Question

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{-5}{x} $$

Short answer

The graph of the function \(g(x)=\frac{-5}{x}\) is a reflection of the graph of \(f(x)=\frac{1}{x}\) in the x-axis, then scaled vertically by a factor of 5. This means that any point from the graph of \(f(x)=\frac{1}{x}\) will have its y-value multiplied by -5 on the graph of \(g(x)=\frac{-5}{x}\).

Step-by-step solution

Understanding the function

The given function is \(g(x)=\frac{-5}{x}\). This is a variation of the standard function \(f(x)=\frac{1}{x}\). By analyzing the function, you can determine that it's a rational function, and the graph will have similar characteristics to the graph of \(f(x)=\frac{1}{x}\) but will be scaled by a factor of -5.

Plotting the function

Choose enough values for the x-coordinate in the function \(g(x)=\frac{-5}{x}\) to get an accurate plot. Remember not to include x = 0 since the function is undefined at x = 0. Plot these values on a graph. You can plot more points if you want a more accurate graph.

Comparing the graphs

Once you've plotted the function \(g(x)=\frac{-5}{x}\), you should compare it to the graph of \(f(x)=\frac{1}{x}\). You'll see that the graph of \(g(x)=\frac{-5}{x}\) is just the graph of \(f(x)=\frac{1}{x}\) scaled vertically by a factor of -5. This means it is a reflection of the graph of \(f(x)=\frac{1}{x}\) in the x-axis and then scaled vertically by a factor of 5. Therefore, any point from the graph of \(f(x)=\frac{1}{x}\) will have its y-value multiplied by -5 on the graph of \(g(x)=\frac{-5}{x}\).

Key concepts

Vertical Scaling

Vertical scaling is a transformation that changes the y-values of a function without affecting the x-values. This process makes the graph of a function appear taller or shorter.

For rational functions like our example, if we have a multiplier such as -5, it scales the graph vertically. This means each point on the original graph has its y-coordinate multiplied by this factor.

In the case of the function \( g(x) = \frac{-5}{x} \), compared to \( f(x) = \frac{1}{x} \), the y-values are scaled by -5. This causes the graph to stretch or compress along the y-axis depending on the magnitude of the scaling factor. If the value is greater than 1, the graph stretches; if it's less than 1, it compresses. Here, the graph stretches since we multiply by 5 (ignoring the negative for now).

Remembering these transformations is key:

• A factor greater than 1 stretches the graph.
• A factor between 0 and 1 compresses the graph.

Reflection Across Axes

Reflection is a transformation that flips a graph across a specific line. In our example, the graph of \( g(x) = \frac{-5}{x} \) reflects across the x-axis.

When a function is multiplied by a negative number, it flips (or reflects) the graph over the x-axis. This changes the direction of the y-values from positive to negative or vice versa. With \( \frac{-5}{x} \), each positive y-value from \( f(x) = \frac{1}{x} \) becomes negative, and vice versa.

Thus, reflection combined with vertical scaling results in a graph that is both stretched and mirrored.

• Positive values turn negative, and negative values turn positive.
• Each point is mirrored from its position across the x-axis.

This gives the visual effect of flipping the entire graph upside down compared to the original.

Rational Functions Characteristics

Rational functions are defined as the ratio of two polynomials. These functions often display specific behaviors which we can analyze for their graphs.

One key characteristic is their vertical asymptotes, which occur where the denominator equals zero. For \( g(x) = \frac{-5}{x} \), the graph has a vertical asymptote at \( x = 0 \). This is the line the graph approaches but never touches or crosses.

Other typical features include horizontal asymptotes or slant asymptotes, depending on the degrees of the numerator and denominator. However, for \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{-5}{x} \), there is a horizontal asymptote at \( y = 0 \), which the graph approaches as \( x \) goes to infinity or negative infinity.

Remember these key points when analyzing rational functions:

• Identify vertical and horizontal asymptotes.
• Analyze the behavior near these asymptotes.
• Look for transformations that alter the shape of the graph.

Knowing these details can help you accurately graph and interpret rational functions.