Problem 35 - Algebra and Trigonometry

Question

Use transformations of the graph of \(y=\frac{1}{x}\) to graph the rational function, as in Example \(2 .\) $$ s(x)=\frac{3}{x+1} $$

Step-by-Step Solution

Answer

Shift left to \(x = -1\), vertical stretch by factor of 3.

Step 1: Identify the Parent Function

The parent function for this problem is \( y = \frac{1}{x} \), which is a basic rational function with vertical and horizontal asymptotes along the x-axis and y-axis, respectively.

Step 2: Horizontal Shift

Observe the function \( s(x) = \frac{3}{x+1} \). The \(x\) term in the denominator is \(x+1\). This results in a horizontal shift, also known as a phase shift. Specifically, the graph of the function is shifted 1 unit to the left. This modification changes the vertical asymptote from \(x = 0\) to \(x = -1\).

Step 3: Vertical Stretch

The numerator of the function, \(3\), modifies the graph by applying a vertical stretch. Compared to the parent function, where the constant is \(1\), multiplying the fraction by \(3\) increases the amplitude of changes in the graph's y-values. This affects the steepness and height of the branches, but not the location of the asymptotes.

Step 4: Drawing the Transformed Graph

After the horizontal shift and vertical stretch, sketch the graph of \( s(x) = \frac{3}{x+1} \). The vertical asymptote is now at \(x = -1\), and the horizontal asymptote remains \(y = 0\). The graph is steeper than that of \(y = \frac{1}{x}\) due to the larger constant factor (3 instead of 1). It retains the characteristic hyperbolic shape seen in rational functions.

Key Concepts

Understanding Rational Functions

Rational functions are a particular kind of function, represented as the quotient of two polynomial functions. A standard form of a rational function is given by \( y = \frac{f(x)}{g(x)} \), where both \( f(x) \) and \( g(x) \) are polynomials, and \( g(x) eq 0 \). These functions can exhibit a wide variety of behaviors depending on the degrees and coefficients of the polynomials involved. A key feature of rational functions is their asymptotic behavior. Rational functions often have vertical asymptotes, which occur when the denominator equals zero as the function approaches infinity or negative infinity at certain points. They can also have horizontal or oblique asymptotes, determined by the degrees of the polynomials in the numerator and denominator. For instance, the parent function \( y = \frac{1}{x} \) is a simple rational function with both a vertical and horizontal asymptote. Any transformation of such a function will affect the location or orientation of these asymptotes.

Exploring Horizontal Shifts

A horizontal shift in graphing is essential when manipulating the graph of a function. By altering the function's equation to include or subtract a constant from the \( x \)-variable, we can move the graph left or right across the coordinate plane. For example, the function \( s(x) = \frac{3}{x+1} \) includes the \( +1 \) inside the denominator, which causes the entire graph to shift horizontally. This modification to the parent function results in moving the graph one unit to the left.

• When you replace \( x \) with \( x+c \), the graph shifts \( c \) units to the left if \( c \) is positive, or \(-c\) units to the right if \( c \) is negative.

As a consequence of this shift, the vertical asymptote that was originally at \( x = 0 \) now appears at \( x = -1 \). Horizontal shifts do not affect the horizontal asymptote of the graph, which in this case remains the line \( y = 0 \).

Vertical Stretches and Their Effects

Vertical stretching is a transformation that affects the steepness of a graph, making it taller or shorter without altering the location of asymptotes. This kind of transformation involves multiplying the entire function by a constant. Consider the function \( s(x) = \frac{3}{x+1} \), where the numerator value of \( 3 \) indicates a vertical stretch. In the context of rational functions, this kind of modification enhances the rate at which the values of the function increase or decrease.

• If \( a > 1 \), the graph experiences a vertical stretch, making it steeper and further away from the x-axis.
• If \( 0 < a < 1 \), the graph undergoes a vertical compression, appearing flatter.

While the vertical stretch affects the graph's overall steepness, it does not influence the graph's horizontal shift or the placement of vertical asymptotes. Consequently, although the functional value changes are magnified, the asymptotes remain at \( x = -1 \) and \( y = 0 \).