Problem 41 - Algebra 2

Question

Graph each pair of functions. Find the approximate point(s) of intersection. \(y=\frac{6}{x-2}, y=6\)

Step-by-Step Solution

Answer

The functions intersect at the point (3, 6)

Step 1: Graph the first function

To graph the function \(y=\frac{6}{x-2}\), treat \(y\) as 'f(x)' and graph accordingly. Since this is a rational function, the graph will be a hyperbola. At \(x=2\), the function is undefined, indicating a vertical asymptote.

Step 2: Graph the second function

Now graph the function \(y=6\), which is a horizontal line at y-coordinate 6.

Step 3: Identify the points of intersection

The points of intersection are the points where the graphs of the two functions meet. These will be the points where \(y=\frac{6}{x-2}\) equals \(y=6\). Solving \(\frac{6}{x-2}=6\) gives \(x=3\). So, the point of intersection is when \(x=3\) and \(y=6\), hence (3,6).

Key Concepts

Graphing Functions

Graphing functions involves plotting the set of all possible points (x, y) on a coordinate plane. For the given exercise, we need to graph two specific functions. First, the rational function \( y=\frac{6}{x-2} \), which is a hyperbola. Then, \( y=6 \), which is a horizontal line.

• For \( y=\frac{6}{x-2} \), we recognize that this function is not defined at \( x=2 \) because the denominator becomes zero, creating a vertical asymptote.
• The hyperbola will approach but never reach this vertical asymptote, reflecting the function's undefined nature at that point.
• Conversely, the function \( y=6 \) is straightforward. It's a horizontal line that runs parallel to the x-axis and passes through the point (0,6).

When graphing, it's essential to carefully plot these functions to visualize where they intersect or approach certain boundaries.

Points of Intersection

The points of intersection between two functions occur where their graphs meet on the coordinate plane. In this context, we're looking at where \( y=\frac{6}{x-2} \) and \( y=6 \) intersect. To find such points, set the functions equal to each other.

Solve \( \frac{6}{x-2} = 6 \) to find when the two functions have the same y-value:

• Multiply both sides by \( x-2 \) to get \( 6 = 6(x-2) \).
• Simplify to obtain \( 6 = 6x - 12 \).
• Add 12 to both sides to get \( 18 = 6x \).
• Divide by 6 to find \( x = 3 \).

This calculation shows the intersection point is at \( (3,6) \). It's crucial for understanding how the behaviors of different functions relate visually and algebraically.

Vertical Asymptote

A vertical asymptote in a function graph indicates where the function is undefined as the value reaches infinity. For the rational function \( y=\frac{6}{x-2} \), a vertical asymptote occurs at \( x=2 \). This happens because the denominator \( x-2 \) becomes zero.

• At \( x=2 \): the function is undefined as division by zero is not possible.
• The graph will never cross or touch this line, it will instead split into two branches, going upwards or downwards towards infinity approaching this invisible barrier.
• The function behavior near \( x=2 \) is key in assessing the graph's overall structure since it can influence where other critical features, like intersections or horizontal asymptotes, appear.

Understanding vertical asymptotes helps in identifying how the function values behave as they approach undefined regions.

Hyperbola

A hyperbola is a type of curve formed on a plane defined by rational functions such as \( y=\frac{6}{x-2} \). They are characterized by two distinct branches. Each branch has its curve bending away from a specified point or line such as an asymptote.

• Hyperbolas appear as the result of rational functions where the denominator contains variables which might make them go to infinity at certain points.
• They are symmetric about their asymptotes, meaning the graph mirrors itself across these lines.
• The hyperbola linked with \( y=\frac{6}{x-2} \) dissects into branches because of the vertical asymptote at \( x=2 \), meaning the graph curves away from this point, reporting no value at precisely \( x=2 \).

In essence, hyperbolas play a significant role in illustrating complex functions visually. They highlight the dynamic change in function values, particularly crucial for comprehending rational functions with problematic points.