Question

Sketch the graph of the equation. Use intercepts, extrema, and asymptotes as sketching aids. \(x y^{2}=4\)

Short answer

The graph of the equation \(x y^{2}=4\) is a hyperbola with branches in the first and third quadrants and asymptotic to the x and y axis, with no x or y intercepts.

Step-by-step solution

Find the X-intercepts

To find x-intercepts, set \(y=0\) and solve for \(x\). In this case, as we set \(y=0\), the expression \(4÷x\) becomes undefined. Therefore, the graph has no x-intercepts.

Find the Y-intercepts

To find y-intercepts, set \(x=0\) and solve for \(y\). This leads to the equation \(0 = 4\), thus the equation has no y-intercepts as well.

Plot Asymptotes

The graph will be asymptotic to both the x and y axis (i.e., it will approach but not touch these axes). Therefore, sketch dotted lines along \(x=0\) (the y-axis) and \(y=0\) (the x-axis) to represent these asymptotes.

Plot the Hyperbola

Sketch the portions of the hyperbola in each quadrant. Since there are no x-intercepts or y-intercepts, the curve doesn't cross either axis. The graph has two disconnected parts or 'branches': one in the first quadrant (\(x, y > 0\)) and the other in the third quadrant (\(x, y < 0\)). The branches approach but do not touch the x-axis and y-axis, which are the asymptotes of the curve.

Key concepts

X-Intercepts and Y-Intercepts

Understanding where a graph intersects the axes is crucial for sketching an accurate representation of any equation. The points where a graph crosses the x-axis are called x-intercepts, and where it crosses the y-axis are known as y-intercepts. These intercepts can provide valuable information about the shape and location of a graph.

For the equation given, when we search for the x-intercepts by setting the variable y to zero, we find that the equation becomes undefined, indicating that our hyperbola doesn't intersect the x-axis at any point. Similarly, for y-intercepts, setting x to zero yields no solution, which means the hyperbola also doesn't intersect the y-axis.

Without real intercepts, a visual plot of this hyperbola must rely on other characteristics such as its asymptotes and behavior in specific quadrants to illustrate its properties accurately.

Asymptotes

The concept of asymptotes is fundamental when dealing with hyperbolas. An asymptote is a line that a curve approaches as it heads towards infinity but never actually reaches. These lines can provide a framework within which the hyperbola is contained and are key in graphing it accurately.

In the case of the equation \(x y^{2}=4\), the hyperbola approaches but does not touch the x-axis and y-axis, hence, these axes are its asymptotes. While plotting the graph, these asymptotes are represented by dotted lines along \(y=0\) (the x-axis) and \(x=0\) (the y-axis). The presence of asymptotes informs us that the hyperbola will have two disjoint branches that get closer and closer to the axes without ever intersecting them, helping us to visualize the general shape of the graph.

Extrema

In the context of graphing functions, extrema refer to the highest and lowest points on a graph, known as maximums and minimums, respectively. These points often represent important characteristics of the function's behavior and can be critical in understanding the overall shape of a graph.

However, it's important to note that not all graphs have extrema. In the case of hyperbolas, like the one represented by the equation \(x y^{2}=4\), there are no maximum or minimum points since the branches of a hyperbola extend infinitely in their respective directions without turning back on themselves. Unlike parabolas, which open either upwards or downwards and have a clear apex, hyperbolas open side-to-side or up-and-down and continue to widen indefinitely, precluding the presence of extrema. This absence is another distinguishing feature of hyperbolas that differentiates them from other conic sections.