((0,0)\) with a radius of 5, because \(\sqrt{25} = 5\). The \(\leq\) symbol indicates that we include the area inside the circle.

Step 2: Graph the Circle

To graph this inequality, first graph the circle. Begin by plotting the center of the circle at the origin \((0,0)\). Then draw a circle with a radius of 5, making sure it passes through points such as \((5,0), (-5,0), (0,5),\) and \((0,-5)\).

Step 3: Shade the Interior

Since the inequality symbol is \(\leq\), the region inside the circle, including the boundary, should be shaded. This indicates that any point \((x, y)\) satisfying the inequality will satisfy \(x^2 + y^2 \leq 25\), meaning it lies within or on the boundary of the circle.

Key Concepts

Circle Equations

A circle equation in the coordinate plane is a mathematical expression that represents all the points around a fixed central point, the circle's center. This expression typically takes the form \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius of the circle. However, if the circle is centered at the origin, as in our given exercise, the equation simplifies to \( x^2 + y^2 = r^2 \). In this case, the center is \((0, 0)\) and the radius is 5 units.

• **Center:** It is the fixed middle point, here it’s the origin \((0,0)\).
• **Radius:** The distance from the center to any point on the circle. Calculate it by \( r = \sqrt{r^2} \).
• **Boundary:** Defined by \( x^2 + y^2 = r^2 \), all points satisfying this are on the circle’s edge.

Understanding the circle's equation is crucial for visualizing and graphing its boundary on the coordinate plane.

Inequality Graphing

Graphing inequalities involves showing all points that satisfy the given inequality condition, often resulting in shading regions on a coordinate plane. Our exercise has the inequality \(x^2 + y^2 \leq 25\). This inequality signifies that you're dealing with all the points inside or on the boundary of the circle defined by \(x^2 + y^2 = 25\).

• **Boundary Curve:** Begin by drawing the circle; it's the border for our region.
• **Inequality Sign:** The \(\leq\) symbol tells us the circle's bounds are included – a solid line is used.
• **Shaded Region:** To represent the solution set, shade inside the circle, where all points satisfy the inequality.

Remember, when graphing inequalities like these:- Use a solid line when \(\leq\) or \(\geq\) are part of the inequality, as the boundary is it included.- Always check some points to ensure the shading is correct, such as testing \((0, 0)\) for inclusion.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can graphically represent equations and inequalities. It’s composed of two perpendicular axes, the x-axis (horizontal) and the y-axis (vertical), that intersect at a point called the origin \((0, 0)\).

• **Axes:** Provide a reference frame for positioning points, with units marked at equal intervals.
• **Quadrants:** Divide the plane into four regions, helping identify the sign and position of coordinates.
• **Plotting Points:** Locate any point using ordered pairs \((x, y)\).

When you graph circles or inequalities, understanding the layout of the coordinate plane helps accurately plot shapes and shade regions. For our circle, you begin at the origin and measure outwards the radius distance in each of the four main directions (e.g., \((5,0), (-5,0), (0,5), (0,-5)\)). This method ensures the circle is proportionately drawn on the plane.