Problem 50 - Algebra 1

Question

Graph the function. Describe the domain. $$y=\frac{-5 x+19}{x-3}$$

Step-by-Step Solution

Answer

The domain of the function is (-∞, 3) union (3, ∞). And the function has a vertical asymptote at x = 3 and a horizontal asymptote at y = -5.

Step 1: Identify the Domain

For any rational function in the form \(y= \frac{ax+b}{x-c}\), a point is not in the domain if it makes the denominator zero. Here, x-3 in the denominator will be zero when x = 3. Hence, the domain of the function will be all real numbers except x = 3. Or we can say the domain is (-∞, 3) union (3, ∞).

Step 2: Identify Vertical Asymptote

A vertical asymptote of a function is a vertical line x=d where the function approaches +∞ or -∞ as x approaches d. Here, we have a vertical asymptote at x-3=0, or x=3, the point that is not part of the domain.

Step 3: Identify Horizontal Asymptote

A horizontal asymptote of a function is a horizontal line y=e where the function approaches e as x approaches +∞ or -∞. By looking at the degrees of the polynomial in the numerator and the denominator, we can tell that if the degrees are the same, the horizontal asymptote is y=a, where a is the ratio of the leading coefficients. Here, both the numerator and the denominator are first degree polynomials, so there will be a horizontal asymptote at y=-5.

Step 4: Graph the Function

With the domain, vertical and horizontal asymptotes known, we can then draw the graph. It should be noted that the graph approaches but never touches or crosses the asymptotes. So it is a hyperbola that opens downwards and sideways and is centered at x = 3. It approaches the horizontal asymptote , but never actually meets it.

Key Concepts

Domain of a function

The domain of a function consists of all the possible input values (usually x-values) for which the function is defined. Understanding the domain is crucial when dealing with rational functions, like the given function \[y = \frac{-5x + 19}{x - 3}.\] For rational functions, pay special attention to the denominator because division by zero is undefined. To find the domain, identify any x-values that make the denominator zero and exclude them from the domain. In this case, the denominator is \(x - 3\), which equals zero when \(x = 3\). Thus, the domain consists of all real numbers except \(x = 3\). We write this as

• \(x eq 3\)
• In interval notation: \((-\infty, 3) \cup (3, \infty)\)

By recognizing the domain, you're identifying where the function is applicable and where it can't be evaluated.

Vertical asymptote

Vertical asymptotes occur in rational functions when the denominator equals zero and the function tends toward infinity. These are vertical lines that the graph approaches but never crosses or touches. They're crucial for accurately sketching the behavior of rational functions. For the function \[y = \frac{-5x + 19}{x - 3}\], we find a vertical asymptote at \(x = 3\) because it makes the denominator zero. As \(x\) approaches 3, the function heads towards positive or negative infinity, reinforcing the notion that x = 3 is not included in the domain.

• Vertical asymptote at \(x = 3\)
• The graph of the function gets infinitely close to this vertical line but does not intersect it.

Vertical asymptotes give us insights into where the function can experience dramatic changes in behavior.

Horizontal asymptote

Horizontal asymptotes indicate how a function behaves as \(x\) tends towards infinity or negative infinity. In rational functions, these are horizontal lines the graph approaches but usually doesn't touch except possibly as \(x\) goes to infinity. To find the horizontal asymptote for \[y = \frac{-5x + 19}{x - 3}\], compare the degrees of the polynomials in the numerator and the denominator. Both are of the first degree, so the horizontal asymptote depends on the leading coefficients, which are

• -5 (numerator)
• 1 (denominator)

This ratio is \(y = \frac{-5}{1} = -5\). Therefore, the horizontal asymptote is \(y = -5\). As \(x\) approaches infinity or negative infinity, the function approaches this line.

• Horizontal asymptote at \(y = -5\)
• This helps plot the function’s behavior at extreme right or left of the graph.

Graphing functions

Graphing a rational function involves understanding its domain, asymptotes, and overall shape. For \[y = \frac{-5x + 19}{x - 3}\],the graph is influenced by features identified in the previous concepts.

• The domain excludes \(x = 3\), which is where we have our vertical asymptote. The graph will make a sharp turn or tend towards infinity at this point.
• There is a horizontal asymptote at \(y = -5\), meaning the graph levels out as \(x\) moves far right or left.

When plotting, sketch the asymptotes first. These act as guidelines shaping where the function goes. For this function:

• The behavior around the asymptotes is that the graph approaches these lines closely but never touches them.
• The overall shape looks like a hyperbola, indicative of these types of rational functions, opening downwards and extending to the sides.

Understanding each component helps intuitively and accurately plot the graph.