Find a rational function ƒ having a graph with the given features.
x-intercepts: (1, 0) and (3, 0)
y-intercept: none
vertical asymptotes: x=0 and x=2
horizontal asymptote: y=1

Verified step by step guidance

1. Identify the x-intercepts of the rational function. Since the function has x-intercepts at (1, 0) and (3, 0), the numerator of the rational function must have factors corresponding to these roots: \((x - 1)\) and \((x - 3)\).
2. Determine the vertical asymptotes. Vertical asymptotes occur where the denominator is zero but the numerator is not. Given vertical asymptotes at \(x = 0\) and \(x = 2\), the denominator must have factors \((x)\) and \((x - 2)\).
3. Write the general form of the rational function using the factors found: \[f(x) = \frac{a(x - 1)(x - 3)}{x(x - 2)}\] where \(a\) is a constant to be determined.
4. Use the horizontal asymptote to find the constant \(a\). Since the horizontal asymptote is \(y = 1\), compare the degrees of numerator and denominator (both degree 2) and set the ratio of the leading coefficients equal to 1. The leading coefficient of the numerator is \(a\), and the denominator's leading coefficient is 1, so \(a = 1\).
5. Check the y-intercept by evaluating \(f(0)\). Since there is no y-intercept, \(f(0)\) must be undefined (which it is, due to the vertical asymptote at \(x=0\)), confirming the function satisfies all given conditions.

Key Concepts

Rational Functions and Their Graphs

A rational function is a ratio of two polynomials. Its graph can have intercepts, asymptotes, and discontinuities depending on the zeros of the numerator and denominator. Understanding how these features relate to the function's formula is essential for constructing or analyzing the graph.

Intercepts of Rational Functions

x-intercepts occur where the numerator equals zero (and the denominator is nonzero), indicating points where the graph crosses the x-axis. A y-intercept occurs at x=0 if the function is defined there. If there is no y-intercept, the function is undefined at x=0, often due to a vertical asymptote.

Asymptotes of Rational Functions

Vertical asymptotes occur where the denominator is zero and the numerator is nonzero, causing the function to approach infinity. Horizontal asymptotes describe the end behavior of the function as x approaches infinity or negative infinity, often determined by the degrees of numerator and denominator polynomials.