Problem 54 - Algebra and Trigonometry

Question

Graph each equation. \(y=-\frac{1}{x}\left(\text { Let } x=-2,-1,-\frac{1}{2},-\frac{1}{3}, \frac{1}{3}, \frac{1}{2}, 1, \text { and } 2 .\right)\)

Step-by-Step Solution

Answer

The graph of the function \(y=-\frac{1}{x}\) is a downward opening hyperbola. The given x-values (-2, -1, -1/2, -1/3, 1/3, 1/2, 1, 2) yield corresponding y-values which are plotted on the graph and connected using a curved line to represent the function.

Step 1: Understand the Function

Understand the function \(y=-\frac{1}{x}\). It's important to note that the negative sign means that this is a reflection of the reciprocal function \(\frac{1}{x}\) over the x-axis.

Step 2: Calculate Y-Values

Substitute the given x-values (-2, -1, -1/2, -1/3, 1/3, 1/2, 1, 2) into the function \(y=-\frac{1}{x}\) one by one to find the corresponding y-values. For example, when \(x=-2\), \(y=-\frac{1}{-2}=1/2\). Repeat this process for all x-values.

Step 3: Plot the Points

Plot the points you found by pairing the x-values with their corresponding y-values. Plot these on the coordinate axis.

Step 4: Draw the Curve

Draw a curve that passes through all of these points, keeping in mind the character of the function \(y=-\frac{1}{x}\). It's important to note that the graph will have two parts (or 'branches'), one in the first quadrant and one in the third quadrant.

Key Concepts

Reflection Over X-Axis

When you talk about a reflection over the x-axis in mathematics, it's like seeing yourself in a mirror flipped upside down vertically. Here, our function is the reciprocal function represented as \(y = -\frac{1}{x}\). The negative sign in front of the fraction indicates this reflection. Instead of the graph opening upwards, it's flipped to open downwards. This may change the way the graph looks compared to the original \(+\frac{1}{x}\) function, but it retains its steep approach toward infinity close to the axes. Keep this reflection in mind when plotting the graph on paper.

Plotting Points

Plotting points is a fundamental step in graphing functions on a two-dimensional plane. Once you have determined the y-values by substituting x-values into your function, you get coordinate pairs like (x, y). For instance, if \(x = -2\) yields a \(y\)-value of \(+1/2\), then your coordinate point is \((-2, 1/2)\). After calculating each pair, you plot these points on the graph:

• -2, 1/2
• -1, 1
• -1/2, 2
• -1/3, 3
• 1/3, -3
• 1/2, -2
• 1, -1
• 2, -1/2

Each plotted point helps in visualizing the function's curve on the graph.

Coordinate Plane

The coordinate plane is a two-dimensional space where we plot our functions. It is divided into four quadrants by two axes—the horizontal x-axis and the vertical y-axis. These axes intersect at the origin point (0,0). On this plane:

• Quadrant I is where x and y are both positive.
• Quadrant II is where x is negative, y is positive.
• Quadrant III is where both x and y are negative.
• Quadrant IV is where x is positive, y is negative.

The function \(y = -\frac{1}{x}\) generally places the graph's 'branches' in Quadrants II and IV due to the effects of reflection over the x-axis. Knowing which quadrant to look at guides your expectations while plotting the points and drawing the curve.

Reciprocal Function Characteristics

Reciprocal functions have distinct traits, making them unique and recognizable on graphs. For the base reciprocal function \(y = \frac{1}{x}\), it is crucial to know these features:

• The graph appears in two hyperbolic branches.
• It doesn’t touch either axis, as \(x = 0\) is undefined.
• The curves approach but never meet the x- and y-axes, representing asymptotes.
• It has symmetry, which shifts upon reflection over the x-axis.

With \(y = -\frac{1}{x}\), these characteristics are similarly present but inverted vertically. Recognizing these hallmark traits assists in predicting and verifying the shape of the curve during graphing.