Question

Sketch the graph of the function. $$y=\frac{1}{x-5}+2$$

Short answer

The function \(y=\frac{1}{x-5}+2\) has a vertical asymptote at x = 5, a horizontal asymptote at y = 2, and no intercepts due to the function's undefined behavior at x = 0. The function increases without bound as x approaches 5 from the left and decreases without bound as x approaches 5 from the right. The function tends to approach the horizontal asymptote as x moves towards both negative and positive infinity.

Step-by-step solution

Identify the Asymptotes

One of the properties that distinguishes rational functions is the existence of asymptotes. There are two types of asymptotes: vertical and horizontal. A vertical asymptote occurs when the denominator of the function equals zero, as the function tends toward infinity. Hence the denominator must not be equal to zero and solve \(x=5\) for vertical asymptote. A horizontal asymptote is a y-value to which the function approaches as x approaches infinity. In this case, it is given by the constant term in the function, which is 2.

Identify the Intercepts

The x-intercepts are located wherever the function intersects the x-axis, in other words, wherever y = 0. The equation becomes \(0 = \frac{1}{x-5}+2\). Solving for x gives the x-intercept(s). The y-intercept is found by setting x to 0 in the original equation, if such a point exists, and then solving for y, though in this case, there is no y-intercept as plugging x=0 into the equation does not yield a real number.

Plotting the function and its Asymptotes

Plot the vertical asymptote by drawing a dashed vertical line through x = 5. Then draw the horizontal asymptote with a dashed horizontal line through y = 2. Next, plot the intercepts found in the previous step. To complete the graph, plot additional points to the left and right of the vertical asymptote to understand how the function behaves. The function approaches infinity as x approaches 5 from the left, and as it approaches 5 from the right, it goes to negative infinity.

Key concepts

Asymptotes

Understanding asymptotes is critical when graphing rational functions. Asymptotes are lines that the graph of a function approaches as the inputs or outputs grow very large. There are two types commonly associated with rational functions: vertical and horizontal.

Vertical asymptotes occur where the denominator of the function equals zero, since division by zero is undefined. In our example, setting the denominator, x - 5, to zero gives us the vertical asymptote at x = 5. This means the graph will get infinitely close to the line x = 5 but never actually touch it.

Horizontal asymptotes reflect the value that the function's output approaches as the input grows infinitely large or small. For the function y = 1/(x - 5) + 2, the horizontal asymptote is at y = 2, indicated by the constant term. This line helps us predict the function's end behavior: as x becomes very large or very small, the graph approaches the horizontal line y = 2.

X-intercepts

The x-intercepts of a graph represent the points where the function crosses the x-axis. These occur where the function's output, or y, is zero. To find the x-intercept of the given function y = 1/(x - 5) + 2, we set y to zero and solve: 0 = 1/(x - 5) + 2. However, for this particular function, no real solution exists, and thus, there are no x-intercepts. This is an interesting characteristic as it reveals information about the behavior of the function - it tells us that the graph never touches the x-axis.

Y-intercepts

In contrast, the y-intercept is the point where the graph intersects the y-axis, which occurs when the input, x, is zero. For many functions, calculating the y-intercept is straightforward; simply substitute x with zero and solve for y. However, in our function, when we input zero into x, we get an undefined value because the denominator becomes negative five, indicating that this function does not have a y-intercept either. The lack of a y-intercept further defines the unique graph of our function.

Plotting Functions

Plotting functions allows us to visualize the behavior of functions and is especially important for rational functions due to their complexity. After identifying asymptotes and intercepts, plotting functions requires additional points to determine the curve's shape. For our example, aside from the asymptotes, we need to pick values for x on either side of the vertical asymptote and calculate the corresponding y values. This step-by-step plotting helps us see how the function behaves near the asymptotes and provides a clearer picture of the graph’s overall shape.

Rational Function Behavior

The behavior of a rational function can appear complex due to the presence of asymptotes and the potential for no intercepts, as seen in the example. Rational functions often display distinct characteristics such as approaching different values as x moves towards the asymptotes from different directions. Specifically, for the function y = 1/(x - 5) + 2, as x approaches 5 from the left, y approaches infinity, indicating a sharp upward curve in the graph. Conversely, as x approaches 5 from the right, y plummets towards negative infinity, signifying a sharp descent. Understanding this behavior is essential for correctly sketching the overall graph and anticipating the function's output for given x values.