Question

Use translations to describe how the graph of \(y=\frac{1}{x}\) compares to the graph of each function. a) \(y-4=\frac{1}{x}\) b) \(y=\frac{1}{x+2}\) c) \(y-3=\frac{1}{x-5}\) d) \(y=\frac{1}{x+3}-4\)

Short answer

a) Up 4 units. b) Left 2 units. c) Right 5 units and up 3 units. d) Left 3 units and down 4 units.

Step-by-step solution

Identify parent function

The parent function is given as \(y = \frac{1}{x}\). This will serve as our base graph to which we will apply translations.

Consider equation (a) \(y-4=\frac{1}{x}\)

Rewrite the equation as \(y = \frac{1}{x} + 4\). This equation represents a vertical translation. Specifically, it means the graph of \(y = \frac{1}{x}\) is shifted up by 4 units.

Consider equation (b) \(y=\frac{1}{x+2}\)

Rewrite the given function as \(y = \frac{1}{(x+2)}\). This equation represents a horizontal translation. It means the graph of \(y = \frac{1}{x}\) is shifted to the left by 2 units.

Consider equation (c) \(y-3=\frac{1}{x-5}\)

Rewrite the equation as \(y = \frac{1}{x-5} + 3\). This shows both horizontal and vertical translations. The graph of \(y = \frac{1}{x}\) is shifted to the right by 5 units and up by 3 units.

Consider equation (d) \(y=\frac{1}{x+3}-4\)

Identify the translations separately. The term \(\frac{1}{x+3}\) shifts the graph to the left by 3 units. Then, the \(-4\) shifts the graph down by 4 units. So, the graph of \(y = \frac{1}{x}\) is shifted left by 3 units and down by 4 units.

Key concepts

Parent Function

The parent function is the original function from which other functions in a family are derived. In this exercise, our parent function is defined as: \[ y = \frac{1}{x} \].
This function is known as a rational function. It generates a hyperbola when graphically represented.
The graph of this function has two branches, one in the first quadrant and the other in the third quadrant of the coordinate plane.
The branches never touch the x-axis or y-axis. These axes are known as asymptotes for the graph of the parent function. As x approaches 0 from the positive or negative side, the y-values grow very large in magnitude.

Vertical Translation

A vertical translation involves shifting the graph of the function up or down along the y-axis. Any transformation of the form: \[ y = \frac{1}{x} + k \] where k is a constant, represents a vertical translation.
For example, consider the function: \[ y - 4 = \frac{1}{x} \].
Rewriting, we get: \[ y = \frac{1}{x} + 4 \].
This means that every point on the graph of the parent function is moved up by 4 units. Each y-value in the equation is increased by 4.

Horizontal Translation

A horizontal translation shifts the graph left or right along the x-axis. The transformation is of the form: \[ y = \frac{1}{x - h} \] where h is a constant.
Consider the function: \[ y = \frac{1}{x + 2} \].
This can be rewritten to: \[ y = \frac{1}{x - (-2)} \].
This shows a horizontal shift. The graph moves left by 2 units. Another example is: \[ y - 3 = \frac{1}{x - 5} \].
Rewriting it yields: \[ y = \frac{1}{x - 5} + 3 \].
Here, the graph shifts right by 5 units.

Graph Transformations

Graph transformations incorporate both vertical and horizontal translations to shift the graph accordingly.
This involves understanding combinations of the form \[ y = \frac{1}{x - h} + k \].
For instance, consider: \[ y = \frac{1}{x + 3} - 4 \].
This function embodies both types of transformations. The first part, \[ \frac{1}{x + 3} \], shifts the graph 3 units to the left. Meanwhile, the \[ -4 \] at the end shifts the graph down 4 units. By visualizing it step-by-step, we can more easily anticipate the changes in the graph’s position.