Exploration 8: Shifting And Stretching Rational Functions

Exploration 8 Shifting And Stretching Rational Functions1 Sketch T

Exploration 8 – Shifting and Stretching Rational Functions.

Sketch the graph of each function. For each, determine the domain, range, vertical asymptote, horizontal asymptote, x-intercept, and y-intercept. Explain how to find the domain and vertical asymptote of a rational function, and how to find the range and horizontal asymptote of the given function. Additionally, describe how to find the x-intercept and y-intercept of a function.

Re-write the given rational function as a single fraction to find the domain, range, intercepts, and asymptotes. Explain the process for identifying the horizontal asymptote of the function.

Graph the function \( f(x) = \frac{x}{x-1} \) by shifting and stretching the basic function \( f(x) = \frac{1}{x} \). Determine the horizontal shift, vertical shift, and stretch factor by rewriting the function in the form \( c \cdot \frac{1}{x - h} + k \), and explain each step.

For the group submission, graph the function \( g(x) = \frac{2x + 3}{x - 4} \) after applying shifts and stretches. Identify the horizontal shift, vertical shift, and stretch factor based on its algebraic form. Calculate the x-intercept, y-intercept, asymptotes, domain, and range, explaining your reasoning.

Discuss how shifting and stretching rational functions affect their graphs and asymptotes, providing examples and the general process for analyzing such transformations.

Paper For Above instruction

The analysis of rational functions through shifting and stretching is fundamental in understanding their transformative behavior and their graphical representations. Rational functions, which are ratios of polynomials, exhibit characteristic features such as asymptotes and intercepts, which can be manipulated through algebraic transformations like shifts and stretches. These transformations provide powerful tools for graphing and analyzing complex functions by relating them to simple, well-understood base functions.

To begin, understanding the domain of a rational function is crucial. The domain consists of all real numbers except those that make the denominator zero, since division by zero is undefined. For example, in the function \(f(x) = \frac{x}{x-1}\), the denominator \(x-1\) equals zero when \(x=1\), so the domain is \( \mathbb{R} \setminus \{1\} \). The vertical asymptote occurs at these points where the denominator zeroes out, in this case at \(x=1\). These vertical asymptotes represent the values where the function approaches infinity or negative infinity, indicating the boundary of the domain's continuity.

Finding the horizontal asymptote involves analyzing the degrees and leading coefficients of the numerator and denominator polynomials. For rational functions where the degree of numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. If the numerator degree exceeds the denominator, the function does not have a horizontal asymptote; instead, it has an oblique or slant asymptote. For the function \(f(x) = \frac{x}{x-1}\), rewriting and analyzing the degrees confirms the horizontal asymptote at \(y=1\), because as \(x \to \pm\infty\), \(f(x) \to 1\).

Intercepts are found by plugging in \(x=0\) for y-intercepts and \(y=0\) for x-intercepts. For \(f(x) = \frac{x}{x-1}\), the y-intercept occurs when \(x=0\), giving \(f(0)=0 / -1=0\), so the y-intercept is at (0, 0). The x-intercept is found by setting the numerator to zero: \(x=0\), which confirms the same point.

Graphically, the transformation of basic functions like \(f(x) = \frac{1}{x}\) helps in understanding these shifts. Shifts to the right or left are achieved by replacing \(x\) with \(x - h\), moving the graph horizontally by \(h\) units. Upward or downward shifts involve adding or subtracting \(k\) to the entire function. Vertical stretches are achieved by multiplying the function by \(c\), which vertically compresses or stretches the graph depending on whether \(c1\).

Applying these concepts to the function \(f(x) = \frac{x}{x-1}\), we can see it as a shifted and stretched version of \(f(x)=\frac{1}{x}\). Rewriting \(f(x)\) in the form \(f(x)=\frac{1}{x-1}\) indicates a shift of the basic reciprocal function one unit to the right. If we include a stretch factor, say 2, as in \(g(x) = 2 \times \frac{1}{x-1}\), it signifies a vertical stretch by a factor of 2, changing the y-values accordingly.

In the case of \(g(x)= \frac{2x+3}{x-4}\), the algebraic form reveals the transformations explicitly. Factoring or rewriting this rational function into a form like \(config = c \cdot \frac{1}{x - h} + k\) allows us to identify shifts and stretches clearly. Here, the denominator \(x - 4\) indicates a horizontal shift 4 units to the right, and the numerator determines how the graph is stretched or compressed, as well as any vertical shifts when simplified.

Graphically, these transformations alter the asymptotes and intercepts. Horizontal shifts relocate vertical asymptotes; for example, shifting the basic \(f(x)=\frac{1}{x}\) to the right by 4 units results in the asymptote at \(x=4\). Vertical shifts, achieved by adding or subtracting a constant after removing the fraction components, influence the y-intercept and the overall graph level. The stretch factor modifies the steepness of the graph near the asymptote and intercepts.

In general, understanding the interaction between shifts and stretches allows for precise graphing of complex rational functions from simpler base functions. The process involves rewriting the function to identify the numerator and denominator structure, then determining how each transformation affects the asymptotes, intercepts, and overall shape. This analytical approach simplifies the visualization of functions and enhances comprehension of their behavior across the coordinate plane.

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