Problem 44 - Algebra 2

Question

Graph each pair of functions. Find the approximate point(s) of intersection. \(y=-\frac{2}{x^{2}}, y=-10\)

Step-by-Step Solution

Answer

The two functions \(y=-\frac{2}{x^{2}}\) and \(y=-10\) intersect at the points \(\left(\sqrt{\frac{2}{10}}, -10\right)\) and \(\left(-\sqrt{\frac{2}{10}}, -10\right)\).

Step 1: Graph the Function y=-2/x^2

Start by graphing the function \(y=-\frac{2}{x^{2}}\). This is a basic reciprocal function which has been transformed vertically by a factor of -2. The x-values are squared, and then 2 is divided by the result, leading to a hyperbolic-like shape on the graph, centered at the origin (0,0).

Step 2: Graph the Function y=-10

Next, graph the function \(y=-10\). This is a horizontal line that crosses the y-axis at -10.

Step 3: Find the Points of Intersection

The point(s) of intersection between the two functions can be determined by setting the two equations equal to each other and solving for x. Set \(-\frac{2}{x^{2}} = -10\) and solve, giving \(x = \sqrt{\frac{2}{10}}\) and \(x = -\sqrt{\frac{2}{10}}\). Substitute these x values into either of the two equations to find the y values, which in this case is -10.

Key Concepts

Intersection Points

When graphing two functions, one important goal is to determine the intersection points. These are the places where both functions have the same output for the same input value. Graphically, intersection points are where the two lines or curves meet on the graph.
To find the intersection points mathematically, we set the equations equal to each other and solve for the variable, usually x. In the given exercise, the functions are set to \(-\frac{2}{x^{2}}\) and \(-10\). By doing this, we find the potential x-values that make both equations equal.
Once we have these x-values, plugging them back into either of the original equations gives us their corresponding y-values, thereby determining the full coordinate points of intersection.

Reciprocal Function

A reciprocal function is a type of function where the output is 1 divided by the input variable. The most basic form is \(y = \frac{1}{x}\). In this exercise, the reciprocal function given is slightly more complex: \(y = -\frac{2}{x^{2}}\).
This function has some special characteristics:

• The values are defined everywhere except where x equals zero, as division by zero is undefined.
• The graph of a reciprocal function typically has two branches with asymptotes. These are lines that the graph approaches but never touches or crosses.
• For our exercise, the factor of -2 means that for each x squared, you divide it into -2, flipping the graph compared to a positive reciprocal and forming a hyperbolic shape.

These traits make reciprocal functions unique and important in graphing tasks.

Horizontal Line

In the context of graphing, a horizontal line represents a function where the output is constant, no matter the input value. This is because it doesn't depend on x.
For example, the function \(y = -10\) in the exercise results in a straight line across the graph at -10 on the y-axis.

• The equation of a horizontal line can be written in the format \(y = c\), where \(c\) is a constant.
• This line will intersect with other curves at places where their y-values match the constant value.
• Horizontally, these lines are invaluable for identifying intersection points with more dynamic, variable-dependent functions.

Consequently, horizontal lines often serve as graphical check-points where variable functions can be compared and analyzed.

Hyperbolic Shape

A hyperbolic shape in graphing is often associated with graphs of reciprocal functions. It has distinctive branches that bend and approach the axes without ever touching them.
In the equation \(y = -\frac{2}{x^{2}}\), this produces two separate curved lines, resembling a hyperbola, centered around the origin but opening upwards, since the y-values are negative.

• This shape is due to the squared nature of the denominator, which stretches the graph differently along the x-axis.
• The reflection of the shape downwards is because of the negative coefficient in the function.
• Hyperbolic shapes are split into parts divided by the vertical asymptote, where x is zero.

Understanding the nature of hyperbolic shapes allows for better comprehension of the behavior of these functions on a graph.