Solve each problem. Work each of the following. Sketch the graph of a function that does not intersect its horizontal asymptote y=1, has the line x=3 as a vertical asymptote, and has x-intercepts (2, 0) and (4, 0).

Verified step by step guidance

1. Understand the problem: We need to find a function with a horizontal asymptote at \(y=1\), a vertical asymptote at \(x=3\), and x-intercepts at \((2,0)\) and \((4,0)\).
2. Since the function has a vertical asymptote at \(x=3\), the denominator of the function should have a factor of \((x - 3)\), which causes the function to be undefined at \(x=3\).
3. The x-intercepts at \(x=2\) and \(x=4\) mean the numerator of the function should have factors \((x - 2)\) and \((x - 4)\), so the function equals zero at these points.
4. To have a horizontal asymptote at \(y=1\), the degrees of the numerator and denominator should be the same, and the ratio of their leading coefficients should be 1. This suggests the numerator and denominator should be polynomials of the same degree with leading coefficients equal.
5. Construct the function as \(f(x) = \frac{a(x - 2)(x - 4)}{a(x - 3)(x - b)}\) where \(a\) is a constant and \(b\) is chosen so that the horizontal asymptote is \(y=1\). Then simplify and verify the conditions, adjusting \(a\) and \(b\) if necessary.

Key Concepts

Asymptotes of a Function

Asymptotes are lines that a graph approaches but never touches. Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity, while vertical asymptotes occur where the function is undefined and the graph tends to infinity. Understanding these helps in sketching the function's end behavior and discontinuities.

X-Intercepts of a Function

X-intercepts are points where the graph crosses the x-axis, meaning the function's output is zero at these points. Identifying x-intercepts helps in determining the roots of the function and is essential for accurately plotting the graph.

Graphing Rational Functions

Rational functions are ratios of polynomials and often have vertical and horizontal asymptotes. To graph them, analyze intercepts, asymptotes, and behavior near these lines. This process helps in creating an accurate sketch that reflects the function's key features.