Question

In Exercises 3-10, graph the function. Compare the graph with the graph of \(f(x)=\frac{1}{x}\). $$ g(x)=\frac{-9}{x} $$

Short answer

The graph of the function \(g(x) = \frac{-9}{x}\) is a reflection of the graph of the base function \(f(x) = \frac{1}{x}\) over the x-axis, and each point on the curve of \(g(x)\) is nine times further from the origin due to the multiplication factor of -9.

Step-by-step solution

Understand the base function

The function \(f(x)=\frac{1}{x}\) is the base function here. The graph for the base function \(f(x)=\frac{1}{x}\) will be a curve in the form of a hyperbola where one arm will be in the first quadrant and other in the third quadrant.

Understand the function to be drawn

Next is the function \(g(x)=\frac{-9}{x}\) which needs to be graphed. It is just the base function but multiplied by -9. As any negative number multiplied by a function is equivalent to reflecting it about x-axis, the graph of this function is expected to be similar to the graph of \(f(x)=\frac{1}{x}\), but flipped over the x-axis. The '9' isn't going to affect the shape, but it is going to scale the graph by making the coordinates 9 times further from the origin.

Plot the base function and the new function

Now, plot the graphs of both \(f(x)=\frac{1}{x}\) and \(g(x)=\frac{-9}{x}\). Draw the curve for the base function in the first and third quadrants. Now for \(g(x)=\frac{-9}{x}\), draw a similar curve as the base function but in the second and fourth quadrants. This will demonstrate the reflection of the graph over the x-axis. And make sure that each point on the curve \(g(x)=\frac{-9}{x}\) is nine times further from the origin as a result of scaling by -9.

Key concepts

Graph Transformations

Graph transformations involve changing a function's graph in various ways. In this exercise, we compare the graph of the base function, \(f(x)=\frac{1}{x}\), with that of \(g(x)=\frac{-9}{x}\). For \(f(x)\), the graph is a hyperbola with two branches: one in the first quadrant and one in the third quadrant. Graph transformations allow us to modify the position, orientation, and size of this base graph.

When we introduce \(g(x)\), which is derived from multiplying the base function by a constant, we engage in multiple transformations such as reflections and scaling. This means we not only shift the graph along different axes but also stretch or shrink it depending on the constant's nature. These changes affect how the function appears visually on a standard Cartesian plane.

Reflection Over Axis

Reflection over an axis is a common transformation in graphing functions. In this case, multiplying the base function \(f(x) = \frac{1}{x}\) by \(-1\) in \(g(x) = \frac{-9}{x}\) induces a reflection across the x-axis. What this does is flip each point of the graph of \(f(x)\) to its opposite position in terms of the y-values.

For the base function \(f(x)\), each point that was originally above the x-axis in quadrants one and three is reflected below in quadrants two and four for \(g(x)\). Reflecting over the x-axis means that if the original function \(f(x)\) was at point \((x, y)\), the new function \(g(x)\) will be at \((x, -y)\). This simple negation of the function's output causes the entire graph to appear upside down.

Scaling of Functions

Scaling of functions refers to the stretching or compressing of a graph either horizontally or vertically. In \(g(x) = \frac{-9}{x}\), the constant \(9\) scales the graph vertically. This means that every point on the base function \(f(x) = \frac{1}{x}\) moves nine times farther away from the origin on the y-axis.

This vertical scaling affects the steepness of the hyperbola's arms. In simpler terms, while the basic shape of the curve remains the same, its size increases due to the new factor. If a point on \(f(x)\) was at \((x,y)\), after scaling, the new point on \(g(x)\) would be \((x, 9y)\), depicting how the entire function stretches away from the x-axis. It's important to note that this type of transformation affects how the function's behavior appears, making patterns such as asymptotes and domain/range more pronounced.