Problem 7 - Calculus for Biology and Medicine

Question

sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=3+1 / x $$

Step-by-Step Solution

Answer

The graph of \( y = 3 + \frac{1}{x} \) has a horizontal asymptote at \( y = 3 \), a vertical asymptote at \( x = 0 \), and forms a hyperbola with key points at \((1, 4)\) and \((-1, 2)\).

Step 1: Understand the function form

The given function is \( y = 3 + \frac{1}{x} \). This is a transformation of the basic reciprocal function \( y = \frac{1}{x} \). Here, the graph of \( y = \frac{1}{x} \) is shifted upwards by 3 units.

Step 2: Determine the asymptotes

The function \( y = \frac{1}{x} \) has a vertical asymptote at \( x = 0 \) (since division by zero is undefined) and a horizontal asymptote at \( y = 0 \). In \( y = 3 + \frac{1}{x} \), the horizontal asymptote shifts to \( y = 3 \), while the vertical asymptote remains at \( x = 0 \).

Step 3: Identify key points

For \( y = 3 + \frac{1}{x} \), calculate key points that help sketch the graph. For example, \( x = 1 \) gives \( y = 4 \) and \( x = -1 \) gives \( y = 2 \). These points (\(1, 4\) and \(-1, 2\)) are crucial for drawing the curve.

Step 4: Sketch the graph behavior

As \( x \to \, ext{positive} \, ext{and} \, ext{negative} \, ext{infinity} \), \( y \to 3 \). As \( x \to 0^{+} \), \( y \to \, ext{positive} \, ext{infinity} \), and as \( x \to 0^{-} \), \( y \to \, ext{negative} \, ext{infinity} \). This information combined with key points allows you to sketch two branches of the hyperbola approaching the asymptotes.

Step 5: Draw the graph

Start by plotting the key points and asymptotes on a coordinate plane. Sketch the curve approaching both the horizontal asymptote \( y = 3 \) and the vertical asymptote \( x = 0 \). The hyperbola should consist of two separate branches, one in the first quadrant and another in the third quadrant.

Key Concepts

Asymptotes

Asymptotes are important features of a graph that describe its behavior as it approaches certain lines. They come in two main types: vertical and horizontal. A vertical asymptote occurs when a function approaches infinity as the variable approaches a certain value. For the function \(y = 3 + \frac{1}{x}\), the vertical asymptote is at \(x = 0\). This is because division by zero is undefined, and as \(x\) gets closer to zero from either side, the value of \(\frac{1}{x}\) becomes very large or very small, respectively.

Horizontal asymptotes describe the value the function approaches as \(x\) becomes very large or very small. In our function, the horizontal asymptote is \(y = 3\). This indicates that as \(x\) heads towards positive or negative infinity, \(y\) tends to 3. Asymptotes are the lines the graph will never quite reach but will get infinitely close to as it extends outwards.

Reciprocal Functions

The concept of reciprocal functions is crucial for understanding certain graph shapes. The basic reciprocal function is \(y = \frac{1}{x}\). It creates a hyperbola with two distinct branches, one in the first and the other in the third quadrant. The graph of \(y = \frac{1}{x}\) has a center of symmetry at the origin, due to its characteristic property: reflecting the entire graph across both axes will yield the same graph.

For \(y = 3 + \frac{1}{x}\), we see a transformation of the basic reciprocal function by vertically shifting it upwards by 3 units. This transformation results in the graph having a new horizontal asymptote at \(y = 3\), but retains the same vertical asymptote at \(x = 0\). The overall shape remains a hyperbola, with points symmetric about the new horizontal line.

Transformations

Transformations allow us to modify graphs in predictable ways. They can shift, stretch, compress, or reflect a graph. In the function \(y = 3 + \frac{1}{x}\), a vertical shift transformation has been applied to the basic \(y = \frac{1}{x}\) graph. This is done by adding 3 to the function, moving every point on the graph up by 3 units.

Vertical shifts like this are one type of transformation. Others include horizontal shifts, which move the graph side to side, and reflections, which flip the graph across an axis. For this function, the transformation is simple yet significant as it repositions the horizontal asymptote and affects the location of key points on the graph. Understanding transformations like these can make graph sketching intuitive.