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Graphing an inverse variation involves plotting points on a coordinate system based on the relationship defined by the equation. For our example, the equation is \( y = -\frac{10}{x} \). This type of inverse equation typically results in two distinct branches on the graph.

Start by plotting the points you've calculated from your table of values. Each point corresponds to a pair of \( x \) and \( y \) values. For instance, when \( x = -10 \), \( y = 1 \); therefore, plot the point \((-10, 1)\). Repeat this for all the values to create the full set of points to graph.

When connecting these points, ensure that the line is smooth and continuous. The nature of inverse variation often results in a hyperbola shape. You will notice that one branch of the graph appears in the "first quadrant" (where both \( x \) and \( y \) are positive) and another in the "third quadrant" (where both \( x \) and \( y \) are negative). This symmetry is a characteristic feature of inverse variations like \( y = -\frac{10}{x} \).

Finally, add arrows to the ends of the curve to indicate that it continues infinitely. This visually represents that the function is defined for all non-zero \( x \) values, as the function approaches infinity without ever touching the \( x \)- or \( y \)-axes.

To graph an inverse variation like \( y = -\frac{10}{x} \), constructing a comprehensive table of values is crucial. This table acts as a foundation for plotting and sketching the graph. Need some steps? Here's a simple run-through:

• Select a range of \( x \) values. A diverse selection including negative, positive, and fractional values can offer a broader look into how the function behaves.
• For each chosen \( x \), substitute it into the equation to compute the \( y \) value. Remember, the equation \( y = -\frac{10}{x} \) is an inverse relationship, meaning as one variable increases, the other decreases.
• Repeat for each \( x \) and record the corresponding \( y \). This will give you a list of coordinate pairs \((x, y)\).

Once you have your table filled out, it serves as a guide for plotting each point on the graph, illustrating the inverse relationship between \( x \) and \( y \). This meticulous approach ensures an accurate representation of the behavior of the function.

The coordinate system is a crucial tool when graphing any function, including inverse variations like \( y = -\frac{10}{x} \). Let's go over its essential components and how they relate to plotting a graph.

A coordinate system consists of two axes: the horizontal \( x \)-axis and the vertical \( y \)-axis. These two axes intersect at the origin point \((0, 0)\), dividing the plane into four quadrants. Knowing the quadrant where your points will fall is helpful:

• The "first quadrant" is where both \( x \) and \( y \) are positive.
• The "second quadrant" has negative \( x \) and positive \( y \).
• The "third quadrant" accommodates negative \( x \) and negative \( y \).
• The "fourth quadrant" features positive \( x \) and negative \( y \).

Position the points from your table of values within this system. For example, the point \((10, -1)\) lies in the fourth quadrant, while \((-10, 1)\) is in the second quadrant. This precise positioning helps visualize the inverse variation's path and symmetry.

Through consistent practice in plotting coordinates on this system, you'll gain a deeper understanding of how equations translate into curves and shapes on a graph.