Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{x-2}{x-3}$$

Short answer

The function \(f(x)=\frac{x-2}{x-3}\) has no symmetry, it has a x-intercept at \(x=2\), a vertical asymptote at \(x=3\) and a horizontal asymptote at \(y=1\). The graph will approach these asymptotes but never reach them.

Step-by-step solution

Find the intercepts

To find the x-intercepts, set \(f(x)\) equal to zero and solve for \(x\), and for the y-intercepts, replace \(x\) with zero. In the function \(f(x)=\frac{x-2}{x-3}\), setting \(f(x)\) equal to zero gives \(x=2\) as x-intercept, and as \(x\) approaches infinity the function evaluates to \(1\) which is the y-intercept.

Check for symmetry

To find if the function is even or odd (symmetry), it's checked if replacing \(x\) with \(-x\) results in the original function (even) or its inverse (odd). Substituting \(x\) for \(-x\) in \(f(x)=\frac{x-2}{x-3}\) does not result in the original or inverse function, so the function is not symmetrical.

Find vertical asymptotes

Vertical asymptotes occur when the denominator is equal to zero and the numerator is not zero. So, set \(x-3=0\) to find \(x=3\) which is a vertical asymptote.

Find horizontal asymptotes

A horizontal asymptote in a rational function can be determined by considering the degrees of the numerator and denominator. Functions where the degree of the numerator is less than or equal to that of the denominator have a horizontal asymptote. In \(f(x)=\frac{x-2}{x-3}\), both numerator and denominator are of degree 1, hence we can get the horizontal asymptote by having the ratio of the leading coefficients which is \(1\).

Key concepts

X-Intercepts and Y-Intercepts

Understanding where a graph crosses the axes is fundamental when sketching rational functions. For the function f(x) = \(\frac{x-2}{x-3}\), the x-intercept is the point where the graph meets the x-axis. To find it, we set the entire function equal to zero, because this is where the y-value of the graph is zero. Solving for x, we get the x-intercept at \(x=2\).

The y-intercept, where the graph meets the y-axis, is found by setting x to zero in the function. But for our given function, plugging zero into x yields \(-\frac{2}{-3}\), which simplifies to \(\frac{2}{3}\). This means the y-intercept is at the point (0, \(\frac{2}{3}\)). Remember, x-intercepts reveal the roots of the equation, while y-intercepts provide a starting point to draw the function.

Symmetry in Functions

Symmetry helps in understanding the behavior of a function and can simplify graphing. A function is even if \(f(x) = f(-x)\), showing symmetry about the y-axis. It's odd if \(-f(x) = f(-x)\), indicating symmetry about the origin. Our example f(x) = \(\frac{x-2}{x-3}\) is neither even nor odd because the function value changes upon substituting \(x\) with \(-x\). Recognizing the lack of symmetry informs us that the graph will not reflect across the y-axis or the origin, which helps prevent incorrect assumptions when plotting the function's shape.

Vertical Asymptotes

A vertical asymptote is a vertical line where the function approaches but never touches or crosses. This happens when the denominator equals zero and the numerator is non-zero, as the function value becomes infinitely large. In the function f(x) = \(\frac{x-2}{x-3}\), setting the denominator to zero gives us \(x-3=0\), thus \(x=3\) is a vertical asymptote. It acts as a boundary on the graph that the curve infinitely approaches, helping to shape the function and its behavior as x moves towards the vertical asymptote from either side.

Horizontal Asymptotes

A horizontal asymptote reflects the behavior of a function as x approaches infinity or negative infinity. For the function f(x) = \(\frac{x-2}{x-3}\), we identify horizontal asymptotes by comparing the degrees of the numerator and denominator. As both are polynomials of the first degree, their ratio of leading coefficients, which in our case is 1, determines the y-value of the horizontal asymptote, being y=1. This means that as x becomes very large or very small, the function will approach but not necessarily reach or cross the line y=1.