Problem 3 - Algebra and Trigonometry Real Mathematics, Real People

Question

What feature of the graph of \(y=\frac{9}{x-3}\) can you find by solving \(x-3=0 ?\)

Step-by-Step Solution

Answer

The feature of the graph of \(y=\frac{9}{x-3}\) found by solving \(x-3=0\) is the vertical asymptote at \(x=3\).

Step 1: Understand the function and the problem

The function given is \(y=\frac{9}{x-3}\), a rational function, and it has a feature when the denominator equals nonzero. The aim is to find this feature.

Step 2: Solve the equation

The equation to solve is \(x-3=0\). Solving for x we get \(x=3\)

Step 3: Identify the feature

With \(x=3\), we find that this is the value for which the denominator of the function is zero. This means that \(x=3\) is a vertical asymptote of the function.

Key Concepts

Rational Functions

Rational functions are special because they are expressed as the ratio of two polynomials. The general form of a rational function is \( y = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) \) is not the zero polynomial.
What makes these functions interesting is how their graphs behave differently than other types of functions like linear or quadratic functions.
Key features of rational functions include:

• Vertical asymptotes, which occur where the denominator is zero, creating a point of undefined value.
• Horizontal or oblique asymptotes, determined by the degrees of the numerator and denominator.
• Intercepts, found where the graph crosses the x-axis or y-axis, providing insight into the function's behavior in two dimensions.

Understanding rational functions requires analyzing their functional form, which often presents unique graphing challenges and characteristics.

Denominator

The denominator of a rational function plays a crucial role in understanding the graph and its features.
For \( y = \frac{9}{x-3} \), the denominator is \( x-3 \). This expression in the denominator determines points where the function is not defined, such as when it equals zero.

Here's what stands out about denominators:

• A zero in the denominator makes the function undefined, leading to vertical asymptotes.
• Changes in the denominator affect the shape and features of the graph, like asymptotes and division of regions.

For the given function, setting \( x-3 = 0 \) shows that the function is undefined at \( x = 3 \), indicating a vertical asymptote at this x-value.
Understanding how to manipulate and solve the denominator is crucial for mastering rational functions.

Graph Features

Graph features of rational functions include significant points and lines that reveal the behavior and characteristics of the function itself.
These features help in visualizing how the function behaves as x approaches specific values, such as vertical asymptotes.

For example:

• Vertical asymptotes appear where the denominator is zero, such as \( x = 3 \) for \( y= \frac{9}{x-3} \). These lines show where the function shoots off to infinity, breaking the graph.
• Horizontal asymptotes, if present, indicate the function's behavior as x leads to infinity or negative infinity.
• Intercepts are where the graph crosses the axes, revealing additional insights into the function's intersections with x and y.

By analyzing these features, students can predict how the graph will look and behave, making it easier to solve graph-related problems and understand the deep behavior of rational functions.