Problem 9 - Algebra 2

Question

Draw a graph of each function. Describe properties of the graph. \(y=\frac{-0.1}{x}\)

Step-by-Step Solution

Answer

The graph of function \(y=\frac{-0.1}{x}\) is a reflection of the graph of \(y=\frac{1}{x}\) over the x-axis, with horizontal and vertical asymptotes being the x and y axis respectively. The domain and range are all real numbers except 0. The function decreases as \(x\) increases or decreases.

Step 1: Identify the function

The function given is \(y=\frac{-0.1}{x}\), which is a reciprocal function. This type of function can be rewritten as \(y=-0.1x^{-1}\). When \(x\) goes to 0, \(y\) tends to infinity, and when \(x\) goes to infinity, \(y\) goes to zero.

Step 2: Sketch the function

To sketch the reciprocal function, mark points on the x and y axis and draw two curves approaching these axes. The curve in the second quadrant should be drawn from the y axis bending towards the x axis, and the curve in the fourth quadrant should start from the x axis and bend towards the y axis.

Step 3: Determine asymptotes

There will be two asymptotes for this function. The horizontal asymptote is \(y=0\) (which is the x-axis) and the vertical asymptote is \(x=0\) (which is the y-axis). The asymptotes are lines that the function will approach but never quite reach.

Step 4: Establish domain and range

The domain of the function (possible x values) will be all real numbers except 0, due to the fact that division by zero is undefined. Hence domain is \(-\infty < x <0\) and \(0 < x < \infty\). The range (possible y values) will also be all real numbers except 0, for the same reason. Hence range is \(-\infty < y <0\) and \(0 < y < \infty\).

Step 5: Identify the behavior of the function

Because the coefficient of \(x\) in our function is negative, the graph of the function is a reflection of the graph of \(y=\frac{1}{x}\) over the x-axis. The function decreases as \(x\) increases or decreases. Therefore, the function is a decreasing function.

Key Concepts

Understanding Asymptotes in Reciprocal Functions

Reciprocal functions are fascinating because they have unique behavior around certain lines known as "asymptotes." For the function \(y = \frac{-0.1}{x}\), there are two main asymptotes to consider. An asymptote is a line that the graph of the function approaches but never actually touches.

• Horizontal Asymptote: This is found on the x-axis or \(y = 0\). As \(x\) becomes very large or very small, the value of \(y\) approaches zero. Hence, the graph gets closer to the x-axis but never intersects it.
• Vertical Asymptote: Present on the y-axis or \(x = 0\). Here, as \(x\) gets closer to zero, \(y\) moves towards positive or negative infinity without actually reaching \(x=0\).

The concept of asymptotes helps us understand the boundary behavior of the function, indicating where the graph will stretch toward infinity or where it flattens out at zero.

Exploring the Domain and Range

When dealing with functions like \(y = \frac{-0.1}{x}\), determining the domain and range gives us insights into the possible inputs (x-values) and outputs (y-values). The core idea is identifying where the function behaves normally versus where it files into undefined or infinite territory.

• Domain: The domain of \(y = \frac{-0.1}{x}\) includes all real numbers except zero. Division by zero is undefined, so values must be \(-\infty < x < 0\) or \(0 < x < \infty\). This excludes the exact point at \(x=0\).
• Range: Similarly, this function can produce any real number except zero. As \(x\) moves, \(y\) can stretch indefinitely towards infinity in either the positive or negative direction, covering \(-\infty < y < 0\) and \(0 < y < \infty\).

Understanding the domain and range allows us to recognize where the function can go and helps pinpoint its limitations. It also highlights the continuity of the function across different sections of the graph.

Graphing and Visualizing Reciprocal Functions

Graphing reciprocal functions can be an insightful way to visualize their properties in a more tangible form. The function \(y = \frac{-0.1}{x}\) can be sketched by focusing on its behavior in relation to its asymptotes and quadrants.First, note that the graph of \(y = \frac{-0.1}{x}\) has two distinct parts due to its negative coefficient. Compared to \(y = \frac{1}{x}\), this graph is reflected across the x-axis, resulting in an inverted image.

• Second Quadrant: In this quadrant, \(x\) values are negative, and \(y\) values become less negative as \(x\) moves away from zero toward the negative. The graph curves from a high negative y-intercept, descending toward the x-axis.
• Fourth Quadrant: Here, the graph behaves similarly but in the positive section of the \(x\)-axis, descending from positive values of \(y\) as \(x\) increases. It mirrors the curve from the second quadrant.

Graphing this function gives a hands-on way to observe how \(y\) values tend to flatten (reflectively across the x-axis) towards the asymptotes as \(x\) stretches toward infinity, demonstrating the constant, decreasing behavior of this function.